Original: Function Classes for Identiﬁable Nonlinear Independent Component Analysis
Revised: Function Classes for Identiﬁable Nonlinear Independent Component Analysis

Original: Abstract
Revised: Abstract

Original: Unsupervised learning of latent variable models (LVMs) is widely used to represent data in machine learning.
Revised: Unsupervised learning of latent variable models (LVMs) is widely used to represent data in machine learning.

Original: When such models reﬂect the ground truth factors and the mechanisms mapping them to observations, there is reason to expect that they allow generalization in downstream tasks.
Revised: When such models reﬂect the ground truth factors and the mechanisms mapping them to observations, there is reason to expect that they allow generalization in downstream tasks.

Original: It is however well known that such identiﬁability guaranties are typically not achievable without putting constraints on the model class.
Revised: It is however well known that such identiﬁability guaranties are typically not achievable without putting constraints on the model class.

Original: This is notably the case for nonlinear Independent Component
Revised: This is notably the case for nonlinear Independent Component

Original: Analysis, in which the LVM maps statistically independent variables to observations via a deterministic nonlinear function.
Revised: Analysis, in which the LVM maps statistically independent variables to observations via a deterministic nonlinear function.

Original: Several families of spurious solutions ﬁtting perfectly the data, but that do not correspond to the ground truth factors can be constructed in generic settings.
Revised: Several families of spurious solutions ﬁtting perfectly the data, but that do not correspond to the ground truth factors can be constructed in generic settings.

Original: However, recent work suggests that constraining the function class of such models may promote identiﬁability.
Revised: However, recent work suggests that constraining the function class of such models may promote identiﬁability.

Original: Speciﬁcally, function classes with constraints on their partial derivatives, gathered in the Jacobian matrix, have been proposed, such as orthogonal coordinate transformations (OCT), which impose orthogonality of the Jacobian columns.
Revised: Speciﬁcally, function classes with constraints on their partial derivatives, gathered in the Jacobian matrix, have been proposed, such as orthogonal coordinate transformations (OCT), which impose orthogonality of the Jacobian columns.

Original: In the present work, we prove that a subclass of these transformations, conformal maps, is identiﬁable and provide novel theoretical results suggesting that OCTs have properties that prevent families of spurious solutions to spoil identiﬁability in a generic setting.
Revised: In the present work, we prove that a subclass of these transformations, conformal maps, is identiﬁable and provide novel theoretical results suggesting that OCTs have properties that prevent families of spurious solutions to spoil identiﬁability in a generic setting.

Original:  Introduction
Revised:  Introduction

Original: Unsupervised representation learning methods can ﬁt Latent Variables Models (LVM) to complex real world data.
Revised: Unsupervised representation learning methods can ﬁt Latent Variables Models (LVM) to complex real world data.

Original: While those latent representations allow to create realistic novel samples or represent the data in a compact way [31, 15], they are a priori not related to the ground truth generative factors of the data.
Revised: While those latent representations allow to create realistic novel samples or represent the data in a compact way [31, 15], they are a priori not related to the ground truth generative factors of the data.

Original: It is highly desirable to recover the true underlying source distribution because those are expected to help with various downstream tasks, e.g., out of distribution generalization [44, 2].
Revised: It is highly desirable to recover the true underlying source distribution because those are expected to help with various downstream tasks, e.g., out of distribution generalization [44, 2].

Original: One principled framework for representation learning is Independent Component Analysis (ICA) where one tries to recover unobserved sources s ∈ R d from observations x
Revised: One principled framework for representation learning is Independent Component Analysis (ICA) where one tries to recover unobserved sources s ∈ R d from observations x

Original: = f ( s ) and one assumes that the components s i are independent.
Revised: = f ( s ) and one assumes that the components s i are independent.

Original: An important result is that for linear functions
Revised: An important result is that for linear functions

Original: f
Revised: f

Original: it is possible to recover s from observations x up to certain symmetries, i.e., the model is identiﬁable [7].
Revised: it is possible to recover s from observations x up to certain symmetries, i.e., the model is identiﬁable [7].

Original: In contrast, for general non-linear models f is highly non-identiﬁable [24].
Revised: In contrast, for general non-linear models f is highly non-identiﬁable [24].

Original: This has important consequences for representation learning, in particular the learning of disentangled representations is also unidentiﬁable without some access to the underlying sources [36].
Revised: This has important consequences for representation learning, in particular the learning of disentangled representations is also unidentiﬁable without some access to the underlying sources [36].

Original: Notably, this makes theoretical analysis of a large body of methods (see, e.g., [19, 29, 41]) that enforces disentanglement difﬁcult.
Revised: Notably, this makes theoretical analysis of a large body of methods (see, e.g., [19, 29, 41]) that enforces disentanglement difﬁcult.

Original: Several additional assumptions were suggested to make the ICA problem identiﬁable.
Revised: Several additional assumptions were suggested to make the ICA problem identiﬁable.

Original: Broadly, there are two directions.
Revised: Broadly, there are two directions.

Original: First, some works imposed additional or different restrictions on the distribution of the sources.
Revised: First, some works imposed additional or different restrictions on the distribution of the sources.

Original: One line of research adds temporal structure by considering time series data [18, 22, 23].
Revised: One line of research adds temporal structure by considering time series data [18, 22, 23].

Original: More recently, Hyvärinen et al.
Revised: More recently, Hyvärinen et al.

Original: [25] proposed to introduce an observed auxiliary variable u , e.g., a class label, such that the source distribution has independent components conditional on the auxiliary variable.
Revised: [25] proposed to introduce an observed auxiliary variable u , e.g., a class label, such that the source distribution has independent components conditional on the auxiliary variable.

Original: They show that under suitable assumptions on the distribution of u and s arbitrary nonlinear mixing functions can be identiﬁed.
Revised: They show that under suitable assumptions on the distribution of u and s arbitrary nonlinear mixing functions can be identiﬁed.

Original: Several recent works extended this approach [28, 47, 52].
Revised: Several recent works extended this approach [28, 47, 52].

Original: Another possibility is to restrict the class of admissible functions by considering more ﬂexible classes than just linear functions, but not allowing arbitrary non-linear functions.
Revised: Another possibility is to restrict the class of admissible functions by considering more ﬂexible classes than just linear functions, but not allowing arbitrary non-linear functions.

Original: The general aim of this approach is to ﬁnd sufﬁciently “small” function classes such that ICA is identiﬁable within this class while making them as large as possible to allow ﬂexible representation of complex data and being applicable to real world problems.
Revised: The general aim of this approach is to ﬁnd sufﬁciently “small” function classes such that ICA is identiﬁable within this class while making them as large as possible to allow ﬂexible representation of complex data and being applicable to real world problems.

Original: So far results in this direction are rather limited.
Revised: So far results in this direction are rather limited.

Original: It was shown that the post-nonlinear model is identiﬁable [48].
Revised: It was shown that the post-nonlinear model is identiﬁable [48].

Original: Moreover, it has been shown that ICA with conformal maps in dimension 2 is almost identiﬁable [24].
Revised: Moreover, it has been shown that ICA with conformal maps in dimension 2 is almost identiﬁable [24].

Original: More recently, identiﬁability of volume preserving transformations was investigated in the auxiliary variable case in [52] (combining the two possible restrictions) and identiﬁability based on sparsity of the mixing function was studied in [53].
Revised: More recently, identiﬁability of volume preserving transformations was investigated in the auxiliary variable case in [52] (combining the two possible restrictions) and identiﬁability based on sparsity of the mixing function was studied in [53].

Original: In this work, we extend the previous works by proving new identiﬁability results for unconditional
Revised: In this work, we extend the previous works by proving new identiﬁability results for unconditional

Original: ICA.
Revised: ICA.

Original: Our main focus is conformal maps (i.e., maps that locally preserve angles) and Orthogonal
Revised: Our main focus is conformal maps (i.e., maps that locally preserve angles) and Orthogonal

Original: Coordinate Transforms (OCT) (i.e., maps where Df (cid:62) Df is a diagonal matrix where Df denotes the derivative of f ).
Revised: Coordinate Transforms (OCT) (i.e., maps where Df (cid:62) Df is a diagonal matrix where Df denotes the derivative of f ).

Original: OCTs, that we will also call orthogonal maps for simplicity, were recently introduced in the context of representation learning in [16], motivated by the independence of mechanisms assumption from the causality literature.
Revised: OCTs, that we will also call orthogonal maps for simplicity, were recently introduced in the context of representation learning in [16], motivated by the independence of mechanisms assumption from the causality literature.

Original: The main focus of this work is to prove new identiﬁability and partial identiﬁability results for this class of functions.
Revised: The main focus of this work is to prove new identiﬁability and partial identiﬁability results for this class of functions.

Original: An overview of our results can be found in Table 1.
Revised: An overview of our results can be found in Table 1.

Original: We summarize the main contributions and the structure of this paper as follows.
Revised: We summarize the main contributions and the structure of this paper as follows.

Original: • We prove that ICA with conformal maps is identiﬁable in d ≥ 3 and extend the earlier results in dimension 2 (Theorems 2-3 in Section 3).
Revised: • We prove that ICA with conformal maps is identiﬁable in d ≥ 3 and extend the earlier results in dimension 2 (Theorems 2-3 in Section 3).

Original: • We deﬁne a new notion of local identiﬁability (Deﬁnition 5) and prove that ICA with orthogonal maps is locally identiﬁable (Theorem 4 in Section 4).
Revised: • We deﬁne a new notion of local identiﬁability (Deﬁnition 5) and prove that ICA with orthogonal maps is locally identiﬁable (Theorem 4 in Section 4).

Original: • On the contrary we show that ICA with volume preserving maps is not identiﬁable not even in the local sense (Theorem 6 in Section 5).
Revised: • On the contrary we show that ICA with volume preserving maps is not identiﬁable not even in the local sense (Theorem 6 in Section 5).

Original: • We introduce new tools to the ICA ﬁeld: our results are based on connections to rigidity theory (see Section 6), restricting the global structure of functions based on local constraints.
Revised: • We introduce new tools to the ICA ﬁeld: our results are based on connections to rigidity theory (see Section 6), restricting the global structure of functions based on local constraints.

Original: Moreover, we exploit the global structure of partial differential equations related to the identiﬁability problem while most earlier results argue locally using linear algebraic tools.
Revised: Moreover, we exploit the global structure of partial differential equations related to the identiﬁability problem while most earlier results argue locally using linear algebraic tools.

Original:  Setting
Revised:  Setting

Original: Independent component analysis investigates the problem of identifying underlying sources when observing a mixture of them.
Revised: Independent component analysis investigates the problem of identifying underlying sources when observing a mixture of them.

Original: We will consider the following general setting: there exists some random hidden vector of sources s ∈ R d and the observed data is generated by
Revised: We will consider the following general setting: there exists some random hidden vector of sources s ∈ R d and the observed data is generated by

Original: [Equation]
Revised: [Equation]

Original: where f : R d → R d is smooth and invertible.
Revised: where f : R d → R d is smooth and invertible.

Original: The condition on s means that its coordinates (often referred to as factors of variation) are independent.
Revised: The condition on s means that its coordinates (often referred to as factors of variation) are independent.

Original: Formally this means that the distribution P of s satisﬁes P ∈ M 1 ( R ) ⊗ n where M 1 ( R ) denotes the probability measures on R .
Revised: Formally this means that the distribution P of s satisﬁes P ∈ M 1 ( R ) ⊗ n where M 1 ( R ) denotes the probability measures on R .

Original: The goal of ICA is to ﬁnd an unmixing g : R d → R d such that g ( x ) has independent components.
Revised: The goal of ICA is to ﬁnd an unmixing g : R d → R d such that g ( x ) has independent components.

Original: Ideally, this should recover the true underlying factors of variation and achieve Blind Source Separation (BSS), i.e., g = f − 1 up to certain symmetries.
Revised: Ideally, this should recover the true underlying factors of variation and achieve Blind Source Separation (BSS), i.e., g = f − 1 up to certain symmetries.

Original: Identiﬁcation of the true generative factors of variations of the observations is of interest since these provide a causal and interventional understanding of the data.
Revised: Identiﬁcation of the true generative factors of variations of the observations is of interest since these provide a causal and interventional understanding of the data.

Original: An important observation was that, in the generality stated above, identiﬁcation of s is not possible.
Revised: An important observation was that, in the generality stated above, identiﬁcation of s is not possible.

Original: In [24] two general constructions of spurious solutions were given: the well known Darmois construction and a construction based on measure preserving transformations.
Revised: In [24] two general constructions of spurious solutions were given: the well known Darmois construction and a construction based on measure preserving transformations.

Original: The latter one is closer to our work here and we will discuss those in more detail in Section 4 and Appendix C.
Revised: The latter one is closer to our work here and we will discuss those in more detail in Section 4 and Appendix C.

Original: In a nutshell it is based on the observation that for measures P with smooth density one can construct smooth Measure
Revised: In a nutshell it is based on the observation that for measures P with smooth density one can construct smooth Measure

Original: Preserving Transformations (MPT), m : R d → R d (that mix the different coordinates), i.e., maps that leave P invariant, such that m ( s ) D =
Revised: Preserving Transformations (MPT), m : R d → R d (that mix the different coordinates), i.e., maps that leave P invariant, such that m ( s ) D =

Original: s if s ∼ P .
Revised: s if s ∼ P .

Original: This implies that all functions ( f ◦ m ) − 1 recover independent sources since ( f ◦ m ) − 1 ( x ) D
Revised: This implies that all functions ( f ◦ m ) − 1 recover independent sources since ( f ◦ m ) − 1 ( x ) D

Original: = s making BSS impossible.
Revised: = s making BSS impossible.

Original: Thus it is a natural question whether additional assumptions on the mixing function f or distribution of s allow us to identify f .
Revised: Thus it is a natural question whether additional assumptions on the mixing function f or distribution of s allow us to identify f .

Original: Let us deﬁne a framework for identiﬁability.
Revised: Let us deﬁne a framework for identiﬁability.

Original: We assume data is generated according to (1) where f belongs to some function class F ( A, B ) of invertible functions
Revised: We assume data is generated according to (1) where f belongs to some function class F ( A, B ) of invertible functions

Original: A → B , which we will always assume to be diffeomorphisms.
Revised: A → B , which we will always assume to be diffeomorphisms.

Original: 2 and P satisﬁes P ∈ P for some set of probability distributions P ⊂ M 1 ( R ) ⊗ d .
Revised: 2 and P satisﬁes P ∈ P for some set of probability distributions P ⊂ M 1 ( R ) ⊗ d .

Original: Finally, let S be a group of transformations g : R d → R d that encodes the allowed symmetries up to which the sources can be identiﬁed as follows.
Revised: Finally, let S be a group of transformations g : R d → R d that encodes the allowed symmetries up to which the sources can be identiﬁed as follows.

Original: The function class will be simply denoted F when domain/codomain information is irrelevant.
Revised: The function class will be simply denoted F when domain/codomain information is irrelevant.

Original: Deﬁnition 1.
Revised: Deﬁnition 1.

Original: (Identiﬁability)
Revised: (Identiﬁability)

Original: We say that independent component analysis in ( F , P ) is identiﬁable up to S if for functions f, f (cid:48) ∈ F and distributions P , P (cid:48)
Revised: We say that independent component analysis in ( F , P ) is identiﬁable up to S if for functions f, f (cid:48) ∈ F and distributions P , P (cid:48)

Original: ∈ P the relation f ( s ) D = f (cid:48) ( s (cid:48) ) where s ∼ P and s (cid:48) ∼ P (cid:48) (2) implies that there is h ∈ S
Revised: ∈ P the relation f ( s ) D = f (cid:48) ( s (cid:48) ) where s ∼ P and s (cid:48) ∼ P (cid:48) (2) implies that there is h ∈ S

Original: such that h = f (cid:48)− 1 ◦ f on the support of P .
Revised: such that h = f (cid:48)− 1 ◦ f on the support of P .

Original: Note that we require the identity h
Revised: Note that we require the identity h

Original: = f (cid:48)− 1 ◦ f only to hold on the support of P because, for complex classes F , there is in general no unique extension of f beyond the support of P and without data the extension cannot be identiﬁed.
Revised: = f (cid:48)− 1 ◦ f only to hold on the support of P because, for complex classes F , there is in general no unique extension of f beyond the support of P and without data the extension cannot be identiﬁed.

Original: We do not always make this explicit in the following.
Revised: We do not always make this explicit in the following.

Original: Put differently, identiﬁability means that given observations of x = f ( s ) and knowledge of ( F , P ) , we can ﬁnd g such that g ◦ f ∈ S , in particular the reconstructed sources s (cid:48)
Revised: Put differently, identiﬁability means that given observations of x = f ( s ) and knowledge of ( F , P ) , we can ﬁnd g such that g ◦ f ∈ S , in particular the reconstructed sources s (cid:48)

Original: =
Revised: =

Original: g ( x ) and the true sources s are related by a symmetry transformation in S .
Revised: g ( x ) and the true sources s are related by a symmetry transformation in S .

Original: We discuss in Appendix C how to identify the set S and how spurious solutions to the identiﬁcation problem can be constructed.
Revised: We discuss in Appendix C how to identify the set S and how spurious solutions to the identiﬁcation problem can be constructed.

Original: In the following, it will be convenient to use the notation f ∗ P which denotes the push-forward of the measure P along the function f .
Revised: In the following, it will be convenient to use the notation f ∗ P which denotes the push-forward of the measure P along the function f .

Original: We refer to Appendix A for a formal deﬁnition, but we note here that the distribution of f ( s ) equals f ∗ P whenever s ∼ P .
Revised: We refer to Appendix A for a formal deﬁnition, but we note here that the distribution of f ( s ) equals f ∗ P whenever s ∼ P .

Original: Therefore, (2) can be equivalently written as f ∗ P = f
Revised: Therefore, (2) can be equivalently written as f ∗ P = f

Original: (cid:48)∗ P (cid:48) .
Revised: (cid:48)∗ P (cid:48) .

Original: We illustrate Deﬁnition 1 through the well known example of linear maps
Revised: We illustrate Deﬁnition 1 through the well known example of linear maps

Original: [Equation]
Revised: [Equation]

Original: i.e., x = As for some invertible matrix A ∈ R n × n .
Revised: i.e., x = As for some invertible matrix A ∈ R n × n .

Original: We further deﬁne
Revised: We further deﬁne

Original: [Equation]
Revised: [Equation]

Original: It is easy to check that S is a group.
Revised: It is easy to check that S is a group.

Original: Then the following identiﬁability result for F lin is well known.
Revised: Then the following identiﬁability result for F lin is well known.

Original: Theorem 1.
Revised: Theorem 1.

Original: (Theorem 11 in [7])
Revised: (Theorem 11 in [7])

Original: The pair ( F lin , P lin ) is identiﬁable up to S lin .
Revised: The pair ( F lin , P lin ) is identiﬁable up to S lin .

Original: This result is optimal as the ordering and scale of the s i is unidentiﬁable and the restriction to at most one Gaussian component is required to avoid linear MPTs of multivariate Gaussians.
Revised: This result is optimal as the ordering and scale of the s i is unidentiﬁable and the restriction to at most one Gaussian component is required to avoid linear MPTs of multivariate Gaussians.

Original: We provide a proof of this result in Appendix D as this serves as a preparation for the more involved Theorembelow.
Revised: We provide a proof of this result in Appendix D as this serves as a preparation for the more involved Theorembelow.

Original: An important observation which was also made in [21] is that with minor differences one can also consider the case where f : R d → M maps to a d -dimensional Riemannian manifold M .
Revised: An important observation which was also made in [21] is that with minor differences one can also consider the case where f : R d → M maps to a d -dimensional Riemannian manifold M .

Original: An
Revised: An

Original: A diffeomorphism
Revised: A diffeomorphism

Original: [Table]
Revised: [Table]

Original: important example for this setting is the case where M ⊂ R n is a submanifold of a higher dimensional
Revised: important example for this setting is the case where M ⊂ R n is a submanifold of a higher dimensional

Original: Euclidean space.
Revised: Euclidean space.

Original: This covers the standard setting of unsupervised representation learning where high dimensional observations (often images) are created from low dimensional factors of variation mirroring the well known manifold hypothesis [49].
Revised: This covers the standard setting of unsupervised representation learning where high dimensional observations (often images) are created from low dimensional factors of variation mirroring the well known manifold hypothesis [49].

Original: Note that this setting essentially covers the case of undercomplete ICA, where we consider f : R d → R n with n > d .
Revised: Note that this setting essentially covers the case of undercomplete ICA, where we consider f : R d → R n with n > d .

Original: The only difference is that we assume that we already know the submanifold M that f maps to.
Revised: The only difference is that we assume that we already know the submanifold M that f maps to.

Original: This manifold can, however, be identiﬁed from the observations x = f ( s ) under minor regularity assumption on f and the support of the data distribution.
Revised: This manifold can, however, be identiﬁed from the observations x = f ( s ) under minor regularity assumption on f and the support of the data distribution.

Original: To avoid technical difﬁculties we assume that the manifold is already known.
Revised: To avoid technical difﬁculties we assume that the manifold is already known.

Original: Note that we restrict our attention to the case where the factors of variations are parametrized by a
Revised: Note that we restrict our attention to the case where the factors of variations are parametrized by a

Original: Euclidean space.
Revised: Euclidean space.

Original: An extension to product manifolds and a combination with the approach in [20] is an interesting question left to future work.
Revised: An extension to product manifolds and a combination with the approach in [20] is an interesting question left to future work.

Original: In the next sections we discuss our results on identiﬁability of ICA for different function classes.
Revised: In the next sections we discuss our results on identiﬁability of ICA for different function classes.

Original: An illustration of the considered classes can be found in Figure 1.
Revised: An illustration of the considered classes can be found in Figure 1.

Original: They are all characterized by a local condition on their gradient.
Revised: They are all characterized by a local condition on their gradient.

Original: Previously, in [21] it was shown that the function class of local isometries is identiﬁable.
Revised: Previously, in [21] it was shown that the function class of local isometries is identiﬁable.

Original: Local isometries have been used frequently in machine learning, and more speciﬁcally in representation learning [43, 49, 10].
Revised: Local isometries have been used frequently in machine learning, and more speciﬁcally in representation learning [43, 49, 10].

Original: Our main results consider two generalizations of these function classes, conformal maps and orthogonal coordinate transformations.
Revised: Our main results consider two generalizations of these function classes, conformal maps and orthogonal coordinate transformations.

Original: Conformal maps preserve angles locally and have been used in computer vision [45, 50, 17].
Revised: Conformal maps preserve angles locally and have been used in computer vision [45, 50, 17].

Original: For d = 2 , conformal maps essentially consist of all biholomorphic mappings of simply connected open domains of the complex plane, and thus constitute a “large”, non-parametric family, as a consequence of the Riemann
Revised: For d = 2 , conformal maps essentially consist of all biholomorphic mappings of simply connected open domains of the complex plane, and thus constitute a “large”, non-parametric family, as a consequence of the Riemann

Original: Mapping Theorem [37].
Revised: Mapping Theorem [37].

Original: For d ≥ 3 , Liouville’s Theorem implies that this class contains relatively “few” functions in ﬁxed dimension (i.e., mapping R d → R d ), in the sense that it is a parametric family, with parameter dimension quadratic in d (see Theorem 7).
Revised: For d ≥ 3 , Liouville’s Theorem implies that this class contains relatively “few” functions in ﬁxed dimension (i.e., mapping R d → R d ), in the sense that it is a parametric family, with parameter dimension quadratic in d (see Theorem 7).

Original: However, this is a rich class when the target space is higher dimensional than the domain (i.e., R d → R n , d < n ).
Revised: However, this is a rich class when the target space is higher dimensional than the domain (i.e., R d → R n , d < n ).

Original: OCT are an even more general class that were motivated based on the principle of independent causal mechanisms in [16].
Revised: OCT are an even more general class that were motivated based on the principle of independent causal mechanisms in [16].

Original: Notably, it contains all conformal maps precomposed with nonlinear entrywise reparametrizations of the source components (see Corollary 1).
Revised: Notably, it contains all conformal maps precomposed with nonlinear entrywise reparametrizations of the source components (see Corollary 1).

Original: It is however much larger, as one can for example concatenate arbitrary functions from the large family of 2d conformal mappings to obtain higher dimensional OCTs.
Revised: It is however much larger, as one can for example concatenate arbitrary functions from the large family of 2d conformal mappings to obtain higher dimensional OCTs.

Original: Moreover, many works showed that training VAEs promotes orthogonality of the columns of the input Jacobian [42, 54, 32, 39, 46] and this has been empirically shown to be a good inductive bias for disentanglement.
Revised: Moreover, many works showed that training VAEs promotes orthogonality of the columns of the input Jacobian [42, 54, 32, 39, 46] and this has been empirically shown to be a good inductive bias for disentanglement.

Original: Indeed, these algorithms are widely used in representation learning and often recover semantically meaningful representations [33, 4, 29, 19].
Revised: Indeed, these algorithms are widely used in representation learning and often recover semantically meaningful representations [33, 4, 29, 19].

Original:  Results for conformal maps
Revised:  Results for conformal maps

Original: Our ﬁrst main result is an extension of Theorem 1 to conformal maps.
Revised: Our ﬁrst main result is an extension of Theorem 1 to conformal maps.

Original: A conformal map is a map that locally preserves angles, i.e. locally it looks like a scaled rotation.
Revised: A conformal map is a map that locally preserves angles, i.e. locally it looks like a scaled rotation.

Original: It can be shown that this is equivalent to the following deﬁnition.
Revised: It can be shown that this is equivalent to the following deﬁnition.

Original: Deﬁnition 2.
Revised: Deﬁnition 2.

Original: (Conformal map)
Revised: (Conformal map)

Original: We deﬁne for domains Ω ⊂ R d the set of conformal maps by
Revised: We deﬁne for domains Ω ⊂ R d the set of conformal maps by

Original: F conf = { f ∈ C 1 (Ω , R d ) : Df ( x ) = λ ( x ) O ( x ) } where λ : Ω → R \ { 0 } is a scalar function and
Revised: F conf = { f ∈ C 1 (Ω , R d ) : Df ( x ) = λ ( x ) O ( x ) } where λ : Ω → R \ { 0 } is a scalar function and

Original: O : Ω → O( d ) is a map to orthogonal matrices (i.e., O ( x ) − 

Revised: O : Ω → O( d ) is a map to orthogonal matrices (i.e., O ( x ) −

Original: = O ( x ) (cid:62) ).
Revised: = O ( x ) (cid:62) ).

Original: All our results also hold for the more general class of conformal maps f : Ω → M where M is a
Revised: All our results also hold for the more general class of conformal maps f : Ω → M where M is a

Original: Riemannian manifold.
Revised: Riemannian manifold.

Original: The complete deﬁnition can be found in Appendix B.
Revised: The complete deﬁnition can be found in Appendix B.

Original: For convenience we deﬁne signed permutation matrices by
Revised: For convenience we deﬁne signed permutation matrices by

Original: Perm ± ( d
Revised: Perm ± ( d

Original: [Equation]
Revised: [Equation]

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: R
Revised: R

Original: [Equation]
Revised: [Equation]

Original: d
Revised: d

Original: [Equation]
Revised: [Equation]

Original: Q ∈ R d × d , such that Q ij = | P ij | , is a permutation matrix } , (6) i.e. the set of matrices whose entry-wise absolute value is a permutation.
Revised: Q ∈ R d × d , such that Q ij = | P ij | , is a permutation matrix } , (6) i.e. the set of matrices whose entry-wise absolute value is a permutation.

Original: Later we will also use the notation Diag( d ) and Perm( d ) for d × d diagonal and permutation matrices, respectively.
Revised: Later we will also use the notation Diag( d ) and Perm( d ) for d × d diagonal and permutation matrices, respectively.

Original: We deﬁne
Revised: We deﬁne

Original: S conf = { x → κPx + a where P ∈ Perm ± ( d ) , a ∈ R d , κ ∈ R \ { 0 }} (7) and
Revised: S conf = { x → κPx + a where P ∈ Perm ± ( d ) , a ∈ R d , κ ∈ R \ { 0 }} (7) and

Original: P conf =
Revised: P conf =

Original: P ⊗ n∩ P lin , where
Revised: P ⊗ n∩ P lin , where

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: ) , there is ∅ (cid:54) =
Revised: ) , there is ∅ (cid:54) =

Original: O ⊂ R open, s.t. µ has positive C 2 density
Revised: O ⊂ R open, s.t. µ has positive C 2 density

Original: [Equation]
Revised: [Equation]

Original: O
Revised: O

Original: [Equation]
Revised: [Equation]

Original: While this condition might appear a bit technical it actually only rules out pathological cases like the cantor measure or densities which are nowhere differentiable and probably it could be relaxed further.
Revised: While this condition might appear a bit technical it actually only rules out pathological cases like the cantor measure or densities which are nowhere differentiable and probably it could be relaxed further.

Original: In particular P 1 contains all probability measures with piecewise smooth densities.
Revised: In particular P 1 contains all probability measures with piecewise smooth densities.

Original: Then the following identiﬁability for conformal maps in dimension d > 2 holds.
Revised: Then the following identiﬁability for conformal maps in dimension d > 2 holds.

Original: Theorem 2.
Revised: Theorem 2.

Original: For d > 2 , ICA with respect to the pair ( F conf , P conf ) is identiﬁable up to S conf .
Revised: For d > 2 , ICA with respect to the pair ( F conf , P conf ) is identiﬁable up to S conf .

Original: This means that we can identify conformal maps up to three symmetries, namely constant shifts of the distributions, rescaling of all coordinates by the same constant factor, and permutations of the coordinates.
Revised: This means that we can identify conformal maps up to three symmetries, namely constant shifts of the distributions, rescaling of all coordinates by the same constant factor, and permutations of the coordinates.

Original: The proof is in Appendix E.
Revised: The proof is in Appendix E.

Original: The main ingredient in the proof is that conformal maps in dimension d > 2 are very rigid and can be characterized explicitly, as we will discuss in Section 6.
Revised: The main ingredient in the proof is that conformal maps in dimension d > 2 are very rigid and can be characterized explicitly, as we will discuss in Section 6.

Original: We remark that it might be more natural to not ﬁx the scale of the sources and allow arbitrary coordinate-wise rescalings.
Revised: We remark that it might be more natural to not ﬁx the scale of the sources and allow arbitrary coordinate-wise rescaling.

Original: The result can be easily extended to accommodate this.
Revised: The result can be easily extended to accommodate this.

Original: We deﬁne
Revised: We deﬁne

Original: S reparam =
Revised: S reparam =

Original: { g : R d → R d | g =
Revised: { g : R d → R d | g =

Original: P ◦ h where P ∈ Perm ± ( d ) and h : R d → R d with h ( x ) =
Revised: P ◦ h where P ∈ Perm ± ( d ) and h : R d → R d with h ( x ) =

Original: ( h 1 ( x 1 ) , . . .
Revised: ( h 1 ( x 1 ) , . . .

Original: , h d ( x d )) (cid:62) for some h i ∈ C 1 ( R , R ) with h (cid:48)
Revised: , h d ( x d )) (cid:62) for some h i ∈ C 1 ( R , R ) with h (cid:48)

Original: i > 0 } .
Revised: i > 0 } .

Original: It is easy to see that S reparam is a group.
Revised: It is easy to see that S reparam is a group.

Original: We deﬁne the class of reparameterized conformal maps by
Revised: We deﬁne the class of parameterized conformal maps by

Original: F r − conf =
Revised: F r − conf =

Original: { f ◦ h | f ∈ F conf , h ∈ S reparam } ∩ C 3 (Ω , M ) and then get the following Corollary.
Revised: { f ◦ h | f ∈ F conf , h ∈ S reparam } ∩ C 3 (Ω , M ) and then get the following Corollary.

Original: Corollary 1.
Revised: Corollary 1.

Original: For d > 2 , ICA with respect to the pair ( F r − conf , P conf ) is identiﬁable up to S reparam if we assume in addition that the observational distribution cannot be expressed as f ∗ P for some f ∈ F conf and P ∈ M 1 ( R ) ⊗ d which has at least two Gaussian components.
Revised: For d > 2 , ICA with respect to the pair ( F r − conf , P conf ) is identiﬁable up to S reparam if we assume in addition that the observational distribution cannot be expressed as f ∗ P for some f ∈ F conf and P ∈ M 1 ( R ) ⊗ d which has at least two Gaussian components.

Original: The additional restriction on the observational distribution is clearly necessary to exclude the nonidentiﬁability of Gaussian distributions.
Revised: The additional restriction on the observational distribution is clearly necessary to exclude the nonidentiﬁability of Gaussian distributions.

Original: For dimension d = 2 it was shown [24] that conformal maps can be identiﬁed up to a rotation when ﬁxing one point of the conformal map (setting f (0) = 0 ).
Revised: For dimension d = 2 it was shown [24] that conformal maps can be identiﬁed up to a rotation when ﬁxing one point of the conformal map (setting f (0) = 0 ).

Original: The authors also claim, without proof, that the remaining ambiguity can be removed for typical probability distributions.
Revised: The authors also claim, without proof, that the remaining ambiguity can be removed for typical probability distributions.

Original: We extend their result by removing the condition that one point is ﬁxed and prove full identiﬁability with a minor assumption on the involved densities.
Revised: We extend their result by removing the condition that one point is ﬁxed and prove full identiﬁability with a minor assumption on the involved densities.

Original: We deﬁne the following set of probability measures on R
Revised: We deﬁne the following set of probability measures on R

Original: [Equation]
Revised: [Equation]

Original: supp
Revised: supp

Original: [Equation]
Revised: [Equation]

Revised: P

Revised: [Equation]

Original:  is a bounded interval
Revised: ) is a bounded interval

Original: I i and
Revised: I [Equation] and

Original: [Equation]
Revised: [Equation]

Original: i has density bounded above and below on I i } (10)
Revised: i has density bounded above and below on I i } (10)

Original: Then we get the following result.
Revised: Then we get the following result.

Original: Theorem 3.
Revised: Theorem 3.

Original: For d = 2 , ICA with respect to the pair ( F conf , P conf2 ) is identiﬁable up to S conf .
Revised: For d = 2 , ICA with respect to the pair ( F conf , P conf2 ) is identiﬁable up to S conf .

Original: This means that we can identify conformal maps on compact domains in dimension 2 up to shifts, permutations of coordinates, and scale.
Revised: This means that we can identify conformal maps on compact domains in dimension 2 up to shifts, permutations of coordinates, and scale.

Original: Note that we can also identify conformal maps if P has full support R 2 using the same proof as for d > 2 (see Lemma 2 in the supplement) and an extension as in Corollary 1 is possible.
Revised: Note that we can also identify conformal maps if P has full support R 2 using the same proof as for d > 2 (see Lemma 2 in the supplement) and an extension as in Corollary 1 is possible.

Original: The proof of this result is in Appendix E.
Revised: The proof of this result is in Appendix E.

Original:  Results for orthogonal maps
Revised:  Results for orthogonal maps

Original: Recently, in [16], the more general class of OCTs was considered in the context of ICA.
Revised: Recently, in [16], the more general class of OCTs was considered in the context of ICA.

Original: They referred to orthogonal coordinates as IMA maps, referencing to independent mechanisms.
Revised: They referred to orthogonal coordinates as IMA maps, referencing to independent mechanisms.

Original: This nomenclature was motivated by the causality literature and we refer to their paper for an extensive motivation and further results.
Revised: This nomenclature was motivated by the causality literature and we refer to their paper for an extensive motivation and further results.

Original: As we focus on theoretical results for this function class we stick to the more common term of OCTs.
Revised: As we focus on theoretical results for this function class we stick to the more common term of OCTs.

Original: Orthogonal coordinate transformations are deﬁned as the set of functions whose derivative have orthogonal columns, i.e., the vectors ∂ i f and ∂ j f are orthogonal for i (cid:54)
Revised: Orthogonal coordinate transformations are deﬁned as the set of functions whose derivative have orthogonal columns, i.e., the vectors ∂ i f and ∂ j f are orthogonal for i (cid:54)

Original: =
Revised: =

Original: j .
Revised: j .

Original: Deﬁnition 3.
Revised: Deﬁnition 3.

Original: (OCT maps)
Revised: (OCT maps)

Original: We deﬁne for domains Ω ⊂ R d the set of OCT maps (orthogonal coordinates) by F OCT = { f ∈ C 1 (Ω , R d ) : Df ( x ) (cid:62) Df ( x
Revised: We deﬁne for domains Ω ⊂ R d the set of OCT maps (orthogonal coordinates) by F OCT = { f ∈ C 1 (Ω , R d ) : Df ( x ) (cid:62) Df ( x

Original: ) ∈ Diag( d ) } .
Revised: ) ∈ Diag( d ) } .

Original: OCTs constitute a rich class of functions.
Revised: OCTs constitute a rich class of functions.

Original: The study of OCTs has a long history and already in the 19th century the structure of all OCTs deﬁned in a neighbourhood of a point were characterized [9, 3].
Revised: The study of OCTs has a long history and already in the 19th century the structure of all OCTs deﬁned in a neighbourhood of a point were characterized [9, 3].

Original: Later, also the set of global orthogonal coordinate systems on R d was characterized [27].
Revised: Later, also the set of global orthogonal coordinate systems on R d was characterized [27].

Original: As those results are not easily accesible we will provide here a simple argument showing that OCTs constitute a rich class of functions.
Revised: As those results are not easily accessible we will provide here a simple argument showing that OCTs constitute a rich class of functions.

Original: We ﬁrst note that F OCT contains the above F r − conf , as functions in the later class have a Df ( x ) that takes the form of a Jacobian of a conformal map whose columns are rescaled by derivatives of the entry-wise reparameterizations, such that they remain orthogonal.
Revised: We ﬁrst note that F OCT contains the above F r − conf , as functions in the later class have a Df ( x ) that takes the form of a Jacobian of a conformal map whose columns are rescaled by derivatives of the entry-wise reparametrizations, such that they remain orthogonal.

Original: However, F OCT is much bigger than F r − conf .
Revised: However, F OCT is much bigger than F r − conf .

Original: For example, take a n -tuple ( f 1 , ..., f n ) of arbitrary injective 2D conformal maps f k : Ω k
Revised: For example, take a n -tuple ( f 1 , ..., f n ) of arbitrary injective 2D conformal maps f k : Ω k

Original: → R 2 ∈ F conf where Ω k ⊂ R 2 and build the “concatenated” map f conc : Ω 1 × · · · × Ω n → R 2 n given by
Revised: → R 2 ∈ F conf where Ω k ⊂ R 2 and build the “concatenated” map f conc : Ω 1 × · · · × Ω n → R 2 n given by

Original: [Equation]
Revised: [Equation]

Original: The Jacobian of f conc is block diagonal, such that columns associated to different diagonal blocks are obviously orthogonal, and columns pertaining to the same k-th diagonal block are orthogonal by conformality of f k .
Revised: The Jacobian of f conc is block diagonal, such that columns associated to different diagonal blocks are obviously orthogonal, and columns pertaining to the same k-th diagonal block are orthogonal by conformality of f k .

Original: With such a construction, that we can also further post-compose with transformations in F conf on R 2 n , we can thus build a large non-pararametric subclass of F OCT on R 2 n .
Revised: With such a construction, that we can also further post-compose with transformations in F conf on R 2 n , we can thus build a large non-parametric subclass of F OCT on R 2 n .

Original: This construction can be easily adapted to the case of odd dimensions.
Revised: This construction can be easily adapted to the case of odd dimensions.

Original: Setting for identiﬁability with OCTs.
Revised: Setting for identiﬁability with OCTs.

Original: OCTs can also be generalized to maps whose target is a d -dimensional manifold (see deﬁnition in Appendix B), and the following results will also apply to such case.
Revised: OCTs can also be generalized to maps whose target is a d -dimensional manifold (see deﬁnition in Appendix B), and the following results will also apply to such case.

Original: First, we note that we can only hope to identify a mechanism f ∈ F
Revised: First, we note that we can only hope to identify a mechanism f ∈ F

Original: OCT up to coordinate-wise transformations and permutations, i.e., maps in S reparam .
Revised: OCT up to coordinate-wise transformations and permutations, i.e., maps in S reparam .

Original: Indeed, if f ∈ F OCT and g ∈ S reparam then f ◦ g ∈ F OCT .
Revised: Indeed, if f ∈ F OCT and g ∈ S reparam then f ◦ g ∈ F OCT .

Original: Thus, in particular f ∗ P =
Revised: Thus, in particular f ∗ P =

Original: ( f ◦ g ) ∗ ( g − 1 ) ∗ P .
Revised: ( f ◦ g ) ∗ ( g − 1 ) ∗ P .

Original: This implies that given observations from f ∗ P we can identify f and P only up to g ∈ S reparam .
Revised: This implies that given observations from f ∗ P we can identify f and P only up to g ∈ S reparam .

Original: More precisely, for any (sufﬁciently smooth) P (cid:48) there is f (cid:48) such that f ∗ P = f
Revised: More precisely, for any (sufﬁciently smooth) P (cid:48) there is f (cid:48) such that f ∗ P = f

Original: (cid:48)∗ P (cid:48) where we pick g such that P (cid:48)
Revised: (cid:48)∗ P (cid:48) where we pick g such that P (cid:48)

Original: =
Revised: =

Original: g − 1 ∗ P .
Revised: g − 1 ∗ P .

Original: As the distribution of the s i is not identiﬁable, we map it to a ﬁxed reference distribution that we choose to be the uniform distribution on (0 , 1) d .
Revised: As the distribution of the s i is not identiﬁable, we map it to a ﬁxed reference distribution that we choose to be the uniform distribution on (0 , 1) d .

Original: We introduce the shorthand C d = (0 , 1) d for the standard open unit cube (exclusion of the boundary will be important for our result) and denote by ν the uniform (Lebesgue) measure on C d .
Revised: We introduce the shorthand C d = (0 , 1) d for the standard open unit cube (exclusion of the boundary will be important for our result) and denote by ν the uniform (Lebesgue) measure on C d .

Original: For ﬁxed base measure ν the symmetry group is reduced to permutations and reﬂections, i.e., maps in P ∈ Perm ± ( d ) .
Revised: For ﬁxed base measure ν the symmetry group is reduced to permutations and reﬂections, i.e., maps in P ∈ Perm ± ( d ) .

Original: We conjecture that for ’typical’ pairs ( f, P ) ∈
Revised: We conjecture that for ’typical’ pairs ( f, P ) ∈

Original: F OCT × P OCT , ICA is identiﬁable with respect to
Revised: F OCT × P OCT , ICA is identiﬁable with respect to

Original: S reparam (with a suitable deﬁnition of P OCT , e.g., P OCT = P conf ).
Revised: S reparam (with a suitable deﬁnition of P OCT , e.g., P OCT = P conf ).

Original: However, we leave a precise statement for future work.
Revised: However, we leave a precise statement for future work.

Original: Below we will show a weaker notion of identiﬁability for OCTs, but, before that, we ﬁrst exhibit exceptional classes of spurious solutions for ICA with OCTs.
Revised: Below we will show a weaker notion of identiﬁability for OCTs, but, before that, we ﬁrst exhibit exceptional classes of spurious solutions for ICA with OCTs.

Original: Spurious solutions for ICA with OCTs.
Revised: Spurious solutions for ICA with OCTs.

Original: We ﬁrst note that just as for linear maps and conformal maps (see Thm. 1) Gaussian distributions can hamper identiﬁcation.
Revised: We ﬁrst note that just as for linear maps and conformal maps (see Thm. 1) Gaussian distributions can hamper identiﬁcation.

Original: This is because arbitrary measures with a factorized density can be pushed forward into multivariate Gaussians using a suitable coordinate-wise transformation.
Revised: This is because arbitrary measures with a factorized density can be pushed forward into multivariate Gaussians using a suitable coordinate-wise transformation.

Original: Fact 1 .
Revised: Fact 1 .

Original: Let P be a probability measure on R with bounded density p and cumulative distribution function F P .
Revised: Let P be a probability measure on R with bounded density p and cumulative distribution function F P .

Original: Denote the cumulative distribution function of the standard normal by F N .
Revised: Denote the cumulative distribution function of the standard normal by F N .

Original: Let h P =
Revised: Let h P =

Original: F − 1 N ◦ F P .
Revised: F − 1 N ◦ F P .

Original: Then ( h P ) ∗ P has a standard normal distribution.
Revised: Then ( h P ) ∗ P has a standard normal distribution.

Original: This implies that if P = P 1 ⊗ . . .
Revised: This implies that if P = P 1 ⊗ . . .

Original: ⊗ P d and x = f ( s )
Revised: ⊗ P d and x = f ( s )

Original: =
Revised: =

Original: ( h P 1 ( s 1 ) , . . .
Revised: ( h P 1 ( s 1 ) , . . .

Original: , h P d ( s d )) then f ∗ P follows a standard normal distribution.
Revised: , h P d ( s d )) then f ∗ P follows a standard normal distribution.

Original: In particular, for every A ∈ O( d ) the distribution of Ax is standard normal and its components are independent.
Revised: In particular, for every A ∈ O( d ) the distribution of Ax is standard normal and its components are independent.

Original: Note that, in contrast to conformal and linear maps, it is not sufﬁcient to exclude Gaussian source distributions: due to the ﬂexibility of the function class, we have to exclude that the pair ( f, P ) has a Gaussian observational distribution f ∗ P .
Revised: Note that, in contrast to conformal and linear maps, it is not sufﬁcient to exclude Gaussian source distributions: due to the ﬂexibility of the function class, we have to exclude that the pair ( f, P ) has a Gaussian observational distribution f ∗ P .

Original: Next we show that a more general construction using OCTs is possible.
Revised: Next we show that a more general construction using OCTs is possible.

Original: Proposition 1.
Revised: Proposition 1.

Original: Let P be a rotation invariant distribution on R d with smooth density.
Revised: Let P be a rotation invariant distribution on R d with smooth density.

Original: Then there is a smooth and invertible (on its image) function f : C d → R d with f ∈ F OCT such that f ∗ ν = P .
Revised: Then there is a smooth and invertible (on its image) function f : C d → R d with f ∈ F OCT such that f ∗ ν = P .

Original: The proof of this result can be found in Appendix F.
Revised: The proof of this result can be found in Appendix F.

Original: The main idea in the proof is that d -dimensional polar coordinates do the trick up to coordinate-wise rescaling.
Revised: The main idea in the proof is that d -dimensional polar coordinates do the trick up to coordinate-wise rescaling.

Original: This proposition implies that the entire family { f R =
Revised: This proposition implies that the entire family { f R =

Original: R ◦ f | R ∈ O ( n ) } satisﬁes ( f R ) ∗ ν = f ∗ ν and, by deﬁnition of F OCT we have f R ∈ F OCT because f ∈ F OCT implies R ◦ f ∈ F OCT .
Revised: R ◦ f | R ∈ O ( n ) } satisﬁes ( f R ) ∗ ν = f ∗ ν and, by deﬁnition of F OCT we have f R ∈ F OCT because f ∈ F OCT implies R ◦ f ∈ F OCT .

Original: In particular all inverses g R = f − 1 R recover independent sources because ( g R ) ∗ f ∗ ν = ν
Revised: In particular all inverses g R = f − 1 R recover independent sources because ( g R ) ∗ f ∗ ν = ν

Original: and BSS is not possible in a meaningful way in this (special) case.
Revised: and BSS is not possible in a meaningful way in this (special) case.

Original: The construction in Proposition 1 gives spurious solutions for substantially more observational distributions than just the Gaussian (this is indicated in Table 1).
Revised: The construction in Proposition 1 gives spurious solutions for substantially more observational distributions than just the Gaussian (this is indicated in Table 1).

Original: Nevertheless we do not view this as a general obstacle to identiﬁability results for OCTs for two reasons.
Revised: Nevertheless we do not view this as a general obstacle to identiﬁability results for OCTs for two reasons.

Original: Firstly the spurious solutions only apply to carefully chosen pairs of function and base measure such that we obtain a still very non-generic (radial) observational distribution.
Revised: Firstly the spurious solutions only apply to carefully chosen pairs of function and base measure such that we obtain a still very non-generic (radial) observational distribution.

Original: Moreover, the function constructed in Proposition 1 cannot be extended to C d = [0 , 1] d such that it remains invertible (in the language of differential geometry: f is only an immersion not an embedding of a submanifold).
Revised: Moreover, the function constructed in Proposition 1 cannot be extended to C d = [0 , 1] d such that it remains invertible (in the language of differential geometry: f is only an immersion not an embedding of a submanifold).

Original: In this sense the main message of Proposition 1 is that an identiﬁability result for F OCT needs to contain assumptions ruling out those spurious solutions (just like Gaussians are excluded for linear ICA).
Revised: In this sense the main message of Proposition 1 is that an identiﬁability result for F OCT needs to contain assumptions ruling out those spurious solutions (just like Gaussians are excluded for linear ICA).

Original: Local Stability of OCTs.
Revised: Local Stability of OCTs.

Original: We now give partial results towards identiﬁability of OCTs.
Revised: We now give partial results towards identiﬁability of OCTs.

Original: While we do not prove general identiﬁability for this class, we demonstrate their local rigidity : OCTs cannot be continuously deformed to obtain spurious solutions.
Revised: While we do not prove general identiﬁability for this class, we demonstrate their local rigidity : OCTs cannot be continuously deformed to obtain spurious solutions.

Original: This is in stark contrast to the general nonlinear case, which we will discuss for comparison below.
Revised: This is in stark contrast to the general nonlinear case, which we will discuss for comparison below.

Original: Actually, we will show the following slightly stronger statement.
Revised: Actually, we will show the following slightly stronger statement.

Original: Suppose we know some initial data generating mechanism f 0 because we have, e.g., access to samples ( s, f 0 ( s )) .
Revised: Suppose we know some initial data generating mechanism f 0 because we have, e.g., access to samples ( s, f 0 ( s )) .

Original: Now we assume that the mixing f t depends smoothly on some parameter t which could be, e.g., time or an environment.
Revised: Now we assume that the mixing f t depends smoothly on some parameter t which could be, e.g., time or an environment.

Original: Then we can identify f t if f t ∈ F OCT for all t and given access to samples f t ( s ) .
Revised: Then we can identify f t if f t ∈ F OCT for all t and given access to samples f t ( s ) .

Original: This would not be true if f t ’s would be unconstrained nonlinear mixing functions.
Revised: This would not be true if f t ’s would be unconstrained nonlinear mixing functions.

Original: We now make these statements precise.
Revised: We now make these statements precise.

Original: We ﬁrst deﬁne smooth deformations of a mixing function.
Revised: We ﬁrst deﬁne smooth deformations of a mixing function.

Original: Deﬁnition 4.
Revised: Deﬁnition 4.

Original: (Smooth invariant deformations) Let F be some function class.
Revised: (Smooth invariant deformations) Let F be some function class.

Original: Consider a family of differentiable transformations Φ ∈ C 1 (( − T, T ) × R d , M ) for some T > 0 and a smooth, ddimensional manifold M such that Φ t =
Revised: Consider a family of differentiable transformations Φ ∈ C 1 (( − T, T ) × R d , M ) for some T > 0 and a smooth, ddimensional manifold M such that Φ t =

Original: Φ( t, · ) is a diffeomorphism onto its image.
Revised: Φ( t, · ) is a diffeomorphism onto its image.

Original: We call Φ t a smooth invariant deformation if Φ t ( · ) ∈ F for all t .
Revised: We call Φ t a smooth invariant deformation if Φ t ( · ) ∈ F for all t .

Original: An illustration of this deﬁnition can be found in Figure 3.
Revised: An illustration of this deﬁnition can be found in Figure 3.

Original: Based on invariant deformations we can now deﬁne a local identiﬁability property of ICA in a given function class.
Revised: Based on invariant deformations we can now deﬁne a local identiﬁability property of ICA in a given function class.

Original: Deﬁnition 5.
Revised: Deﬁnition 5.

Original: (Local identiﬁability of ICA).
Revised: (Local identiﬁability of ICA).

Original: Consider a function class F .
Revised: Consider a function class F .

Original: Let f t be a smooth invariant deformation in F that is analytic in t .
Revised: Let f t be a smooth invariant deformation in F that is analytic in t .

Original: Let Φ t be another smooth invariant deformation in F analytic in t such that Φ 

Revised: Let Φ t be another smooth invariant deformation in F analytic in t such that Φ

Original: = f 0 and (Φ t ) ∗ ν = ( f t ) ∗ ν for all t .
Revised: = f 0 and (Φ t ) ∗ ν = ( f t ) ∗ ν for all t .

Original: Assume that there is ε > 0 such that Φ t ( s ) =
Revised: Assume that there is ε > 0 such that Φ t ( s ) =

Original: f t ( s ) if dist( s, ∂C d ) < ε .
Revised: f t ( s ) if dist( s, ∂C d ) < ε .

Original: Then we say that ICA in F is locally identiﬁable at ( f t , ν ) if these assumptions imply Φ t = f t for all t .
Revised: Then we say that ICA in F is locally identiﬁable at ( f t , ν ) if these assumptions imply Φ t = f t for all t .

Original: We call F locally identiﬁable if it is locally identiﬁable at ( f t , ν ) for all analytic local deformations f t .
Revised: We call F locally identiﬁable if it is locally identiﬁable at ( f t , ν ) for all analytic local deformations f t .

Original: Local identiﬁability of a function class means that we can identify smooth deformations f t ∈ F from some initial mixing f 0 ∈ F if f 0 is known on the whole latent domain, and the behaviour of f t close to the domain’s boundary is known for all t .
Revised: Local identiﬁability of a function class means that we can identify smooth deformations f t ∈ F from some initial mixing f 0 ∈ F if f 0 is known on the whole latent domain, and the behaviour of f t close to the domain’s boundary is known for all t .

Original: This can be interpreted as plain identiﬁability in the concept drift setting [51, 14].
Revised: This can be interpreted as plain identiﬁability in the concept drift setting [51, 14].

Original: Formulated differently, it means that an adversary cannot smoothly deform the function f 0 in a subset whose boundary is away from the boundary of the domain, such that the outcome is ambiguous given the resulting observational distributions.
Revised: Formulated differently, it means that an adversary cannot smoothly deform the function f 0 in a subset whose boundary is away from the boundary of the domain, such that the outcome is ambiguous given the resulting observational distributions.

Original: An experimental illustration of this setting and Theorem 4 below can be found in Appendix H.
Revised: An experimental illustration of this setting and Theorem 4 below can be found in Appendix H.

Original: The notion of locality in Deﬁnition 5 should be understood as “non-global” and notably does not imply restrictions to a small neighborhood, as local properties often do.
Revised: The notion of locality in Deﬁnition 5 should be understood as “non-global” and notably does not imply restrictions to a small neighbourhood, as local properties often do.

Original: The non-globality manifests itself in two ways: we consider smooth transformations of the ground truth, i.e., small changes of the data generating function f 

Revised: The non-globality manifests itself in two ways: we consider smooth transformations of the ground truth, i.e., small changes of the data generating function f

Original: and in addition we assume that the changes are not everywhere in s , i.e., sources s close to the boundary are kept invariant.
Revised: and in addition we assume that the changes are not everywhere in s , i.e., sources s close to the boundary are kept invariant.

Original: An extension to other source distributions beyond ﬁxed ν (and possibly changing with t ) is possible but not necessary in the context of OCTs as explained above.
Revised: An extension to other source distributions beyond ﬁxed ν (and possibly changing with t ) is possible but not necessary in the context of OCTs as explained above.

Original: We can show that local identiﬁability holds true in F OCT .
Revised: We can show that local identiﬁability holds true in F OCT .

Original: Theorem 4.
Revised: Theorem 4.

Original: The function class F OCT is locally identiﬁable.
Revised: The function class F OCT is locally identiﬁable.

Original: The proof, in Appendix F, introduces new tools to the ﬁeld of ICA.
Revised: The proof, in Appendix F, introduces new tools to the ﬁeld of ICA.

Original: The main idea is to consider the vector ﬁeld X that generates the deformation Φ t and then rewrite the assumption as systems of partial differential equations for X .
Revised: The main idea is to consider the vector ﬁeld X that generates the deformation Φ t and then rewrite the assumption as systems of partial differential equations for X .

Original: The proof is then completed by showing that the only solution of this system vanishes.
Revised: The proof is then completed by showing that the only solution of this system vanishes.

Original: Let us state one simple consequence of this theorem.
Revised: Let us state one simple consequence of this theorem.

Original: Corollary 2.
Revised: Corollary 2.

Original: Let Φ t be a smooth analytic invariant deformation in F OCT such that (Φ t ) ∗ ν = f ∗ ν for all t and there is ε > 

Revised: Let Φ t be a smooth analytic invariant deformation in F OCT such that (Φ t ) ∗ ν = f ∗ ν for all t and there is ε >

Original: such that Φ t ( s ) =
Revised: such that Φ t ( s ) =

Original: f ( s ) if dist( s, ∂C d ) < ε .
Revised: f ( s ) if dist( s, ∂C d ) < ε .

Original: Then Φ t = f for all t .
Revised: Then Φ t = f for all t .

Original: Let us reiterate what this corollary shows: we cannot smoothly and locally transform the function f such that (1) the observational distribution remains invariant, i.e., equal to f ∗ ν , and (2) the deformed functions remain OCTs.
Revised: Let us reiterate what this corollary shows: we cannot smoothly and locally transform the function f such that (1) the observational distribution remains invariant, i.e., equal to f ∗ ν , and (2) the deformed functions remain OCTs.

Original: At a high level this result suggests that OCTs can be identiﬁed if we know f close to the boundary of the support of s , e.g., by having, in addition to unlabelled data f ( s ) , labelled data ( s, f ( s )) for those s where one coordinate s i is extremal.
Revised: At a high level this result suggests that OCTs can be identiﬁed if we know f close to the boundary of the support of s , e.g., by having, in addition to unlabelled data f ( s ) , labelled data ( s, f ( s )) for those s where one coordinate s i is extremal.

Original: Note that we actually do not show this result as there might be further solutions which are not connected by smooth transformations.
Revised: Note that we actually do not show this result as there might be further solutions which are not connected by smooth transformations.

Original: We expect that those results can be generalized substantially.
Revised: We expect that those results can be generalized substantially.

Original: In particular, we conjecture that for “most” functions
Revised: In particular, we conjecture that for “most” functions

Original: f the boundary condition can be removed thus giving a stronger local identiﬁability result up to the boundary of the support of s .
Revised: f the boundary condition can be removed thus giving a stronger local identiﬁability result up to the boundary of the support of s .

Original: As a partial result in this direction we prove the following theorem.
Revised: As a partial result in this direction we prove the following theorem.

Original: Theorem 5.
Revised: Theorem 5.

Original: Let f : C d → R d be given by f ( x ) = RDx , with R ∈ O
Revised: Let f : C d → R d be given by f ( x ) = RDx , with R ∈ O

Original: ( d ) and D =
Revised: ( d ) and D =

Original: Diag( µ 1 , . .
Revised: Diag( µ 1 , . .

Original: . , µ d ) where µ i are i.i.d.
Revised: . , µ d ) where µ i are i.i.d.

Original: samples from a distribution supported on the positive reals R + which has a density.
Revised: samples from a distribution supported on the positive reals R + which has a density.

Original: Suppose that Φ t is a smooth invariant deformation in F OCT such that Φ 0 = f , (Φ t ) ∗ ν = f ∗ ν , and Φ t is analytic in t .
Revised: Suppose that Φ t is a smooth invariant deformation in F OCT such that Φ 0 = f , (Φ t ) ∗ ν = f ∗ ν , and Φ t is analytic in t .

Original: Then for almost all µ i (i.e., with probability one) this implies Φ t = f on C d , i.e., Φ t is constant in time.
Revised: Then for almost all µ i (i.e., with probability one) this implies Φ t = f on C d , i.e., Φ t is constant in time.

Original: In Appendix F we show that this theorem follows from a slightly stronger result stated as Theoremwhich has a similar proof as Theorem 4.
Revised: In Appendix F we show that this theorem follows from a slightly stronger result stated as Theoremwhich has a similar proof as Theorem 4.

Original: We do expect that the conclusion of the Theorem actually holds for all µ i not just almost all, but we are unable to show this.
Revised: We do expect that the conclusion of the Theorem actually holds for all µ i not just almost all, but we are unable to show this.

Original: Comparison with ICA for general nonlinear functions.
Revised: Comparison with ICA for general nonlinear functions.

Original: Let us emphasize that those results are non-trivial as they establish a large difference between ICA with generic nonlinear maps and ICA with OCTs.
Revised: Let us emphasize that those results are non-trivial as they establish a large difference between ICA with generic nonlinear maps and ICA with OCTs.

Original: To clarify this, we state that no result similar to Theorem 4 holds without the assumption that Φ t ∈ F OCT .
Revised: To clarify this, we state that no result similar to Theorem 4 holds without the assumption that Φ t ∈ F OCT .

Original: Put differently, the function class F nonlinear is not locally identiﬁable.
Revised: Put differently, the function class F nonlinear is not locally identiﬁable.

Original: Fact 2 .
Revised: Fact 2 .

Original: Suppose f : C d → R d is a diffeomorphism on its image.
Revised: Suppose f : C d → R d is a diffeomorphism on its image.

Original: Then there are uncountably many smooth deformations Φ t of ( f, ν ) such that (i) (Φ t ) ∗ ν = f ∗ ν and (ii) there is an ε > 

Revised: Then there are uncountably many smooth deformations Φ t of ( f, ν ) such that (i) (Φ t ) ∗ ν = f ∗ ν and (ii) there is an ε >

Original: such that Φ t ( s ) =
Revised: such that Φ t ( s ) =

Original: f ( s ) whenever dist( s, ∂C d ) < ε .
Revised: f ( s ) whenever dist( s, ∂C d ) < ε .

Original: For completeness, we provide a general construction that is close to our proof of Theorem 4 in
Revised: For completeness, we provide a general construction that is close to our proof of Theorem 4 in

Original: Appendix C in the supplement.
Revised: Appendix C in the supplement.

Original: A very clear construction for this result was given in [24].
Revised: A very clear construction for this result was given in [24].

Original: Lemma 1 (Smoothly varying radius dependent rotations (see [24])) .
Revised: Lemma 1 (Smoothly varying radius dependent rotations (see [24])) .

Original: Let R : R × R + → O( d ) be a smooth function mapping to orthogonal matrices and let a ∈ C d .
Revised: Let R : R × R + → O( d ) be a smooth function mapping to orthogonal matrices and let a ∈ C d .

Original: Assume that R ( t, r ) =
Revised: Assume that R ( t, r ) =

Original: id for r ≥ dist( a, ∂C d ) .
Revised: id for r ≥ dist( a, ∂C d ) .

Original: Then the map s → h R,a ( t, s )
Revised: Then the map s → h R,a ( t, s )

Original: =
Revised: =

Original: R ( | s − a | , t )( s − a ) + a preserves the uniform measure ν on C d for all t so that f ◦ h R,a ( t, s ) D = f ( s ) for all t if s is distributed according to ν .
Revised: R ( | s − a | , t )( s − a ) + a preserves the uniform measure ν on C d for all t so that f ◦ h R,a ( t, s ) D = f ( s ) for all t if s is distributed according to ν .

Original: An illustration of this construction is shown in Figure 2 (see App.
Revised: An illustration of this construction is shown in Figure 2 (see App.

Original: C for details).
Revised: C for details).

Original: Clearly, by concatenation this allows us to create a vast family of spurious solutions.
Revised: Clearly, by concatenation this allows us to create a vast family of spurious solutions.

Original: Note that those solutions are excluded when restricting to OCTs which is a corollary of Theorem 4.
Revised: Note that those solutions are excluded when restricting to OCTs which is a corollary of Theorem 4.

Original: Corollary 3.
Revised: Corollary 3.

Original: Suppose f ∈ F OCT .
Revised: Suppose f ∈ F OCT .

Original: Let Φ t be the smooth invariant deformation deﬁned by Φ t = f ◦ h R,a ( t, · ) where h R,a is as in Lemma 1.
Revised: Let Φ t be the smooth invariant deformation deﬁned by Φ t = f ◦ h R,a ( t, · ) where h R,a is as in Lemma 1.

Original: If Φ t ∈ F OCT for all t this implies that h R,a ( t, s ) =
Revised: If Φ t ∈ F OCT for all t this implies that h R,a ( t, s ) =

Original: s and Φ t = f for all t .
Revised: s and Φ t = f for all t .

Original: We now summarize our view on the results of this section informally (we do not claim that the statements below regarding (inﬁnite dimensional) manifolds can be made rigorous).
Revised: We now summarize our view on the results of this section informally (we do not claim that the statements below regarding (inﬁnite dimensional) manifolds can be made rigorous).

Original: For a given data generating mechanism ( f 0 , ν ) we expect that typically the set of all solutions M OCT = { f ∈
Revised: For a given data generating mechanism ( f 0 , ν ) we expect that typically the set of all solutions M OCT = { f ∈

Original: F OCT | f ∗ ν = ( f 0 ) ∗ ν } ⊂ F OCT is a zero dimensional submanifold, i.e., consists of isolated spurious solutions and we prove this when ﬁxing the boundary (see Theorem 4) while the corresponding submanifold of general nonlinear spurious solutions M nonlinear
Revised: F OCT | f ∗ ν = ( f 0 ) ∗ ν } ⊂ F OCT is a zero dimensional submanifold, i.e., consists of isolated spurious solutions and we prove this when ﬁxing the boundary (see Theorem 4) while the corresponding submanifold of general nonlinear spurious solutions M nonlinear

Original: = { f : C d → R d | f ∗ ν = ( f 0 ) ∗ ν } is inﬁnite dimensional even when requiring f ( s ) =
Revised: = { f : C d → R d | f ∗ ν = ( f 0 ) ∗ ν } is inﬁnite dimensional even when requiring f ( s ) =

Original: f 0 ( s ) close to the boundary of C d .
Revised: f 0 ( s ) close to the boundary of C d .

Original:  Results for volume preserving maps
Revised:  Results for volume preserving maps

Original: Let us ﬁnally consider volume preserving transformations.
Revised: Let us ﬁnally consider volume preserving transformations.

Original: For Ω ⊂ R d , those are deﬁned as the set of functions F vol = { f : Ω → R d | det Df ( x ) = 1 for all x ∈ Ω } .
Revised: For Ω ⊂ R d , those are deﬁned as the set of functions F vol = { f : Ω → R d | det Df ( x ) = 1 for all x ∈ Ω } .

Original: Invertible volume preserving deformations have the property that they preserve the standard (Lebesgue)-measure λ in the sense that f ∗ λ Ω = λ f (Ω) .
Revised: Invertible volume preserving deformations have the property that they preserve the standard (Lebesgue)-measure λ in the sense that f ∗ λ Ω = λ f (Ω) .

Original: Recently it was proposed that volume preserving functions are a suitable function class for ICA.
Revised: Recently it was proposed that volume preserving functions are a suitable function class for ICA.

Original: Here we show that those functions are not sufﬁciently rigid to allow identiﬁability of ICA in the unconditional case.
Revised: Here we show that those functions are not sufﬁciently rigid to allow identiﬁability of ICA in the unconditional case.

Original: Note that Lemma 1 and Fact 2 already show how to construct spurious solutions for the case that the base distribution is the uniform measure ν .
Revised: Note that Lemma 1 and Fact 2 already show how to construct spurious solutions for the case that the base distribution is the uniform measure ν .

Original: However, for an arbitrary distribution P this is slightly more difﬁcult because we need to ﬁnd maps g that preserve P , i.e., g ∗ P = P , and are volume preserving, i.e., preserve the standard measure.
Revised: However, for an arbitrary distribution P this is slightly more difﬁcult because we need to ﬁnd maps g that preserve P , i.e., g ∗ P = P , and are volume preserving, i.e., preserve the standard measure.

Original: Theorem 6.
Revised: Theorem 6.

Original: Let p be a twice differentiable probability density with bounded gradient.
Revised: Let p be a twice differentiable probability density with bounded gradient.

Original: Suppose that x = f ( s ) where the distribution P of s has density p and f is a diffeomorphism with det Df ( x ) =for x ∈ R d .
Revised: Suppose that x = f ( s ) where the distribution P of s has density p and f is a diffeomorphism with det Df ( x ) =for x ∈ R d .

Original: Then there is a family of functions f t : R × R
Revised: Then there is a family of functions f t : R × R

Original: d
Revised: d

Original: → R d with f 0 = f and f t (cid:54) = f 0 for t (cid:54) =such that det Df t ( x ) = 1 and ( f t ) ∗ P = f ∗ P .
Revised: → R d with f 0 = f and f t (cid:54) = f 0 for t (cid:54) =such that det Df t ( x ) = 1 and ( f t ) ∗ P = f ∗ P .

Original: The proof and an illustration are in Appendix G.
Revised: The proof and an illustration are in Appendix G.

Original: It is based on the ﬂows generated by suitable explicit vector ﬁelds.
Revised: It is based on the ﬂows generated by suitable explicit vector ﬁelds.

Original: As those ﬂows can be concatenated we obtain a large family of spurious solutions.
Revised: As those ﬂows can be concatenated we obtain a large family of spurious solutions.

Original: We think that the approach used here is a powerful technique to construct counter-examples to identiﬁability in ICA.
Revised: We think that the approach used here is a powerful technique to construct counter-examples to identiﬁability in ICA.

Original: As local identiﬁability is weaker than identiﬁability we conclude that ICA in
Revised: As local identiﬁability is weaker than identiﬁability we conclude that ICA in

Original: F vol is not identiﬁable.
Revised: F vol is not identiﬁable.

Original: Note that this is even true when we know the distribution of s .
Revised: Note that this is even true when we know the distribution of s .

Original: A rigorous version of this statement is that if ( F vol , P vol ) is identiﬁable with respect to S vol then S vol contains functions mixing coordinates s i , s j for i (cid:54) =
Revised: A rigorous version of this statement is that if ( F vol , P vol ) is identiﬁable with respect to S vol then S vol contains functions mixing coordinates s i , s j for i (cid:54) =

Original: j (i.e., there is h ∈ S vol
Revised: j (i.e., there is h ∈ S vol

Original: such that ∂ i ∂
Revised: such that ∂ i ∂

Original: j h (cid:54) = 0 ).
Revised: j h (cid:54) = 0 ).

Original:  Relation to rigidity theory
Revised:  Relation to rigidity theory

Original: Our results rely on rigidity properties of certain function classes.
Revised: Our results rely on rigidity properties of certain function classes.

Original: Rigidity refers to the property that a local constraint on the derivative of a function implies global restrictions on the shape of the function.
Revised: Rigidity refers to the property that a local constraint on the derivative of a function implies global restrictions on the shape of the function.

Original: These type of results are of interest in the ﬁeld of continuum mechanics [5] where, e.g., the condition Df ∈ SO( d ) is used to describe deformation of rigid solids and the condition Det Df =to describe incompressible ﬂuids.
Revised: These type of results are of interest in the ﬁeld of continuum mechanics [5] where, e.g., the condition Df ∈ SO( d ) is used to describe deformation of rigid solids and the condition Det Df =to describe incompressible ﬂuids.

Original: We now state a well known rigidity result for conformal maps that is the main input in the proof of
Revised: We now state a well known rigidity result for conformal maps that is the main input in the proof of

Original: Theorem 2.
Revised: Theorem 2.

Original: This result shows that there are very few conformal maps in dimension d > 2 .
Revised: This result shows that there are very few conformal maps in dimension d > 2 .

Original: Theorem 7 (Liouville) .
Revised: Theorem 7 (Liouville) .

Original: Let d > 2 , Ω ⊂ R d open and connected, f : Ω → R d conformal.
Revised: Let d > 2 , Ω ⊂ R d open and connected, f : Ω → R d conformal.

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: (12) where b, a ∈ R d , α ∈ R , A ∈ O( d ) , and ε ∈ { 0 , 2 } .
Revised: (12) where b, a ∈ R d , α ∈ R , A ∈ O( d ) , and ε ∈ { 0 , 2 } .

Original: Originally this result was shown by Liouville [35], a modern treatment is [26].
Revised: Originally this result was shown by Liouville [35], a modern treatment is [26].

Original: In particular, this shows that conformal maps are (up to translations) rotations or rotations followed by an inversion.
Revised: In particular, this shows that conformal maps are (up to translations) rotations or rotations followed by an inversion.

Original: To illustrate the strength of Theorem 7 we compare it to the setting of volume preserving maps which satisfy no similar rigidity property.
Revised: To illustrate the strength of Theorem 7 we compare it to the setting of volume preserving maps which satisfy no similar rigidity property.

Original: Intuitively the different rigidity properties are already apparent from the connection to solids, which can merely be rotated and shifted, and ﬂuids which can also be stirred leading to chaotic deformations.
Revised: Intuitively the different rigidity properties are already apparent from the connection to solids, which can merely be rotated and shifted, and ﬂuids which can also be stirred leading to chaotic deformations.

Original: Rigorously the different behaviours can be clariﬁed by the observation that conformal maps have a ﬁnite number of parameters and thus a ﬁnite number of constraints (e.g., of the form f ( x ) =
Revised: Rigorously the different behaviours can be clariﬁed by the observation that conformal maps have a ﬁnite number of parameters and thus a ﬁnite number of constraints (e.g., of the form f ( x ) =

Original: y ) allows to identify them.
Revised: y ) allows to identify them.

Original: In contrast volume preserving maps cannot be identiﬁed from ﬁnitely many constraints as the following proposition shows.
Revised: In contrast volume preserving maps cannot be identiﬁed from ﬁnitely many constraints as the following proposition shows.

Original: Proposition 2.
Revised: Proposition 2.

Original: For d > 2 and { x 1 , . . .
Revised: For d > 2 and { x 1 , . . .

Original: , x n , y 1 , . . .
Revised: , x n , y 1 , . . .

Original: , y n } ⊂ R d , all pairwise different, there is a volume preserving diffeomorphism f : R d → R d , f ∈ F vol such that f ( x i ) =
Revised: , y n } ⊂ R d , all pairwise different, there is a volume preserving diffeomorphism f : R d → R d , f ∈ F vol such that f ( x i ) =

Original: y i .
Revised: y i .

Original: Let us emphasize that the different rigidity properties are not at all surprising when arguing based on degrees of freedom or numbers of constraints.
Revised: Let us emphasize that the different rigidity properties are not at all surprising when arguing based on degrees of freedom or numbers of constraints.

Original: While volume preserving maps enforce only a single scalar constraint on the Jacobian Df the condition for conformal maps gives n ( n + 1) / 2 −constraints on the Jacobian.
Revised: While volume preserving maps enforce only a single scalar constraint on the Jacobian Df the condition for conformal maps gives n ( n + 1) / 2 −constraints on the Jacobian.

Original: Let us ﬁnally comment on OCTs where the picture is not as well understood.
Revised: Let us ﬁnally comment on OCTs where the picture is not as well understood.

Original: As discussed in
Revised: As discussed in

Original: Section 4 it is known [9, 3] that OCTs constitute a rich, non-parametric class of functions and therefore OCTs are much more ﬂexible than conformal maps.
Revised: Section 4 it is known [9, 3] that OCTs constitute a rich, non-parametric class of functions and therefore OCTs are much more ﬂexible than conformal maps.

Original: We illustrated this with example OCT constructions in Section 4 leveraging 2D conformal maps.
Revised: We illustrated this with example OCT constructions in Section 4 leveraging 2D conformal maps.

Original: Nevertheless, it is not known if and what rigidity properties can be derived for OCTs.
Revised: Nevertheless, it is not known if and what rigidity properties can be derived for OCTs.

Original: However, our results suggest that the additional measure preservation condition in the context of ICA gives enough rigidity to (almost) give identiﬁability of
Revised: However, our results suggest that the additional measure preservation condition in the context of ICA gives enough rigidity to (almost) give identiﬁability of

Original: ICA.
Revised: ICA.

Original: In this sense OCTs might be a good function class for ICA as it is rich enough to allow complex representations of data while at the same time being sufﬁciently rigid to still provide a notion of identiﬁability whose strength remains to be determined.
Revised: In this sense OCTs might be a good function class for ICA as it is rich enough to allow complex representations of data while at the same time being sufﬁciently rigid to still provide a notion of identiﬁability whose strength remains to be determined.

Original:  Discussion
Revised:  Discussion

Original: ICA is long known to be identiﬁable for linear maps, baring pathological cases, and highly nonidentiﬁable for general nonlinear ones.
Revised: ICA is long known to be identiﬁable for linear maps, baring pathological cases, and highly nonidentiﬁable for general nonlinear ones.

Original: Surprisingly, similar results for function classes of intermediate complexity remain scarce.
Revised: Surprisingly, similar results for function classes of intermediate complexity remain scarce.

Original: In this work we address this question with several identiﬁability results for different function classes.
Revised: In this work we address this question with several identiﬁability results for different function classes.

Original: Our ﬁrst main result is that ICA is identiﬁable in the class of conformal maps (up to classical ambiguities).
Revised: Our ﬁrst main result is that ICA is identiﬁable in the class of conformal maps (up to classical ambiguities).

Original: This considerably extends previous claims, limited to a speciﬁc 2D setting [24], and ruling out several families of spurious solutions [16].
Revised: This considerably extends previous claims, limited to a speciﬁc 2D setting [24], and ruling out several families of spurious solutions [16].

Original: On the negative side we show that the ICA problem for volume preserving maps admits a large class of spurious solutions.
Revised: On the negative side we show that the ICA problem for volume preserving maps admits a large class of spurious solutions.

Original: Finally, we show that OCTs satisfy certain weaker notions of local identiﬁability.
Revised: Finally, we show that OCTs satisfy certain weaker notions of local identiﬁability.

Original: In our proofs, we draw connections to methods and techniques that, to the best of our knowledge, have not been used in the context of ICA before.
Revised: In our proofs, we draw connections to methods and techniques that, to the best of our knowledge, have not been used in the context of ICA before.

Original: We relate the identiﬁability problem in ICA to the rigidity of the considered function class F and use tools from the theory of partial differential equations.
Revised: We relate the identiﬁability problem in ICA to the rigidity of the considered function class F and use tools from the theory of partial differential equations.

Original: These techniques have been applied very successfully to the analysis of elastic solids [6, 5] and we believe that there are many applications of these methods in the ﬁeld of ICA.
Revised: These techniques have been applied very successfully to the analysis of elastic solids [6, 5] and we believe that there are many applications of these methods in the ﬁeld of ICA.

Original: While the main focus of current research after the seminal work of Hyvärinen et al.
Revised: While the main focus of current research after the seminal work of Hyvärinen et al.

Original: [25] is on the auxiliary variable case, there are three reasons to consider unconditional ICA.
Revised: [25] is on the auxiliary variable case, there are three reasons to consider unconditional ICA.

Original: Firstly, it is a fundamental research question that is, as illustrated by our results, deeply rooted in functional analysis.
Revised: Firstly, it is a fundamental research question that is, as illustrated by our results, deeply rooted in functional analysis.

Original: Secondly there is high application potential for completely unsupervised learning without any auxiliary variables, as the corresponding datasets do not require labelling or speciﬁc experimental settings.
Revised: Secondly there is high application potential for completely unsupervised learning without any auxiliary variables, as the corresponding datasets do not require labelling or speciﬁc experimental settings.

Original: Thirdly, it is very likely that the techniques can be generalised to the auxiliary variable case.
Revised: Thirdly, it is very likely that the techniques can be generalized to the auxiliary variable case.

Original: Another important open problem is assessing the type of constraints on ground truth mechanisms, encoded by function classes, that are relevant for real world data.
Revised: Another important open problem is assessing the type of constraints on ground truth mechanisms, encoded by function classes, that are relevant for real world data.

Original: It is plausible that those mechanisms are typically much more regular than generic nonlinear functions.
Revised: It is plausible that those mechanisms are typically much more regular than generic nonlinear functions.

Original: Recently, Gresele et al.
Revised: Recently, Gresele et al.

Original: [16] suggested, based on arguments from the causality literature that F OCT is a natural class for representation learning (and our results show it also has favourable theoretical properties), but this will require experimental conﬁrmation on real world data.
Revised: [16] suggested, based on arguments from the causality literature that F OCT is a natural class for representation learning (and our results show it also has favourable theoretical properties), but this will require experimental conﬁrmation on real world data.

Original: Finally, a central question from a machine learning perspective is the ability to design learning algorithms that can train LVMs with identiﬁable function class constraints.
Revised: Finally, a central question from a machine learning perspective is the ability to design learning algorithms that can train LVMs with identiﬁable function class constraints.

Original: Interestingly, Gresele et al.
Revised: Interestingly, Gresele et al.

Original: [16] showed that OCT maps can be learnt using a closed from regularized likelihood loss, thereby providing, supported by our result, a full-ﬂedged identiﬁable nonlinear ICA framework.
Revised: [16] showed that OCT maps can be learnt using a closed from regularized likelihood loss, thereby providing, supported by our result, a full-ﬂedged identiﬁable nonlinear ICA framework.

Original: Acknowledgments and Disclosure of Funding
Revised: Acknowledgments and Disclosure of Funding

Original: We thank Luigi Gresele and Julius von Kügelgen for helpful discussions and feedback on an earlier version of this paper.
Revised: We thank Luigi Gresele and Julius von Kügelgen for helpful discussions and feedback on an earlier version of this paper.

Original: We thank the anonymous reviewers whose comments helped us to make the paper more accessible.
Revised: We thank the anonymous reviewers whose comments helped us to make the paper more accessible.

Original: This work was in part supported by the German Federal Ministry of
Revised: This work was in part supported by the German Federal Ministry of

Original: Education and Research (BMBF) through the Tübingen AI Center (FKZ: 01IS18039B) and
Revised: Education and Research (BMBF) through the Tübingen AI Center (FKZ: 01IS18039B) and

Original: the
Revised: the

Original: Machine Learning Cluster of Excellence, EXC number 2064/1 – Project number 390727645.
Revised: Machine Learning Cluster of Excellence, EXC number 2064/1 – Project number 390727645.

Original: A Mathematical background
Revised: A Mathematical background

Original: For the convenience of the reader we collect some mathematical deﬁnitions and notations.
Revised: For the convenience of the reader we collect some mathematical deﬁnitions and notations.

Original: All deﬁnitions can be found in standard textbooks.
Revised: All deﬁnitions can be found in standard textbooks.

Original: Pushforward of measures.
Revised: Pushforward of measures.

Original: For a measure µ on a (measurable) space X and a measurable map, f : X → Y the pushforward measure f ∗ µ is deﬁned by ( f ∗ µ )( A ) =
Revised: For a measure µ on a (measurable) space X and a measurable map, f : X → Y the pushforward measure f ∗ µ is deﬁned by ( f ∗ µ )( A ) =

Original: µ ( f − 1 ( A )) for any measurable set A ⊂ Y .
Revised: µ ( f − 1 ( A )) for any measurable set A ⊂ Y .

Original: Here f − 1 ( A ) = { x ∈ X | f ( x ) ∈ A } denotes the preimage of A under f .
Revised: Here f − 1 ( A ) = { x ∈ X | f ( x ) ∈ A } denotes the preimage of A under f .

Original: Sometimes the pushforward measure is denoted by f # µ .
Revised: Sometimes the pushforward measure is denoted by f # µ .

Original: Note that no further restrictions on f are necessary, in particular f does not need to be invertible.
Revised: Note that no further restrictions on f are necessary, in particular f does not need to be invertible.

Original: One important property that we will use frequently is the relation ( g ◦ f ) ∗ µ = f ∗ ( g ∗ µ ) .
Revised: One important property that we will use frequently is the relation ( g ◦ f ) ∗ µ = f ∗ ( g ∗ µ ) .

Original: Note that if s ∼ P , i.e., s has distribution P then f ( s ) ∼ f ∗ P .
Revised: Note that if s ∼ P , i.e., s has distribution P then f ( s ) ∼ f ∗ P .

Original: Indeed, this is obvious as P ( f ( s ) ∈
Revised: Indeed, this is obvious as P ( f ( s ) ∈

Original: A ) =
Revised: A ) =

Original: P ( s ∈ f − 1 ( A ))
Revised: P ( s ∈ f − 1 ( A ))

Original: =
Revised: =

Original: P ( f − 1 ( A )) .
Revised: P ( f − 1 ( A )) .

Original: In the context of ICA it is convenient to mostly talk about distributions and push-forwards as we typically never observe pairs ( s, f ( s )) .
Revised: In the context of ICA it is convenient to mostly talk about distributions and push-forwards as we typically never observe pairs ( s, f ( s )) .

Original: For later usage we also recall the transformation formula for random variables.
Revised: For later usage we also recall the transformation formula for random variables.

Original: If f ∈ C 1 ( R n , R n ) is an invertible diffeomorphism and Q = f ∗ P where P and Q have density p and q respectively then
Revised: If f ∈ C 1 ( R n , R n ) is an invertible diffeomorphism and Q = f ∗ P where P and Q have density p and q respectively then

Original: [Equation]
Revised: [Equation]

Original: Note that the potentially more familiar version for random variables reads as follows.
Revised: Note that the potentially more familiar version for random variables reads as follows.

Original: Let X and Y be random variables satisfying Y =
Revised: Let X and Y be random variables satisfying Y =

Original: f ( X ) , then their densities are related by
Revised: f ( X ) , then their densities are related by

Original: [Equation]
Revised: [Equation]

Original: Diffeomorphisms.
Revised: Diffeomorphisms.

Original: Let U, V ⊂ R d .
Revised: Let U, V ⊂ R d .

Original: A diffeomorphism from U to V is a bijective map f : U → V such that f ∈ C 1 ( U, V ) and f − 1 ∈ C 1 ( V, U ) .
Revised: A diffeomorphism from U to V is a bijective map f : U → V such that f ∈ C 1 ( U, V ) and f − 1 ∈ C 1 ( V, U ) .

Original: Note that it is not sufﬁcient to assume that f is bijective and in C 1 ( U, V ) , the classic counterexample being f ( x ) = x 3 .
Revised: Note that it is not sufﬁcient to assume that f is bijective and in C 1 ( U, V ) , the classic counterexample being f ( x ) = x 3 .

Original: A sufﬁcient condition is that Df ( x ) is an invertible matrix for all x .
Revised: A sufﬁcient condition is that Df ( x ) is an invertible matrix for all x .

Original: Sometimes we loosely speak about diffeomorphisms f : U → R d which should be understood as f being a diffeomorphism on its image f ( U ) .
Revised: Sometimes we loosely speak about diffeomorphisms f : U → R d which should be understood as f being a diffeomorphism on its image f ( U ) .

Original: Vector ﬁelds and ﬂows Vector ﬁelds can be introduced nicely in the language of differential geometry.
Revised: Vector ﬁelds and ﬂows Vector ﬁelds can be introduced nicely in the language of differential geometry.

Original: However, we think that for the purpose of this paper it is more appropriate to give a more down to earth discussion focused on R d .
Revised: However, we think that for the purpose of this paper it is more appropriate to give a more down to earth discussion focused on R d .

Original: We refer to [34] for a general introduction.
Revised: We refer to [34] for a general introduction.

Original: A vector ﬁeld is a map
Revised: A vector ﬁeld is a map

Original: X : R d → R d .
Revised: X : R d → R d .

Original: Our reason to consider vector ﬁelds is that they can be used to describe smooth transformations of R d .
Revised: Our reason to consider vector ﬁelds is that they can be used to describe smooth transformations of R d .

Original: We will assume that X is Lipschitz continuous.
Revised: We will assume that X is Lipschitz continuous.

Original: We deﬁne the ﬂow of a vector ﬁeld as a map Φ : R × R d → R d such that
Revised: We deﬁne the ﬂow of a vector ﬁeld as a map Φ : R × R d → R d such that

Original: [Equation]
Revised: [Equation]

Original: Let us remark concerning the notation that it is convenient to put the t argument below, i.e., we write Φ t ( x ) =
Revised: Let us remark concerning the notation that it is convenient to put the t argument below, i.e., we write Φ t ( x ) =

Original: Φ( t, x ) .
Revised: Φ( t, x ) .

Original: Moreover, when applying differential operators D they will by default only act on the spatial variable x , i.e., D Φ t ( x ) denotes the derivative of the function x
Revised: Moreover, when applying differential operators D they will by default only act on the spatial variable x , i.e., D Φ t ( x ) denotes the derivative of the function x

Original: → Φ t ( x ) for a ﬁxed t .
Revised: → Φ t ( x ) for a ﬁxed t .

Original: Note that the solutions of differential equations can exhibit blow-up so Φ t ( x ) might not be deﬁned for all times t .
Revised: Note that the solutions of differential equations can exhibit blow-up so Φ t ( x ) might not be deﬁned for all times t .

Original: However, when we assume that X is bounded the ﬂow exists globally.
Revised: However, when we assume that X is bounded the ﬂow exists globally.

Original: It can be shown that if X is k times differentiable then so is Φ t and one can conclude that then Φ t is a diffeomorphism ( Φ is bijective by the uniqueness of ordinary differential equations).
Revised: It can be shown that if X is k times differentiable then so is Φ t and one can conclude that then Φ t is a diffeomorphism ( Φ is bijective by the uniqueness of ordinary differential equations).

Original: We will also consider time-dependent vector ﬁelds X : ( − T, T )
Revised: We will also consider time-dependent vector ﬁelds X : ( − T, T )

Original: × R d → R d where the ﬂows can be deﬁned similarly, replacing X by X t .
Revised: × R d → R d where the ﬂows can be deﬁned similarly, replacing X by X t .

Original: We are particularly interested in the action of ﬂows on probability measures, i.e., we consider the measures (Φ t ) ∗ P for some initial measure P .
Revised: We are particularly interested in the action of ﬂows on probability measures, i.e., we consider the measures (Φ t ) ∗ P for some initial measure P .

Original: It can be shown that if the density of P is p then the density p t of (Φ t ) ∗ P satisﬁes the continuity equation
Revised: It can be shown that if the density of P is p then the density p t of (Φ t ) ∗ P satisﬁes the continuity equation

Original: [Equation]
Revised: [Equation]

Original: Here Div denotes the divergence which is deﬁned by Div f = (cid:80) i ∂ i f i =
Revised: Here Div denotes the divergence which is deﬁned by Div f = (cid:80) i ∂ i f i =

Original: Tr Df .
Revised: Tr Df .

Original: One important consequence is that the ﬂow of the vector ﬁeld preserves the measure, i.e., (Φ t ) ∗ P iff Div( pX t ) =
Revised: One important consequence is that the ﬂow of the vector ﬁeld preserves the measure, i.e., (Φ t ) ∗ P iff Div( pX t ) =

Original: for all t .
Revised: for all t .

Original: Moreover, the standard Lebesgue measure in R d is preserved if Div( X t ) = 0 , i.e., divergence free vector ﬁelds generate volume preserving ﬂows and vice versa.
Revised: Moreover, the standard Lebesgue measure in R d is preserved if Div( X t ) = 0 , i.e., divergence free vector ﬁelds generate volume preserving ﬂows and vice versa.

Original: Additional notation.
Revised: Additional notation.

Original: We at some places use the notation [ n ] = { 1 , . . . , n } .
Revised: We at some places use the notation [ n ] = { 1 , . . . , n } .

Original: We also use the O notation.
Revised: We also use the O notation.

Original: Recall that f ( x ) =
Revised: Recall that f ( x ) =

Original: O ( g ( x )) as x → ∞ means that there are constants x 0 and C > 

Revised: O ( g ( x )) as x → ∞ means that there are constants x 0 and C >

Original: such that f ( x ) ≤ Cg ( x ) for x ≥ x 0 .
Revised: such that f ( x ) ≤ Cg ( x ) for x ≥ x 0 .

Original: Recall that we introduced the notation C d = (0 , 1) c in the main part and we denoted by ν the uniform measure on C d .
Revised: Recall that we introduced the notation C d = (0 , 1) c in the main part and we denoted by ν the uniform measure on C d .

Original: We write iff as a shorthand for ’if and only if’.
Revised: We write ’iff’ as a shorthand for ’if and only if’.

Original: B Function class deﬁnitions for general manifolds
Revised: B Function class deﬁnitions for general manifolds

Original: Here we extend the deﬁnition of the function classes to Riemannian manifolds.
Revised: Here we extend the deﬁnition of the function classes to Riemannian manifolds.

Original: Riemannian manifolds are manifolds M equipped with a metric g .
Revised: Riemannian manifolds are manifolds M equipped with a metric g .

Original: For complete deﬁnitions we again refer to [34].
Revised: For complete deﬁnitions we again refer to [34].

Original: We denote the differential of a map f : M → N at x ∈ M by (d f ) s : T s M → T f ( s ) N .
Revised: We denote the differential of a map f : M → N at x ∈ M by (d f ) s : T s M → T f ( s ) N .

Original: As before, we use the notation Df ( s
Revised: As before, we use the notation Df ( s

Original: ) ∈ R d × d (cid:48) for the usual derivative of a map f : R d → R d (cid:48) .
Revised: ) ∈ R d × d (cid:48) for the usual derivative of a map f : R d → R d (cid:48) .

Original: The matrix Df ( s ) is the representation of (d f ) s in the standard basis.
Revised: The matrix Df ( s ) is the representation of (d f ) s in the standard basis.

Original: A smooth map f : M → N between Riemannian manifolds ( M, g ) and ( N, h ) is conformal if there is a function λ : M → R + such that
Revised: A smooth map f : M → N between Riemannian manifolds ( M, g ) and ( N, h ) is conformal if there is a function λ : M → R + such that

Original: [Equation]
Revised: [Equation]

Original:  where f ∗ h denotes the pullback metric.
Revised: 17) where f ∗ h denotes the pullback metric.

Original: This means that for X, Y ∈ T x M
Revised: This means that for X, Y ∈ T x M

Original: [Equation]
Revised: [Equation]

Revised: g

Original: [Equation]
Revised: [Equation]

Revised: X

Revised: [Equation]

Revised: [Equation]

Original: Moreover, we observe that the adjoint ( df ) ∗ s satisﬁes by deﬁnition
Revised: Moreover, we observe that the adjoint ( df ) ∗ s satisﬁes by deﬁnition

Original: [Equation]
Revised: [Equation]

Original: We conclude that
Revised: We conclude that

Original: [Equation]
Revised: [Equation]

Original: In the case that M = N = R d both equipped with the standard metric the deﬁnition is seen to be equivalent to Deﬁnition 2 given in the main part.
Revised: In the case that M = N = R d both equipped with the standard metric the deﬁnition is seen to be equivalent to Deﬁnition 2 given in the main part.

Original: When N is a d -dimensional submanifold of R d (cid:48) with d (cid:48) >
Revised: When N is a d -dimensional submanifold of R d (cid:48) with d (cid:48) >

Original: d then we obtain the pointwise condition
Revised: d then we obtain the pointwise condition

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: i.e., Df ( s ) has orthogonal columns with equal norm.
Revised: i.e., Df ( s ) has orthogonal columns with equal norm.

Original: Note that the concatenation of conformal maps is conformal and the inverse of conformal diffeomorphisms is again conformal.
Revised: Note that the concatenation of conformal maps is conformal and the inverse of conformal diffeomorphisms is again conformal.

Original: Next, we extend the deﬁnition of orthogonal coordinate transformations.
Revised: Next, we extend the deﬁnition of orthogonal coordinate transformations.

Original: We remark that orthogonal coordinates are chart dependent so no coordinate free deﬁnition can be given.
Revised: We remark that orthogonal coordinates are chart dependent so no coordinate free deﬁnition can be given.

Original: Thus we focus on the case where the domain of the function is R d which we equip with the standard metric and we consider the standard orthogonal coordinate vector ﬁelds which we denote by e i .
Revised: Thus we focus on the case where the domain of the function is R d which we equip with the standard metric and we consider the standard orthogonal coordinate vector ﬁelds which we denote by e i .

Original: Then we call a smooth map f : R d → M for a Riemannian manifold ( M, g ) an orthogonal coordinate transformation if for i (cid:54) =
Revised: Then we call a smooth map f : R d → M for a Riemannian manifold ( M, g ) an orthogonal coordinate transformation if for i (cid:54) =

Original: j and all s ∈ R d
Revised: j and all s ∈ R d

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Again, this condition can be equivalently written as
Revised: Again, this condition can be equivalently written as

Original: [Equation]
Revised: [Equation]

Original: 23) for i (cid:54) =
Revised: 23) for i (cid:54) =

Original: j .
Revised: j .

Original: If M is a submanifold of R d (cid:48) with the metric induced from the standard Euclidean metric we obtain the natural generalization of Deﬁnition 3, i.e.,[Equation]
Revised: If M is a submanifold of R d (cid:48) with the metric induced from the standard Euclidean metric we obtain the natural generalization of Deﬁnition 3, i.e.,

Original: [Equation]
Revised: [Equation]

Original: An important remark is that orthogonal coordinates do not exist for all manifolds as there are obstructions.
Revised: An important remark is that orthogonal coordinates do not exist for all manifolds as there are obstructions.

Original: Manifolds with this property are called locally diagonalizable.
Revised: Manifolds with this property are called locally diagonalizable.

Original: This is closely related to the representation capability of the function class.
Revised: This is closely related to the representation capability of the function class.

Original: C Measure preserving transformations and spurious solutions
Revised: C Measure preserving transformations and spurious solutions

Original: In this section we review the construction of spurious solutions for the ICA problem.
Revised: In this section we review the construction of spurious solutions for the ICA problem.

Original: We assume that we consider ICA in the class ( F , P ) where F is a function class and P a class of probability measures.
Revised: We assume that we consider ICA in the class ( F , P ) where F is a function class and P a class of probability measures.

Original: We are interested in understanding the set of spurious solutions for a pair ( f, P ) , i.e., the set of pairs ( f (cid:48) , P (cid:48) )
Revised: We are interested in understanding the set of spurious solutions for a pair ( f, P ) , i.e., the set of pairs ( f (cid:48) , P (cid:48) )

Original: ∈ F × P such that f ∗ P = f
Revised: ∈ F × P such that f ∗ P = f

Original: (cid:48)∗ P (cid:48) .
Revised: (cid:48)∗ P (cid:48) .

Original: We now note that if we can construct h such that h ∗ P (cid:48)
Revised: We now note that if we can construct h such that h ∗ P (cid:48)

Original: =
Revised: =

Original: P and set f (cid:48)
Revised: P and set f (cid:48)

Original: =
Revised: =

Original: f ◦ h
Revised: f ◦ h

Original: we have f ∗ P = f
Revised: we have f ∗ P = f

Original: (cid:48)∗ P (cid:48) .
Revised: (cid:48)∗ P (cid:48) .

Original: Next we deﬁne the subset of right composable functions
Revised: Next we deﬁne the subset of right composable functions

Original: [Equation]
Revised: [Equation]

Original: and similarly the subset of left-composable functions
Revised: and similarly the subset of left-composable functions

Original: [Equation]
Revised: [Equation]

Original: We remark that if F is a group then obviously F L = F R = F and the problem reduces to ﬁnding measure preserving transformations in F .
Revised: We remark that if F is a group then obviously F L = F R = F and the problem reduces to ﬁnding measure preserving transformations in F .

Original: The classes F lin , F nonlinear , F conf , and F volume are all groups.
Revised: The classes F lin , F nonlinear , F conf , and F volume are all groups.

Original: We will comment on F OCT below.
Revised: We will comment on F OCT below.

Original: We note that if h ∈ F R satisﬁes h ∗ P (cid:48) =
Revised: We note that if h ∈ F R satisﬁes h ∗ P (cid:48) =

Original: P then f (cid:48)
Revised: P then f (cid:48)

Original: =
Revised: =

Original: f ◦ h ∈ F and f (cid:48)∗ P (cid:48)
Revised: f ◦ h ∈ F and f (cid:48)∗ P (cid:48)

Original: =
Revised: =

Original: f ∗ P so this gives us a spurious solution.
Revised: f ∗ P so this gives us a spurious solution.

Original: A speciﬁc case is given by P = P (cid:48) in which case we are looking for measure preserving transformations (MPTs) h ∈ F R .
Revised: A speciﬁc case is given by P = P (cid:48) in which case we are looking for measure preserving transformations (MPTs) h ∈ F R .

Original: Note that such h for a certain P allows us to construct spurious solutions for all pairs ( f, P ) with arbitrary f .
Revised: Note that such h for a certain P allows us to construct spurious solutions for all pairs ( f, P ) with arbitrary f .

Original: This implies that if ( F , P ) is identiﬁable with respect to S then any h as above satisﬁes h ∈ S (if P (cid:48) has full support, and otherwise there is a h (cid:48)
Revised: This implies that if ( F , P ) is identiﬁable with respect to S then any h as above satisﬁes h ∈ S (if P (cid:48) has full support, and otherwise there is a h (cid:48)

Original: ∈ S such that h and h (cid:48) agree in the support of P (cid:48) ).
Revised: ∈ S such that h and h (cid:48) agree in the support of P (cid:48) ).

Original: We consider an example.
Revised: We consider an example.

Original: Let P be the distribution of the standard Gaussian and h
Revised: Let P be the distribution of the standard Gaussian and h

Original: =
Revised: =

Original: R for some
Revised: R for some

Original: R ∈ O( d ) .
Revised: R ∈ O( d ) .

Original: Then h ∈ F lin and h ∗ P = P as the standard Gaussian is invariant under rotations.
Revised: Then h ∈ F lin and h ∗ P = P as the standard Gaussian is invariant under rotations.

Original: However, such a linear measure preserving transformation does not exist for other P ∈ M 1 ( R ) ⊗ d .
Revised: However, such a linear measure preserving transformation does not exist for other P ∈ M 1 ( R ) ⊗ d .

Original: This is the reason that Gaussian distributions are excluded for linear ICA.
Revised: This is the reason that Gaussian distributions are excluded for linear ICA.

Original: Next we observe (as we discussed in the main part) that if the function class F is stable by right composition with arbitrary component-wise transformation we can turn any admissible latent distribution into another one.
Revised: Next we observe (as we discussed in the main part) that if the function class F is stable by right composition with arbitrary component-wise transformation we can turn any admissible latent distribution into another one.

Original: This means we can ﬁx a reference measure (we will usually use ν ) and then we can ﬁnd (at least under suitable regularity assumptions) for P ∈ M 1 ( R ) ⊗ d maps h P → ν , h ν → P ∈ F R such that h P → ν ∗
Revised: This means we can ﬁx a reference measure (we will usually use ν ) and then we can ﬁnd (at least under suitable regularity assumptions) for P ∈ M 1 ( R ) ⊗ d maps h P → ν , h ν → P ∈ F R such that h P → ν ∗

Original: P = ν , h ν → P ∗ ν = P .
Revised: P = ν , h ν → P ∗ ν = P .

Original: If we can ﬁnd an MPT h ∈ F R such that h ∗ ν = ν mixing the coordinates we can then ﬁnd spurious solutions for any pair ( f, P ) by considering f (cid:48)
Revised: If we can ﬁnd an MPT h ∈ F R such that h ∗ ν = ν mixing the coordinates we can then ﬁnd spurious solutions for any pair ( f, P ) by considering f (cid:48)

Original: =
Revised: =

Original: f ◦ h ν
Revised: f ◦ h ν

Original: → P ◦ h ◦ h P → ν because then f (cid:48)∗ P = f ∗ P and f (cid:48) ∈ F by deﬁnition of F R .
Revised: → P ◦ h ◦ h P → ν because then f (cid:48)∗ P = f ∗ P and f (cid:48) ∈ F by deﬁnition of F R .

Original: For the class F nonlinear such MPTs exist, we gave one construction in Lemma 1.
Revised: For the class F nonlinear such MPTs exist, we gave one construction in Lemma 1.

Original: In Appendix F we will sketch the proof of this result and discuss another construction based on divergence free vector ﬁelds.
Revised: In Appendix F we will sketch the proof of this result and discuss another construction based on divergence free vector ﬁelds.

Original: For the class F volume we cannot arbitrarily transform the input distribution.
Revised: For the class F volume we cannot arbitrarily transform the input distribution.

Original: However, we can still ﬁnd coordinate mixing MPTs for every P (with smooth density).
Revised: However, we can still ﬁnd coordinate mixing MPTs for every P (with smooth density).

Original: This will be proved in
Revised: This will be proved in

Original: Appendix G. These constructions rule out any (meaningful) identiﬁability result for F nonlinear and
Revised: Appendix G. These constructions rule out any (meaningful) identiﬁability result for F nonlinear and

Original: F volume .
Revised: F volume .

Original: There is also a slightly different approach to construct spurious solutions for a pair ( f, P ) .
Revised: There is also a slightly different approach to construct spurious solutions for a pair ( f, P ) .

Original: For any MPT h ∈ F L such that h ∗ f ∗ P = f ∗ P , i.e., transformations that preserve the observational distribution Q = f ∗ P , the function h ◦ f deﬁnes a spurious solution.
Revised: For any MPT h ∈ F L such that h ∗ f ∗ P = f ∗ P , i.e., transformations that preserve the observational distribution Q = f ∗ P , the function h ◦ f deﬁnes a spurious solution.

Original: Note that this gives spurious solutions for all pairs ( f, P ) such that f ∗ P follows the ﬁxed distribution Q .
Revised: Note that this gives spurious solutions for all pairs ( f, P ) such that f ∗ P follows the ﬁxed distribution Q .

Original: This approach will allow us to construct spurious solutions in F OCT for some ﬁne-tuned pairs ( f, P ) with f ∈ F OCT (see the proof of Proposition 1 in Appendix F).
Revised: This approach will allow us to construct spurious solutions in F OCT for some ﬁne-tuned pairs ( f, P ) with f ∈ F OCT (see the proof of Proposition 1 in Appendix F).

Original: Note that because of their ﬁne-tuning, such spurious solutions can arguably be considered pathological cases instead of key non-identiﬁability issues.
Revised: Note that because of their ﬁne-tuning, such spurious solutions can arguably be considered pathological cases instead of key non-identiﬁability issues.

Original: This is inline with how such solutions a considered in the case of linear ICA.
Revised: This is inline with how such solutions a considered in the case of linear ICA.

Original: Let us ﬁnally have a closer look at F OCT .
Revised: Let us ﬁnally have a closer look at F OCT .

Original: It is quite straightforward to see that
Revised: It is quite straightforward to see that

Original: [Equation]
Revised: [Equation]

Original: In particular, all rotations are contained in F L conf .
Revised: In particular, all rotations are contained in F L conf .

Original: More interestingly, we have
Revised: More interestingly, we have

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: some P ( x ) ∈ Perm( d )
Revised: some P ( x ) ∈ Perm( d )

Original: [Equation]
Revised: [Equation]

Original: ( x )
Revised: ( x )

Original: [Equation]
Revised: [Equation]

Original: ( d )
Revised: ( d )

Original: [Equation]
Revised: [Equation]

Original: P ◦ h where P ∈ Perm ± ( d ) and h : R d → R d with h ( x )
Revised: P ◦ h where P ∈ Perm ± ( d ) and h : R d → R d with h ( x )

Original: [Equation]
Revised: [Equation]

Original: ( h
Revised: ( h

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: , h d (
Revised: , h d (

Original: [Equation]
Revised: [Equation]

Original: d )) (cid:62) for some h i ∈ C 1 ( R , R ) with h (cid:48) i > 0 } .
Revised: d )) (cid:62) for some h i ∈ C 1 ( R , R ) with h (cid:48) i > 0 } .

Original: [Equation]
Revised: [Equation]

Original: The ﬁrst step can be seen using the chain rule and the deﬁnition of F OCT .
Revised: The ﬁrst step can be seen using the chain rule and the deﬁnition of F OCT .

Original: The second step follows from the fact that the permutation P is necessarily constant for such a function.
Revised: The second step follows from the fact that the permutation P is necessarily constant for such a function.

Original: This again recovers the fact that OCTs can only be identiﬁed up to permutations and coordinate-wise reparametrisations.
Revised: This again recovers the fact that OCTs can only be identiﬁed up to permutations and coordinate-wise reparametrisations.

Original: However, we also conclude that all h ∈ F R OCT act coordinate-wise (up to a permutation), i.e., do not prevent BSS.
Revised: However, we also conclude that all h ∈ F R OCT act coordinate-wise (up to a permutation), i.e., do not prevent BSS.

Original: This shows that for F OCT no completely generic spurious solution based on a single
Revised: This shows that for F OCT no completely generic spurious solution based on a single

Original: MPT mixing the coordinates exists.
Revised: MPT mixing the coordinates exists.

Original: This is different from F nonlinear .
Revised: This is different from F nonlinear .

Original: We emphasize that this does not rule out the existence of a ﬁne-tuned spurious solution for every (or most) pairs ( f, P ) because then we only need to ﬁnd h such that h ∗ P = P and f ◦ h ∈ F OCT .
Revised: We emphasize that this does not rule out the existence of a ﬁne-tuned spurious solution for every (or most) pairs ( f, P ) because then we only need to ﬁnd h such that h ∗ P = P and f ◦ h ∈ F OCT .

Original: The relation f ◦ h ∈ F OCT of course holds for h ∈ F R OCT but there will, in general, be many more h for a ﬁxed f .
Revised: The relation f ◦ h ∈ F OCT of course holds for h ∈ F R OCT but there will, in general, be many more h for a ﬁxed f .

Original: D Proof of Theorem
Revised: D Proof of Theorem

Original: We here, for completeness, give a proof of Theorem 1.
Revised: We here, for completeness, give a proof of Theorem 1.

Original: While this result is well known we think that it makes sense to include a condensed proof because it contains many of the key steps of the more involved proof for conformal maps in the next section and it is not as well known as the proof based on Darmois-Skitovich Theorem which does not generalise to nonlinear functions.
Revised: While this result is well known we think that it makes sense to include a condensed proof because it contains many of the key steps of the more involved proof for conformal maps in the next section and it is not as well known as the proof based on Darmois-Skitovich Theorem which does not generalize to nonlinear functions.

Original: Proof of Theorem 1.
Revised: Proof of Theorem 1.

Original: We assume that x D =
Revised: We assume that x D =

Original: As D = A (cid:48) s (cid:48) .
Revised: As D = A (cid:48) s (cid:48) .

Original: We ﬁrst assume that the densities p : R n → R and q : R n → R of s and s (cid:48) are C 2 functions and p ( x ) > 0 for all x ∈ R d .
Revised: We ﬁrst assume that the densities p : R n → R and q : R n → R of s and s (cid:48) are C 2 functions and p ( x ) > 0 for all x ∈ R d .

Original: We denote their distributions by P and Q .
Revised: We denote their distributions by P and Q .

Original: By independence of the components we can write p ( x ) =
Revised: By independence of the components we can write p ( x ) =

Original: (cid:81)
Revised: (cid:81)

Original: i p i ( x i ) , q ( x ) =
Revised: i p i ( x i ) , q ( x ) =

Original: (cid:81)
Revised: (cid:81)

Original: i q i ( x i ) for some C 2 functions p i and q i .
Revised: i q i ( x i ) for some C 2 functions p i and q i .

Original: By assumption we conclude that (( A (cid:48) ) − 1 A ) ∗ P =
Revised: By assumption we conclude that (( A (cid:48) ) − 1 A ) ∗ P =

Original: Q .
Revised: Q .

Original: We denote B =
Revised: We denote B =

Original: (( A (cid:48) )
Revised: (( A (cid:48) )

Original: − 1 A ) − 1 =
Revised: − 1 A ) − 1 =

Original: A − 1 A (cid:48) so that B − 1 ∗ P = Q and the transformation formula for densities implies that
Revised: A − 1 A (cid:48) so that B − 1 ∗ P = Q and the transformation formula for densities implies that

Original: [Equation]
Revised: [Equation]

Original: The main idea of the proof is to use the observation that for i (cid:54) =
Revised: The main idea of the proof is to use the observation that for i (cid:54) =

Original: j and all y such that q ( y ) (cid:54) =
Revised: j and all y such that q ( y ) (cid:54) =

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: i.e., mixed second derivatives of the log density vanish.
Revised: i.e., mixed second derivatives of the log density vanish.

Original: We plug (29) into this equation and get
Revised: We plug (29) into this equation and get

Original: [Equation]
Revised: [Equation]

Original: We now denote x =
Revised: We now denote x =

Original: By .
Revised: By .

Original: Then we get (using that B is invertible) that for all x such that p ( x ) (cid:54) =
Revised: Then we get (using that B is invertible) that for all x such that p ( x ) (cid:54) =

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Varying one x k individually we conclude that each summand is constant which implies that either
Revised: Varying one x k individually we conclude that each summand is constant which implies that either

Original: B kj B ki = 0 for all i (cid:54) =
Revised: B kj B ki = 0 for all i (cid:54) =

Original: j or ln( p k ) (cid:48)(cid:48) ( x k ) is constant.
Revised: j or ln( p k ) (cid:48)(cid:48) ( x k ) is constant.

Original: It is straightforward to see that the only probability distribution with ln( p ) (cid:48)(cid:48)
Revised: It is straightforward to see that the only probability distribution with ln( p ) (cid:48)(cid:48)

Original: = κ for some constant κ are Gaussian distributions.
Revised: = κ for some constant κ are Gaussian distributions.

Original: Indeed, note that ln( p ) (cid:48)(cid:48)
Revised: Indeed, note that ln( p ) (cid:48)(cid:48)

Original: = κ for some constant implies that p ( x ) = exp( κx 2 / 2 + c 1 x + c 2 ) .
Revised: = κ for some constant implies that p ( x ) = exp( κx 2 / 2 + c 1 x + c 2 ) .

Original: If p is the density of a probability distribution and in particular integrable we see that this implies κ < 0 and p is a
Revised: If p is the density of a probability distribution and in particular integrable we see that this implies κ < 0 and p is a

Original: Gaussian density with κ = − σ − 2 .
Revised: Gaussian density with κ = − σ − 2 .

Original: Note that by assumption at most one component of P is Gaussian, w.l.o.g., k = 1 .
Revised: Note that by assumption at most one component of P is Gaussian, w.l.o.g., k = 1 .

Original: As p k is not a Gaussian for k > 1 and thus ln( p k ) (cid:48)(cid:48) not constant we conclude
Revised: As p k is not a Gaussian for k > 1 and thus ln( p k ) (cid:48)(cid:48) not constant we conclude

Original: B kj B ki =(34) for i (cid:54) =
Revised: B kj B ki =(34) for i (cid:54) =

Original: j and all k > 1 .
Revised: j and all k > 1 .

Original: Plugging this into (33) we obtain
Revised: Plugging this into (33) we obtain

Original: [Equation]
Revised: [Equation]

Original: Note that if p 1 is a Gaussian density then ln( p 1 ) (cid:48)(cid:48) =
Revised: Note that if p 1 is a Gaussian density then ln( p 1 ) (cid:48)(cid:48) =

Original: − σ − 2 (cid:54) = 

Revised: − σ − 2 (cid:54) =

Original: so we conclude that in any case
Revised: so we conclude that in any case

Original: B 

Revised: B

Original: i B 1 j = 

Revised: i B 1 j =

Original: for i (cid:54) =
Revised: for i (cid:54) =

Original: j , i.e., (34) holds as well for k = 1 .
Revised: j , i.e., (34) holds as well for k = 1 .

Original: In other words, at most one entry of every row of B is non-zero.
Revised: In other words, at most one entry of every row of B is non-zero.

Original: This implies that B = P Λ for some
Revised: This implies that B = P Λ for some

Original: P ∈ Perm( d ) and Λ ∈ Diag( d ) .
Revised: P ∈ Perm( d ) and Λ ∈ Diag( d ) .

Original: This was to be shown.
Revised: This was to be shown.

Original: Let us clarify what happens when there is more than one Gaussian component.
Revised: Let us clarify what happens when there is more than one Gaussian component.

Original: In this case there might be multiple constant non-zero terms in (33) whose contributions can cancel and we cannot conclude that (34) holds for all k .
Revised: In this case there might be multiple constant non-zero terms in (33) whose contributions can cancel and we cannot conclude that (34) holds for all k .

Original: This recovers the well known non-uniqueness for Gaussian variables.
Revised: This recovers the well known non-uniqueness for Gaussian variables.

Original: It remains to extend the result to distributions whose density is not twice differentiable.
Revised: It remains to extend the result to distributions whose density is not twice differentiable.

Original: By standardizing s and s (cid:48) we can assume that B ∈ O( n ) .
Revised: By standardizing s and s (cid:48) we can assume that B ∈ O( n ) .

Original: Indeed, when the covariances of s and s (cid:48) satisfy Σ s =
Revised: Indeed, when the covariances of s and s (cid:48) satisfy Σ s =

Original: Σ s (cid:48)
Revised: Σ s (cid:48)

Original: =
Revised: =

Original: Id d then s =
Revised: Id d then s =

Original: Bs (cid:48) implies B (cid:62)
Revised: Bs (cid:48) implies B (cid:62)

Original: B = B T Σ s (cid:48) B = Σ s =
Revised: B = B T Σ s (cid:48) B = Σ s =

Original: Id .
Revised: Id .

Original: Then s (cid:48)
Revised: Then s (cid:48)

Original: D =
Revised: D =

Original: Bs implies that for an independent standard normal
Revised: Bs implies that for an independent standard normal

Original: [Equation]
Revised: [Equation]

Original: 36) where we used that standard normal variables are invariant under orthogonal maps.
Revised: 36) where we used that standard normal variables are invariant under orthogonal maps.

Original: Note that s + N ∈ P lin .
Revised: Note that s + N ∈ P lin .

Original: Indeed, ( s i , N i ) ⊥⊥ ( s j , N j ) implies ( s i + N i ) ⊥⊥ ( s j + N j ) for i (cid:54) =
Revised: Indeed, ( s i , N i ) ⊥⊥ ( s j , N j ) implies ( s i + N i ) ⊥⊥ ( s j + N j ) for i (cid:54) =

Original: j .
Revised: j .

Original: If s i + N i is
Revised: If s i + N i is

Original: Gaussian then s i is Gaussian (consider, e.g., the Fourier transform)
Revised: Gaussian then s i is Gaussian (consider, e.g., the Fourier transform)

Original: so s + N also has at most one
Revised: so s + N also has at most one

Original: Gaussian component.
Revised: Gaussian component.

Original: The density of s + N is smooth and pointwise positive, so we can apply the reasoning above to s + N and s (cid:48) + N and conclude B ∈ S lin .
Revised: The density of s + N is smooth and pointwise positive, so we can apply the reasoning above to s + N and s (cid:48) + N and conclude B ∈ S lin .

Original: E Proofs for the results on conformal maps
Revised: E Proofs for the results on conformal maps

Original: In this section we give the proofs for Section 3.
Revised: In this section we give the proofs for Section 3.

Original: First, we consider d > 2 and then the special case d = 2 .
Revised: First, we consider d > 2 and then the special case d = 2 .

Original: E.1 Proof of Theorem
Revised: E.1 Proof of Theorem

Original: The proof of Theorem 2 uses similar ideas as the proof for Theorem 1, however, the calculations are more involved.
Revised: The proof of Theorem 2 uses similar ideas as the proof for Theorem 1, however, the calculations are more involved.

Original: The key ingredient is the classiﬁcation of all conformal maps in Theorem 7.
Revised: The key ingredient is the classiﬁcation of all conformal maps in Theorem 7.

Original: From there we see that we already dealt with the linear case in Theorem 1, so it is sufﬁcient to focus on the case of nonlinear transformations.
Revised: From there we see that we already dealt with the linear case in Theorem 1, so it is sufﬁcient to focus on the case of nonlinear transformations.

Original: Recall that the Moebius transformations introduced in (12) are given by
Revised: Recall that the Moebius transformations introduced in (12) are given by

Original: [Equation]
Revised: [Equation]

Original: where b, a ∈ R d , α ∈ R , A ∈ O( d ) , and ε ∈ { 0 , 2 } .
Revised: where b, a ∈ R d , α ∈ R , A ∈ O( d ) , and ε ∈ { 0 , 2 } .

Original: In particular, the Theorem will be an easy consequence of the following lemma.
Revised: In particular, the Theorem will be an easy consequence of the following lemma.

Original: Lemma 2.
Revised: Lemma 2.

Original: Suppose g : R d → R d is a nonlinear Moebius transformation, i.e., a map as in (37) with ε = 2 .
Revised: Suppose g : R d → R d is a nonlinear Moebius transformation, i.e., a map as in (37) with ε = 2 .

Original: Let s , s (cid:48) be random variables whose distributions are in P conf .
Revised: Let s , s (cid:48) be random variables whose distributions are in P conf .

Original: Then s D (cid:54) =
Revised: Then s D (cid:54) =

Original: g ( s (cid:48) ) .
Revised: g ( s (cid:48) ) .

Original: Let us quickly show how it implies Theorem 2 before we prove this lemma.
Revised: Let us quickly show how it implies Theorem 2 before we prove this lemma.

Original: Proof of Theorem 2.
Revised: Proof of Theorem 2.

Original: We use the same notation as in the proof of Theorem 1.
Revised: We use the same notation as in the proof of Theorem 1.

Original: We denote the distribution of s and s (cid:48) by P and Q and we assume
Revised: We denote the distribution of s and s (cid:48) by P and Q and we assume

Original: x D = f ( s ) D = f (cid:48)
Revised: x D = f ( s ) D = f (cid:48)

Original: ( s (cid:48) ) with f, f (cid:48) :
Revised: ( s (cid:48) ) with f, f (cid:48) :

Original: R d → M conformal (see Appendix B for the deﬁnition).
Revised: R d → M conformal (see Appendix B for the deﬁnition).

Original: This implies that ( f (cid:48)− 1 f ) ∗ P = Q .
Revised: This implies that ( f (cid:48)− 1 f ) ∗ P = Q .

Original: We denote g =
Revised: We denote g =

Original: (( f (cid:48) ) − 1 f ) − 1 = f − 1 f (cid:48) so that P = g ∗ Q .
Revised: (( f (cid:48) ) − 1 f ) − 1 = f − 1 f (cid:48) so that P = g ∗ Q .

Original: Now we make the simple but important observation that g as a concatenation of a conformal map and the inverse of a conformal map is a conformal map from R d to R d .
Revised: Now we make the simple but important observation that g as a concatenation of a conformal map and the inverse of a conformal map is a conformal map from R d to R d .

Original: Thus, we can apply Liouville’s Theorem to g (recalled in Theorem 7) which implies that
Revised: Thus, we can apply Liouville’s Theorem to g (recalled in Theorem 7) which implies that

Original: [Equation]
Revised: [Equation]

Original: (38) where b, a ∈ R d , α ∈ R , A ∈ O( d ) , ε ∈ { 0 , 2 } .
Revised: (38) where b, a ∈ R d , α ∈ R , A ∈ O( d ) , ε ∈ { 0 , 2 } .

Original: Using Lemma 2 we conclude that ε
Revised: Using Lemma 2 we conclude that ε

Original: = 

Revised: =

Original: and g is linear.
Revised: and g is linear.

Original: Then we can apply Theorem 1 (using that P conf ⊂
Revised: Then we can apply Theorem 1 (using that P conf ⊂

Original: P lin )
Revised: P lin )

Original: and conclude that αA = P Λ for a permutation matrix and a diagonal matrix Λ .
Revised: and conclude that αA = P Λ for a permutation matrix and a diagonal matrix Λ .

Original: Since g is conformal we have that A is orthogonal and all eigenvalues have absolute value 1 which implies that Λ ii = ±
Revised: Since g is conformal we have that A is orthogonal and all eigenvalues have absolute value 1 which implies that Λ ii = ±

Original: α for 1 ≤ i ≤ d and thus
Revised: α for 1 ≤ i ≤ d and thus

Original: A ∈ M perm , ± ( R d × d ) .
Revised: A ∈ M perm , ± ( R d × d ) .

Original: To prove Lemma 2 we need one technical result that shows that local properties of the density p
Revised: To prove Lemma 2 we need one technical result that shows that local properties of the density p

Original: i (i.e., properties that hold for x i ∈
Revised: i (i.e., properties that hold for x i ∈

Original: O i for some non-empty open sets
Revised: O i for some non-empty open sets

Original: O i ) in fact hold for all x i (cid:54) = 0 .
Revised: O i ) in fact hold for all x i (cid:54) = 0 .

Original: This will be based on the nonlinearity of the map g combined with the factorisation p ( x ) =
Revised: This will be based on the nonlinearity of the map g combined with the factorization p ( x ) =

Original: (cid:81)
Revised: (cid:81)

Original: i p i ( x i ) of the densities.
Revised: i p i ( x i ) of the densities.

Original: An illustration can be found below in Figure 4.
Revised: An illustration can be found below in Figure 4.

Original: Lemma 3.
Revised: Lemma 3.

Original: Let O
Revised: Let O

Original: =
Revised: =

Original: O 1 × . . .
Revised: O 1 × . . .

Original: ×
Revised: ×

Original: O d ⊂ R d and U
Revised: O d ⊂ R d and U

Original: =
Revised: =

Original: U 1 × . . .
Revised: U 1 × . . .

Original: ×
Revised: ×

Original: U d ⊂ R d
Revised: U d ⊂ R d

Original: where O i and U i are non-empty open sets.
Revised: where O i and U i are non-empty open sets.

Original: Let g ( x ) =
Revised: Let g ( x ) =

Original: Ax/ | x | 2 for an orthogonal matrix A .
Revised: Ax/ | x | 2 for an orthogonal matrix A .

Original: Assume that g ( O ) =
Revised: Assume that g ( O ) =

Original: U .
Revised: U .

Original: Then
Revised: Then

Original: U i is either (0 , ∞ ) , ( −∞ , 0) ( −∞ , 0) ∪ (0 , ∞ ) , or ( −∞ , ∞ ) .
Revised: U i is either (0 , ∞ ) , ( −∞ , 0) ( −∞ , 0) ∪ (0 , ∞ ) , or ( −∞ , ∞ ) .

Original: Moreover, if the i -th row of A is not equal to a (signed) standard basis vector then U
Revised: Moreover, if the i -th row of A is not equal to a (signed) standard basis vector then U

Original: i =
Revised: i =

Original: R .
Revised: R .

Original: Informally the result follows from the fact that coordinate planes { x i = c } are mapped to spheres by g except for c = 0 .
Revised: Informally the result follows from the fact that coordinate planes { x i = c } are mapped to spheres by g except for c = 0 .

Original: We assume that the boundary of O and U is the union of subsets of hyperplanes.
Revised: We assume that the boundary of O and U is the union of subsets of hyperplanes.

Original: However, g ( O ) =
Revised: However, g ( O ) =

Original: U then implies that the boundaries of O and U are mapped to each other.
Revised: U then implies that the boundaries of O and U are mapped to each other.

Original: Since the boundaries of both sets are the union of subsets of hyperplanes that are mapped to hyperplanes we conclude their boundaries must be a subset of the coordinate axes hyperplanes H
Revised: Since the boundaries of both sets are the union of subsets of hyperplanes that are mapped to hyperplanes we conclude their boundaries must be a subset of the coordinate axes hyperplanes H

Original: i = { x : x i = 0 } (because they would be mapped to spherical caps by g ).
Revised: i = { x : x i = 0 } (because they would be mapped to spherical caps by g ).

Original: For completeness, we give a careful proof below.
Revised: For completeness, we give a careful proof below.

Original: Let us highlight that this lemma essentially rules out counterexamples with ﬁnite support because we can apply the lemma to the set { x ∈ R d : p ( x ) > 0 } .
Revised: Let us highlight that this lemma essentially rules out counterexamples with ﬁnite support because we can apply the lemma to the set { x ∈ R d : p ( x ) > 0 } .

Original: This is in contrast to the 2dimensional case where conformal maps between any two rectangles exist making the proof of the corresponding statement below more difﬁcult.
Revised: This is in contrast to the 2dimensional case where conformal maps between any two rectangles exist making the proof of the corresponding statement below more difﬁcult.

Original: We now prove Lemma 2.
Revised: We now prove Lemma 2.

Original: Proof of Lemma 2.
Revised: Proof of Lemma 2.

Original: The proof is a bit lengthy, so we ﬁrst give an informal overview of the main steps.
Revised: The proof is a bit lengthy, so we ﬁrst give an informal overview of the main steps.

Original: As before, we denote the distribution of s and s (cid:48) by P and Q .
Revised: As before, we denote the distribution of s and s (cid:48) by P and Q .

Original: We argue by contradiction, so we assume that P = g ∗ Q .
Revised: We argue by contradiction, so we assume that P = g ∗ Q .

Original: By assumption g ( x ) = b + αA ( x − a ) / | x − a | 2 .
Revised: By assumption g ( x ) = b + αA ( x − a ) / | x − a | 2 .

Original: We now proceed in several steps that constrain the structure of g .
Revised: We now proceed in several steps that constrain the structure of g .

Original: Let us brieﬂy describe the steps of the proof.
Revised: Let us brieﬂy describe the steps of the proof.

Original: In Step 1 and 2 we eliminate the trivial symmetries of the measure and show that the mild local regularity assumption on the measures imply global regularity.
Revised: In Step 1 and 2 we eliminate the trivial symmetries of the measure and show that the mild local regularity assumption on the measures imply global regularity.

Original: Then, in Steps 3 and 4 we derive in (48) a condition similar to (33) but more involved.
Revised: Then, in Steps 3 and 4 we derive in (48) a condition similar to (33) but more involved.

Original: To exploit this condition we look in Steps 5 and 6 at certain limiting regimes where the terms become much simpler and almost reduce to the condition of the linear case.
Revised: To exploit this condition we look in Steps 5 and 6 at certain limiting regimes where the terms become much simpler and almost reduce to the condition of the linear case.

Original: This allows us to conclude that A is a permutation matrix in Step 7.
Revised: This allows us to conclude that A is a permutation matrix in Step 7.

Original: Using that A is a permutation matrix in (48) we get in Step 8 a much simpler relation that almost factorizes.
Revised: Using that A is a permutation matrix in (48) we get in Step 8 a much simpler relation that almost factorizes.

Original: This allows us to derive a simple differential equation in Step 9 which restricts the potential densities to a simple parametric form.
Revised: This allows us to derive a simple differential equation in Step 9 which restricts the potential densities to a simple parametric form.

Original: This allows us to conclude.
Revised: This allows us to conclude.

Original: Step 1: Elimination of trivial symmetries.
Revised: Step 1: Elimination of trivial symmetries.

Original: First we show that we can assume a = b = 0 and α = 1 .
Revised: First we show that we can assume a = b = 0 and α = 1 .

Original: We denote the shifts on R d by T a ( x ) =
Revised: We denote the shifts on R d by T a ( x ) =

Original: x + a and the dilations D α ( x ) = αx .
Revised: x + a and the dilations D α ( x ) = αx .

Original: Then we can rewrite g = T b
Revised: Then we can rewrite g = T b

Original: ◦ D α ◦ g 0 ◦ T − a with g 0 ( x ) =
Revised: ◦ D α ◦ g 0 ◦ T − a with g 0 ( x ) =

Original: Ax/ | x | 2 and therefore ( g 0 ) ∗ ( T − a ) ∗ Q
Revised: Ax/ | x | 2 and therefore ( g 0 ) ∗ ( T − a ) ∗ Q

Original: =
Revised: =

Original: ( D α − 1 ◦ T − b ) ∗ P .
Revised: ( D α − 1 ◦ T − b ) ∗ P .

Original: Since shifts and dilations preserve the class P conf it is sufﬁcient to show the result for g = g 0 .
Revised: Since shifts and dilations preserve the class P conf it is sufﬁcient to show the result for g = g 0 .

Original: To simplify the notation we drop the 0 in the following and just assume g ( x ) =
Revised: To simplify the notation we drop the 0 in the following and just assume g ( x ) =

Original: Ax/ | x | 2 .
Revised: Ax/ | x | 2 .

Original: Step 2: Support and smoothness of distributions.
Revised: Step 2: Support and smoothness of distributions.

Original: The goal of this step is to show that under the assumption of Lemma 2 the density of P and similarly of Q is positive and C 2 away from the coordinate hyperplanes { x i = 0 } for a union of quadrants, while it vanishes on the remaining quadrants.
Revised: The goal of this step is to show that under the assumption of Lemma 2 the density of P and similarly of Q is positive and C 2 away from the coordinate hyperplanes { x i = 0 } for a union of quadrants, while it vanishes on the remaining quadrants.

Original: This will be a consequence of Lemma 3.
Revised: This will be a consequence of Lemma 3.

Original: Let U
Revised: Let U

Original: i
Revised: i

Original: =
Revised: =

Original: Int(supp P i ) .
Revised: Int(supp P i ) .

Original: By assumption, P ∈ P conf which entails U i (cid:54) = ∅ .
Revised: By assumption, P ∈ P conf which entails U i (cid:54) = ∅ .

Original: Then U
Revised: Then U

Original: = U 1 × . . .
Revised: = U 1 × . . .

Original: × U n = Int(supp P ) .
Revised: × U n = Int(supp P ) .

Original: Deﬁne O i similarly for Q .
Revised: Deﬁne O i similarly for Q .

Original: The relation g ∗ Q = P implies g ( O )
Revised: The relation g ∗ Q = P implies g ( O )

Original: =
Revised: =

Original: U .
Revised: U .

Original: Applying Lemma 3 we conclude that U is the union of quadrants.
Revised: Applying Lemma 3 we conclude that U is the union of quadrants.

Original: The same argument will imply that p is actually C 2 away from the coordinate planes.
Revised: The same argument will imply that p is actually C 2 away from the coordinate planes.

Original: We consider the interior of the set of points where P has a twice differentiable and positive density and call it
Revised: We consider the interior of the set of points where P has a twice differentiable and positive density and call it

Original: U (cid:48) .
Revised: U (cid:48) .

Original: For x ∈ U (cid:48) there is a density p in a neighbourhood of x
Revised: For x ∈ U (cid:48) there is a density p in a neighbourhood of x

Original: and it factorizes by the independence assumption.
Revised: and it factorizes by the independence assumption.

Original: The relation
Revised: The relation

Original: [Equation]
Revised: [Equation]

Original: 39) implies that then p i is twice differentiable at x i .
Revised: 39) implies that then p i is twice differentiable at x i .

Original: Vice-versa, if all p i are twice differentiable at x i then p is twice differentiable at x = ( x 1 , . . .
Revised: Vice-versa, if all p i are twice differentiable at x i then p is twice differentiable at x = ( x 1 , . . .

Original: , x d ) .
Revised: , x d ) .

Original: This implies that U (cid:48) =
Revised: This implies that U (cid:48) =

Original: U (cid:48) 1 × . . .
Revised: U (cid:48) 1 × . . .

Original: × U (cid:48)
Revised: × U (cid:48)

Original: d for some open sets U (cid:48) i .
Revised: d for some open sets U (cid:48) i .

Original: By deﬁnition of P conf the sets U (cid:48)
Revised: By deﬁnition of P conf the sets U (cid:48)

Original: i are non-empty.
Revised: i are non-empty.

Original: Similarly, we deﬁne O (cid:48) .
Revised: Similarly, we deﬁne O (cid:48) .

Original: The relation (14) for the densities p and q implies, together with the smoothness of g , that g ( O (cid:48) )
Revised: The relation (14) for the densities p and q implies, together with the smoothness of g , that g ( O (cid:48) )

Original: =
Revised: =

Original: U (cid:48) .
Revised: U (cid:48) .

Original: Then we apply again Lemma 3 to conclude that p and q are C 2 functions away from the hyperplanes { x i = 0 } and if the density is non-zero at a point in a quadrant then it is positive in the complete interior of the quadrant.
Revised: Then we apply again Lemma 3 to conclude that p and q are C 2 functions away from the hyperplanes { x i = 0 } and if the density is non-zero at a point in a quadrant then it is positive in the complete interior of the quadrant.

Original: Finally, U (cid:48) i =
Revised: Finally, U (cid:48) i =

Original: R if the i -th row of A has more than one non-zero entry (again by Lemma 3).
Revised: R if the i -th row of A has more than one non-zero entry (again by Lemma 3).

Original: By deﬁnition this means that p i ∈ C 2 ( R ) and p i ( x ) > 0 for such i and all x .
Revised: By deﬁnition this means that p i ∈ C 2 ( R ) and p i ( x ) > 0 for such i and all x .

Original: Let us emphasize here, that we already ﬁnished the proof of Lemma 2 for probability distributions with bounded support.
Revised: Let us emphasize here, that we already ﬁnished the proof of Lemma 2 for probability distributions with bounded support.

Original: This is more difﬁcult in dimension 2 because there are two-dimensional conformal functions mapping rectangles to rectangles.
Revised: This is more difﬁcult in dimension 2 because there are two-dimensional conformal functions mapping rectangles to rectangles.

Original: We will consider this in the proof of Theorembelow.
Revised: We will consider this in the proof of Theorembelow.

Original: A major step in the proof is to show that A is a permutation matrix under the assumptions of the lemma.
Revised: A major step in the proof is to show that A is a permutation matrix under the assumptions of the lemma.

Original: We deﬁne the index set I ⊂ [ d ] as the set of all indices i such that the i -th row of the matrix
Revised: We deﬁne the index set I ⊂ [ d ] as the set of all indices i such that the i -th row of the matrix

Original: A has only one non-zero entry.
Revised: A has only one non-zero entry.

Original: Our goal is to show that I =
Revised: Our goal is to show that I =

Original: [ d ] .
Revised: [ d ] .

Original: We have shown so far that p k is positive and twice differentiable if k / ∈ I .
Revised: We have shown so far that p k is positive and twice differentiable if k / ∈ I .

Original: The proof in the linear case relied on the relation (33).
Revised: The proof in the linear case relied on the relation (33).

Original: We now derive a similar relation for non-linear
Revised: We now derive a similar relation for non-linear

Original: Moebius transformations.
Revised: Moebius transformations.

Original: Step 3: Derivative formulas.
Revised: Step 3: Derivative formulas.

Original: For future reference we note (using A ∈ O( n ) )
Revised: For future reference we note (using A ∈ O( n ) )

Original: that (for i (cid:54) = j )
Revised: that (for i (cid:54) = j )

Original: [Equation]
Revised: [Equation]

Original: Step 4: Derivation of a condition for the densities We apply the same reasoning as in the proof of Theorem 1 to derive partial differential equations for the density p .
Revised: Step 4: Derivation of a condition for the densities We apply the same reasoning as in the proof of Theorem 1 to derive partial differential equations for the density p .

Original: The condition s =
Revised: The condition s =

Original: g ( s (cid:48) ) , or equivalently g − 1 ( s ) =
Revised: g ( s (cid:48) ) , or equivalently g − 1 ( s ) =

Original: s (cid:48) and the density relation (14) imply
Revised: s (cid:48) and the density relation (14) imply

Original: [Equation]
Revised: [Equation]

Original: This implies
Revised: This implies

Original: [Equation]
Revised: [Equation]

Original: We calculate for i (cid:54) =
Revised: We calculate for i (cid:54) =

Original: j
Revised: j

Original: [Equation]
Revised: [Equation]

Original: y
Revised: y

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation]
Revised: [Equation]

Original: Evaluating the derivatives using (40) and (41) we get
Revised: Evaluating the derivatives using (40) and (41) we get

Original: [Equation]
Revised: [Equation]

Original: Finally we express the variable y through x = g ( y ) =
Revised: Finally we express the variable y through x = g ( y ) =

Original: Ay | y | − 2 .
Revised: Ay | y | − 2 .

Original: Note that then | y | = | x | − 1 and y =
Revised: Note that then | y | = | x | − 1 and y =

Original: A − 1 x | x | − 2 .
Revised: A − 1 x | x | − 2 .

Original: Plugging this in the last equation we get
Revised: Plugging this in the last equation we get

Original: [Equation]
Revised: [Equation]

Original: where we used A − 

Revised: where we used A −

Original: = A (cid:62) as A is orthogonal and we used Einstein summation convention to sum over indices that appear twice (we kept the sum over k for better readability).
Revised: = A (cid:62) as A is orthogonal and we used Einstein summation convention to sum over indices that appear twice (we kept the sum over k for better readability).

Original: Note that this expression is not homogeneous in x .
Revised: Note that this expression is not homogeneous in x .

Original: Step 5: Simpliﬁcations as x r → ∞ .
Revised: Step 5: Simpliﬁcations as x r → ∞ .

Original: We ﬁx an index 1 ≤ r ≤ d .
Revised: We ﬁx an index 1 ≤ r ≤ d .

Original: The strategy is now to send x r → ∞ while keeping the other coordinates bounded.
Revised: The strategy is now to send x r → ∞ while keeping the other coordinates bounded.

Original: We can assume by reﬂecting coordinates that the quadrant { x i > 0 , ∀ i } is contained in the support of P and p has a positive C 2 density there.
Revised: We can assume by reﬂecting coordinates that the quadrant { x i > 0 , ∀ i } is contained in the support of P and p has a positive C 2 density there.

Original: We can then rewrite (48)
Revised: We can then rewrite (48)

Original: [Equation]
Revised: [Equation]

Original: We conclude that
Revised: We conclude that

Original: [Equation]
Revised: [Equation]

Original: By varying x k (cid:54) = x r this almost implies that A kj A ki ln( p k ) (cid:48)(cid:48) =
Revised: By varying x k (cid:54) = x r this almost implies that A kj A ki ln( p k ) (cid:48)(cid:48) =

Original: c k for some constant whenever p k > 0 and twice differentiable.
Revised: c k for some constant whenever p k > 0 and twice differentiable.

Original: However, for this we need to show that the terms hidden in O ( · ) are really negligible, i.e. ln( p r ) (cid:48)(cid:48) and ln( p r ) (cid:48)
Revised: However, for this we need to show that the terms hidden in O ( · ) are really negligible, i.e. ln( p r ) (cid:48)(cid:48) and ln( p r ) (cid:48)

Original: /x r are bounded as x r → ∞ so that the remainder becomes o (1) which we will establish.
Revised: /x r are bounded as x r → ∞ so that the remainder becomes o (1) which we will establish.

Original: Note that if such a relation could be derived we could conclude, similarly to the linear case, that A is a permutation matrix.
Revised: Note that if such a relation could be derived we could conclude, similarly to the linear case, that A is a permutation matrix.

Original: Step 6: Boundedness of ln( p r ) (cid:48)(cid:48) and ln( p r ) (cid:48)
Revised: Step 6: Boundedness of ln( p r ) (cid:48)(cid:48) and ln( p r ) (cid:48)

Original: /x r .
Revised: /x r .

Original: Recall that I is the set of indices such that the i -th row of A has only one non-zero entry.
Revised: Recall that I is the set of indices such that the i -th row of A has only one non-zero entry.

Original: Let r / ∈
Revised: Let r / ∈

Original: I and pick j, i such that A rj A ri (cid:54) = 0 .
Revised: I and pick j, i such that A rj A ri (cid:54) = 0 .

Original: Fix all coordinates x k except x r so that p k is positive and twice differentiable at x k .
Revised: Fix all coordinates x k except x r so that p k is positive and twice differentiable at x k .

Original: Then we can express (50) as
Revised: Then we can express (50) as

Original: [Equation]
Revised: [Equation]

Original: where the remainder term R contains the remaining terms.
Revised: where the remainder term R contains the remaining terms.

Original: The expression R of course depends on the other coordinates x k for k (cid:54) =
Revised: The expression R of course depends on the other coordinates x k for k (cid:54) =

Original: r
Revised: r

Original: but since they are considered ﬁxed here we can view R as a function of x r alone.
Revised: but since they are considered ﬁxed here we can view R as a function of x r alone.

Original: Equation (50) then implies that there is M > 0 sufﬁciently large such that for x r > M the remainder term R ( x ) satisﬁes for some constant c > 0 .
Revised: Equation (50) then implies that there is M > 0 sufﬁciently large such that for x r > M the remainder term R ( x ) satisﬁes for some constant c > 0 .

Original: [Equation]
Revised: [Equation]

Original: Here the last constant term bounds the x − 1 r contribution.
Revised: Here the last constant term bounds the x − 1 r contribution.

Original: Suppose ln( p r ) (cid:48) ≤ 0 .
Revised: Suppose ln( p r ) (cid:48) ≤ 0 .

Original: Then we can bound
Revised: Then we can bound

Original: [Equation]
Revised: [Equation]

Original: We ﬁnd that ln( p r ) (cid:48)(cid:48) ≥ −
Revised: We ﬁnd that ln( p r ) (cid:48)(cid:48) ≥ −

Original: [Equation]
Revised: [Equation]

Original: c (for ln( p r ) (cid:48)(cid:48) >
Revised: c (for ln( p r ) (cid:48)(cid:48) >

Original: this is clear, and otherwise we can absorb the absolute value part).
Revised: this is clear, and otherwise we can absorb the absolute value part).

Original: We conclude by integration that for all x r > M (note that the bound is trivially true if ln( p r ) (cid:48) >
Revised: We conclude by integration that for all x r > M (note that the bound is trivially true if ln( p r ) (cid:48) >

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation]
Revised: [Equation]

Original: Similarly we can bound for ln( p r ) (cid:48) ( x r ) ≥
Revised: Similarly we can bound for ln( p r ) (cid:48) ( x r ) ≥

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: implying ln( p r ) (cid:48)(cid:48) ( x r ) ≤ 2 c for x r ≥ M such that ln( p r ) (cid:48) ( x r ) > 0 .
Revised: implying ln( p r ) (cid:48)(cid:48) ( x r ) ≤ 2 c for x r ≥ M such that ln( p r ) (cid:48) ( x r ) > 0 .

Original: We obtain ln( p r ) (cid:48)
Revised: We obtain ln( p r ) (cid:48)

Original: [Equation]
Revised: [Equation]

Original: p
Revised: p

Original: [Equation]
Revised: [Equation]

Original: Together the last two steps imply that | ln( p r ) (cid:48) ( x r ) | ≤ C + Cx r for some C > 0 and x r > M .
Revised: Together the last two steps imply that | ln( p r ) (cid:48) ( x r ) | ≤ C + Cx r for some C > 0 and x r > M .

Original: Going back to (51) we conclude that there is C > 0 such that | ln
Revised: Going back to (51) we conclude that there is C > 0 such that | ln

Original: ( p r ) (cid:48)(cid:48) ( x r ) | ≤ C for x r > M .
Revised: ( p r ) (cid:48)(cid:48) ( x r ) | ≤ C for x r > M .

Original: We conclude that for r / ∈ I and all i (cid:54) =
Revised: We conclude that for r / ∈ I and all i (cid:54) =

Original: j 

Revised: j

Original: = (cid:88)
Revised: = (cid:88)

Original: [Equation]
Revised: [Equation]

Original: Step 7: A is a permutation matrix.
Revised: Step 7: A is a permutation matrix.

Original: If I =
Revised: If I =

Original: [ n ] we are done.
Revised: [ n ] we are done.

Original: So there is r / ∈ I and using (57) we conclude by varying x k (cid:54) =
Revised: So there is r / ∈ I and using (57) we conclude by varying x k (cid:54) =

Original: x r that
Revised: x r that

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: for some constant (depending on k , i , and j ).
Revised: for some constant (depending on k , i , and j ).

Original: Note that if we assumed that at most one p k is a
Revised: Note that if we assumed that at most one p k is a

Original: Gaussian density we could conclude as in the linear case.
Revised: Gaussian density we could conclude as in the linear case.

Original: However, this assumption is not necessary, as we will now show.
Revised: However, this assumption is not necessary, as we will now show.

Original: By deﬁnition, A kj
Revised: By deﬁnition, A kj

Original: [Equation]
Revised: [Equation]

Original: ln( p k ) (cid:48)(cid:48)
Revised: ln( p k ) (cid:48)(cid:48)

Original: = 0 for k ∈ I because there is only one non-zero entry in row k of
Revised: = 0 for k ∈ I because there is only one non-zero entry in row k of

Original: A .
Revised: A .

Original: We have seen in Step 2 that for k / ∈
Revised: We have seen in Step 2 that for k / ∈

Original: I the density p k ∈ C 2 ( R ) and is positive.
Revised: I the density p k ∈ C 2 ( R ) and is positive.

Original: By assumption, we can ﬁnd
Revised: By assumption, we can ﬁnd

Original: i (cid:54) =
Revised: i (cid:54) =

Original: j such that A kj A ki (cid:54) = 0 for k / ∈ I .
Revised: j such that A kj A ki (cid:54) = 0 for k / ∈ I .

Original: The relation (58) then implies for k / ∈ I that ln( p k ) (cid:48)(cid:48)
Revised: The relation (58) then implies for k / ∈ I that ln( p k ) (cid:48)(cid:48)

Original: ( x k ) =
Revised: ( x k ) =

Original: β k for some constant β k < 

Revised: β k for some constant β k <

Original: ( p k is a probability density) and all x k ∈ R .
Revised: ( p k is a probability density) and all x k ∈ R .

Original: Then ln( p k ) (cid:48) ( x k ) =
Revised: Then ln( p k ) (cid:48) ( x k ) =

Original: β k x k + γ k for some constant γ k .
Revised: β k x k + γ k for some constant γ k .

Original: With x r → ∞ we conclude from (57) that for r / ∈
Revised: With x r → ∞ we conclude from (57) that for r / ∈

Original: I
Revised: I

Original: [Equation]
Revised: [Equation]

Original: Summing this over r / ∈
Revised: Summing this over r / ∈

Original: I we get
Revised: I we get

Original: [Equation]
Revised: [Equation]

Original: Dividing (60) by d − | I | + 4 and subtracting it from (59) we conclude that
Revised: Dividing (60) by d − | I | + 4 and subtracting it from (59) we conclude that

Original: [Equation]
Revised: [Equation]

Original: for all r / ∈ I and all i (cid:54)
Revised: for all r / ∈ I and all i (cid:54)

Original: =
Revised: =

Original: j .
Revised: j .

Original: Since β r is non-zero this implies that A ri A rj = 

Revised: Since β r is non-zero this implies that A ri A rj =

Original: for i (cid:54) =
Revised: for i (cid:54) =

Original: j and thus r ∈ I , a contradiction.
Revised: j and thus r ∈ I , a contradiction.

Original: This establishes I =
Revised: This establishes I =

Original: [ d ] and thus A is a signed permutation matrix.
Revised: [ d ] and thus A is a signed permutation matrix.

Original: By permuting and reﬂecting the coordinates of P we can assume A = Id in the following.
Revised: By permuting and reﬂecting the coordinates of P we can assume A = Id in the following.

Original: Step 8: Simpliﬁcations in (48) for A
Revised: Step 8: Simpliﬁcations in (48) for A

Original: = Id .
Revised: = Id .

Original: First we remark that for A
Revised: First we remark that for A

Original: = Id the function g leaves the quadrants invariant.
Revised: = Id the function g leaves the quadrants invariant.

Original: It is thus sufﬁcient to consider the case where p and q vanish outside { x i > 0 ∀ i } and show that no solutions of s =
Revised: It is thus sufﬁcient to consider the case where p and q vanish outside { x i > 0 ∀ i } and show that no solutions of s =

Original: g ( s (cid:48) ) exist under this condition.
Revised: g ( s (cid:48) ) exist under this condition.

Original: Using step 2 we can assume that p i ( x i ) > 0 for all x i > 0 .
Revised: Using step 2 we can assume that p i ( x i ) > 0 for all x i > 0 .

Original: In the following all domain are assumed to be the positive half-line.
Revised: In the following all domain are assumed to be the positive half-line.

Original: For A = Id the condition (48) becomes for x = ( x 1 , . . .
Revised: For A = Id the condition (48) becomes for x = ( x 1 , . . .

Original: , x d ) such that x i >
Revised: , x d ) such that x i >

Original: [Equation]
Revised: [Equation]

Original: Dividing this by 2 x i x j we obtain
Revised: Dividing this by 2 x i x j we obtain

Original: [Equation]
Revised: [Equation]

Original: We now assume that d > 2 .
Revised: We now assume that d > 2 .

Original: Let i , j , and r be pairwise different.
Revised: Let i , j , and r be pairwise different.

Original: Using the last display with i , j and j , r and subtracting the resulting equations we obtain
Revised: Using the last display with i , j and j , r and subtracting the resulting equations we obtain

Original: [Equation]
Revised: [Equation]

Original: Varying x r and x i independently and since i ∈ [ n ] is arbitrary we conclude that there is a constant κ such that
Revised: Varying x r and x i independently and since i ∈ [ n ] is arbitrary we conclude that there is a constant κ such that

Original: [Equation]
Revised: [Equation]

Original: for all i and x i > 0 .
Revised: for all i and x i > 0 .

Original: Step 9: Conclusion for n > 2 .
Revised: Step 9: Conclusion for n > 2 .

Original: The solutions of the ODE y ( x ) /x
Revised: The solutions of the ODE y ( x ) /x

Original: + y (cid:48)
Revised: + y (cid:48)

Original: ( x ) = κ are given by
Revised: ( x ) = κ are given by

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where α is any constant.
Revised: where α is any constant.

Original: We conclude that there are constants α j such that
Revised: We conclude that there are constants α j such that

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: This implies that
Revised: This implies that

Original: [Equation]
Revised: [Equation]

Original: By applying the main argument to q we infer that q j has to have again the same structure as in (67) so we conclude that κ = 0 and (cid:80) j α j = − d .
Revised: By applying the main argument to q we infer that q j has to have again the same structure as in (67) so we conclude that κ = 0 and (cid:80) j α j = − d .

Original: Alternatively, one directly sees that q only factorizes as q ( y ) =
Revised: Alternatively, one directly sees that q only factorizes as q ( y ) =

Original: (cid:81)
Revised: (cid:81)

Original: q i ( y i )
Revised: q i ( y i )

Original: if those conditions hold.
Revised: if those conditions hold.

Original: It is easy to see that those densities satisfy the assumptions.
Revised: It is easy to see that those densities satisfy the assumptions.

Original: However, x α is never integrable so there are no probability distributions satisfying the relations s
Revised: However, x α is never integrable so there are no probability distributions satisfying the relations s

Original: = g ( s (cid:48) )
Revised: = g ( s (cid:48) )

Original: This ends the proof for d > 2 .
Revised: This ends the proof for d > 2 .

Original: Step 10: Conclusion for d = 2 .
Revised: Step 10: Conclusion for d = 2 .

Original: For n = 2 we cannot simplify (63) by considering indices i (cid:54) =
Revised: For n = 2 we cannot simplify (63) by considering indices i (cid:54) =

Original: j (cid:54) =
Revised: j (cid:54) =

Original: k .
Revised: k .

Original: Instead, we directly exploit (63) to obtain a similar conclusion.
Revised: Instead, we directly exploit (63) to obtain a similar conclusion.

Original: Similarly to the argument in Step 6 it can be shown that ln( p r ) (cid:48)(cid:48) and ln( p r ) (cid:48)
Revised: Similarly to the argument in Step 6 it can be shown that ln( p r ) (cid:48)(cid:48) and ln( p r ) (cid:48)

Original: /x
Revised: /x

Original: r are bounded for x r away from 0.
Revised: r are bounded for x r away from 0.

Original: Then we consider x 1 → ∞ in (63) and divide by x 21 .
Revised: Then we consider x 1 → ∞ in (63) and divide by x 21 .

Original: We get (using { i, j } = { 1 , 2 } )
Revised: We get (using { i, j } = { 1 , 2 } )

Original: [Equation]
Revised: [Equation]

Original: By varying x 2 we conclude just as for d > 2 that ln( p 2 ) (cid:48)
Revised: By varying x 2 we conclude just as for d > 2 that ln( p 2 ) (cid:48)

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: is constant.
Revised: is constant.

Original: We conclude as before.
Revised: We conclude as before.

Original: Let us now prove the geometric result from Lemma 3 above.
Revised: Let us now prove the geometric result from Lemma 3 above.

Original: Proof.
Revised: Proof.

Original: The main idea of the proof is that a box contained in U after inversion is distorted so that its convex hull (contained in O ) is strictly bigger than the box image so that inverting backwards gives us a bigger box in U except for some special cases.
Revised: The main idea of the proof is that a box contained in U after inversion is distorted so that its convex hull (contained in O ) is strictly bigger than the box image so that inverting backwards gives us a bigger box in U except for some special cases.

Original: An illustration of this argument is shown in
Revised: An illustration of this argument is shown in

Original: Figure 4.
Revised: Figure 4.

Original: The formal argument below is slightly technical.
Revised: The formal argument below is slightly technical.

Original: An illustration of the actual argument can be found in Figure 5.
Revised: An illustration of the actual argument can be found in Figure 5.

Original: To simplify the notation we write ι for the inversions x → x/ | x | 2 .
Revised: To simplify the notation we write ι for the inversions x → x/ | x | 2 .

Original: Then we get g =
Revised: Then we get g =

Original: A ◦ ι = ι ◦ A .
Revised: A ◦ ι = ι ◦ A .

Original: We consider the projections
Revised: We consider the projections

Original: π i : R d → R projecting on the i -th coordinate.
Revised: π i : R d → R projecting on the i -th coordinate.

Original: We consider a map Π on subsets of R d deﬁned by Π( M ) =
Revised: We consider a map Π on subsets of R d deﬁned by Π( M ) =

Original: π 1 ( M ) × . . .
Revised: π 1 ( M ) × . . .

Original: × π d ( M ) .
Revised: × π d ( M ) .

Original: Let C denote the convex hull of a set.
Revised: Let C denote the convex hull of a set.

Original: Let R ⊂
Revised: Let R ⊂

Original: O be a (connected) box.
Revised: O be a (connected) box.

Original: Then g ( R ) ⊂
Revised: Then g ( R ) ⊂

Original: U implies Π( g ( R )) ⊂
Revised: U implies Π( g ( R )) ⊂

Original: Π( U ) =
Revised: Π( U ) =

Original: U .
Revised: U .

Original: Since g ( R ) is connected Π( g ( R )) is convex and thus C ( g ( R ))
Revised: Since g ( R ) is connected Π( g ( R )) is convex and thus C ( g ( R ))

Original: ⊂
Revised: ⊂

Original: Π( g ( R )) .
Revised: Π( g ( R )) .

Original: We conclude that g − 1 ( C ( g ( R )) ⊂ g − 1 (Π( g ( R )) ⊂ g − 1 ( U ) ⊂
Revised: We conclude that g − 1 ( C ( g ( R )) ⊂ g − 1 (Π( g ( R )) ⊂ g − 1 ( U ) ⊂

Original: O .
Revised: O .

Original: As A is linear we have A − 1 C A ( M )
Revised: As A is linear we have A − 1 C A ( M )

Original: =
Revised: =

Original: C ( M ) for any set M ⊂ R d .
Revised: C ( M ) for any set M ⊂ R d .

Original: Thus, we get g − 1 ( C ( g ( R )) =
Revised: Thus, we get g − 1 ( C ( g ( R )) =

Original: ι C ι ( R ) .
Revised: ι C ι ( R ) .

Original: W.l.o.g.
Revised: W.l.o.g.

Original: we now suppose that there is a box R = ( x 1 , y 1 ) × . . .
Revised: we now suppose that there is a box R = ( x 1 , y 1 ) × . . .

Original: ( x d , y d ) ⊂
Revised: ( x d , y d ) ⊂

Original: O with 0 < x i < y
Revised: O with 0 < x i < y

Original: i .
Revised: i .

Original: We write x (cid:48)
Revised: We write x (cid:48)

Original: = ( x 2 , . . .
Revised: = ( x 2 , . . .

Original: , x d ) , y (cid:48) =
Revised: , x d ) , y (cid:48) =

Original: ( y 2 , . .
Revised: ( y 2 , . .

Original: . , y d ) .
Revised: . , y d ) .

Original: We consider the point
Revised: We consider the point

Original: [Equation]
Revised: [Equation]

Original: Clearly z ∈ C ιR .
Revised: Clearly z ∈ C ιR .

Original: Then ι
Revised: Then ι

Original: [Equation]
Revised: [Equation]

Original: We calculate ι
Revised: We calculate ι

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: We bound (using x (cid:48) 2 < y (cid:48) 2 ) (cid:13)(cid:13)(cid:13)(cid:13) ( y
Revised: We bound (using x (cid:48) 2 < y (cid:48) 2 ) (cid:13)(cid:13)(cid:13)(cid:13) ( y

Original: [Equation]
Revised: [Equation]

Original: x (cid:48)
Revised: x (cid:48)

Original: [Equation]
Revised: [Equation]

Original: (cid:62) y
Revised: (cid:62) y

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: (cid:13)(cid:13)(cid:13)(cid:13)
Revised: (cid:13)(cid:13)(cid:13)(cid:13)

Original: [Equation]
Revised: [Equation]

Original: (cid:13)(cid:13)(cid:13)(cid:13)
Revised: (cid:13)(cid:13)(cid:13)(cid:13)

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Together the last displays imply that
Revised: Together the last displays imply that

Original: [Equation]
Revised: [Equation]

Original: Let z 1 be maximal such that ( x 1 , z 1 ) ⊂
Revised: Let z 1 be maximal such that ( x 1 , z 1 ) ⊂

Original: O 1 .
Revised: O 1 .

Original: Then the reasoning above shows that z 1 = ∞ .
Revised: Then the reasoning above shows that z 1 = ∞ .

Original: The same reasoning for the other coordinates implies that R (cid:48) =
Revised: The same reasoning for the other coordinates implies that R (cid:48) =

Original: ( x 1 , ∞ )
Revised: ( x 1 , ∞ )

Original: × . . .
Revised: × . . .

Original: × ( x d , ∞ ) ⊂
Revised: × ( x d , ∞ ) ⊂

Original: O .
Revised: O .

Original: By applying the same reasoning to sequences of boxes in g ( R ) approaching the origin we conclude that U is the union of quadrants (and the same holds for O ).
Revised: By applying the same reasoning to sequences of boxes in g ( R ) approaching the origin we conclude that U is the union of quadrants (and the same holds for O ).

Original: It remains to prove the last remark.
Revised: It remains to prove the last remark.

Original: As quadrants are invariant under ι we have ι ( O ) =
Revised: As quadrants are invariant under ι we have ι ( O ) =

Original: O and conclude AO =
Revised: O and conclude AO =

Original: U , or equivalently A (cid:62) U
Revised: U , or equivalently A (cid:62) U

Original: = O .
Revised: = O .

Original: It is sufﬁcient to show that 0 ∈ U i .
Revised: It is sufﬁcient to show that 0 ∈ U i .

Original: For simplicity we assume R = (0 , ∞ )
Revised: For simplicity we assume R = (0 , ∞ )

Original: d ⊂ U , the generalisation to other quadrants is immediate.
Revised: d ⊂ U , the generalization to other quadrants is immediate.

Original: Since we assume that the i -th row v i
Revised: Since we assume that the i -th row v i

Original: =
Revised: =

Original: A (cid:62) e
Revised: A (cid:62) e

Original: i of A is not equal to a signed standard basis vector it has at least two non-zero entries.
Revised: i of A is not equal to a signed standard basis vector it has at least two non-zero entries.

Original: Thus we can ﬁnd w such that w · v i = 

Revised: Thus we can ﬁnd w such that w · v i =

Original: and all entries of w are non-zero.
Revised: and all entries of w are non-zero.

Original: Since w is orthogonal to the span of Ae i there is a vector α such that A (cid:62) α = w and α i = 0 .
Revised: Since w is orthogonal to the span of Ae i there is a vector α such that A (cid:62) α = w and α i = 0 .

Original: By adding a suitable vector β we can ensure that ( β + α ) i = 0 , all entries of A (cid:62) ( α + β ) are non-zero and ( β + α ) j > 0 for j (cid:54)
Revised: By adding a suitable vector β we can ensure that ( β + α ) i = 0 , all entries of A (cid:62) ( α + β ) are non-zero and ( β + α ) j > 0 for j (cid:54)

Original: =
Revised: =

Original: i .
Revised: i .

Original: The second condition can be satisﬁed by picking the entries of β one after another.
Revised: The second condition can be satisﬁed by picking the entries of β one after another.

Original: The conditions ( β + α ) j > 0 for j (cid:54) =
Revised: The conditions ( β + α ) j > 0 for j (cid:54) =

Original: i and ( β + α ) i = 0 imply that β + α ∈ U since we assumed (0 , ∞ )
Revised: i and ( β + α ) i = 0 imply that β + α ∈ U since we assumed (0 , ∞ )

Original: d ⊂ U .
Revised: d ⊂ U .

Original: But then A (cid:62) ( α + β ) ∈
Revised: But then A (cid:62) ( α + β ) ∈

Original: O and since A (cid:62) ( α + β ) is strictly contained in a quadrant (all entries are non-zero)
Revised: O and since A (cid:62) ( α + β ) is strictly contained in a quadrant (all entries are non-zero)

Original: we conclude that A (cid:62) ( α + β ) ∈
Revised: we conclude that A (cid:62) ( α + β ) ∈

Original: O and thus α + β ∈ U .
Revised: O and thus α + β ∈ U .

Original: This implies ( α + β ) i = 

Revised: This implies ( α + β ) i =

Original: ∈ U i .
Revised: ∈ U i .

Original: E.2 Proof of Corollary
Revised: E.2 Proof of Corollary

Original: Here we prove the simple extension of Theorem 2 to rescaled conformal maps.
Revised: Here we prove the simple extension of Theorem 2 to rescaled conformal maps.

Original: Proof of Corollary 1.
Revised: Proof of Corollary 1.

Original: Suppose that there are f, f (
Revised: Suppose that there are f, f (

Original: cid:48) ∈ F r − conf and P , P (cid:48)
Revised: cid:48) ∈ F r − conf and P , P (cid:48)

Original: ∈ P conf such that
Revised: ∈ P conf such that

Original: [Equation]
Revised: [Equation]

Original: Then we need to show that f − 1 f (cid:48) ∈ S reparam .
Revised: Then we need to show that f − 1 f (cid:48) ∈ S reparam .

Original: By deﬁnition of F r − conf there are g, g (cid:48) ∈ F conf and h, h (cid:48)
Revised: By deﬁnition of F r − conf there are g, g (cid:48) ∈ F conf and h, h (cid:48)

Original: ∈ S reparam such that f = g ◦ h , f (cid:48) =
Revised: ∈ S reparam such that f = g ◦ h , f (cid:48) =

Original: g (cid:48) ◦ h (cid:48) .
Revised: g (cid:48) ◦ h (cid:48) .

Original: The smoothness condition for F r − conf implies h, h (cid:48) are three times differentiable and thus Q = h ∗ P and Q (cid:48)
Revised: The smoothness condition for F r − conf implies h, h (cid:48) are three times differentiable and thus Q = h ∗ P and Q (cid:48)

Original: =
Revised: =

Original: h (cid:48)∗ P (cid:48) satisfy the smoothness condition of measures in P conf .
Revised: h (cid:48)∗ P (cid:48) satisfy the smoothness condition of measures in P conf .

Original: From (75) we infer g ∗ Q D = g
Revised: From (75) we infer g ∗ Q D = g

Original: (cid:48)∗ Q (cid:48) and this is the observational distribution.
Revised: (cid:48)∗ Q (cid:48) and this is the observational distribution.

Original: By the assumption on the observational distribution, we conclude that Q ∼ g − 1 ( x ) and Q (cid:48) ∼ g (cid:48)− 1 ( x ) have at most one Gaussian component and thus Q , Q (cid:48) ∈ P conf .
Revised: By the assumption on the observational distribution, we conclude that Q ∼ g − 1 ( x ) and Q (cid:48) ∼ g (cid:48)− 1 ( x ) have at most one Gaussian component and thus Q , Q (cid:48) ∈ P conf .

Original: Applying
Revised: Applying

Original: Theorem 2 we infer that g − 1 g (cid:48)
Revised: Theorem 2 we infer that g − 1 g (cid:48)

Original: ∈ S conf ⊂ S reparam .
Revised: ∈ S conf ⊂ S reparam .

Original: Since S reparam is a group, we conclude that f − 1 f (cid:48)
Revised: Since S reparam is a group, we conclude that f − 1 f (cid:48)

Original: =
Revised: =

Original: h − 1 ◦ ( g − 1 g (cid:48) ) ◦ h (cid:48) ∈ S reparam .
Revised: h − 1 ◦ ( g − 1 g (cid:48) ) ◦ h (cid:48) ∈ S reparam .

Original: This ends the proof.
Revised: This ends the proof.

Original: E.3 Proof of Theorem
Revised: E.3 Proof of Theorem

Original: Now we address the case d = 2 .
Revised: Now we address the case d = 2 .

Original: Let us ﬁrst note that conformal maps f : Ω → R 2 for Ω ⊂ Rcan be identiﬁed with holomorphic maps g : Ω → C with non-vanishing derivatives by considering g ( x + iy ) =
Revised: Let us ﬁrst note that conformal maps f : Ω → R 2 for Ω ⊂ Rcan be identiﬁed with holomorphic maps g : Ω → C with non-vanishing derivatives by considering g ( x + iy ) =

Original: f 1 ( x + iy )
Revised: f 1 ( x + iy )

Original: + if 2 ( x + iy ) .
Revised: + if 2 ( x + iy ) .

Original: Moreover,
Revised: Moreover,

Original: [Equation]
Revised: [Equation]

Original: where g (cid:48) denotes the complex derivative.
Revised: where g (cid:48) denotes the complex derivative.

Original: The proof of Theorem 3 is based on the fact that conformal maps between rectangles can be characterized rather explicitly.
Revised: The proof of Theorem 3 is based on the fact that conformal maps between rectangles can be characterized rather explicitly.

Original: In particular, we use the following result.
Revised: In particular, we use the following result.

Original: Theorem 8 (Schwarz-Christoffel mapping) .
Revised: Theorem 8 (Schwarz-Christoffel mapping) .

Original: Conformal maps f that map the unit disk D
Revised: Conformal maps f that map the unit disk D

Original: = { z ∈ C | | z | < 1 } to a polygon with angles
Revised: = { z ∈ C | | z | < 1 } to a polygon with angles

Original: α k π for 1 ≤ k ≤ n can be written as
Revised: α k π for 1 ≤ k ≤ n can be written as

Original: [Equation]
Revised: [Equation]

Original: for two constants C, K ∈ C and w k ∈ C with | w k | = 1 .
Revised: for two constants C, K ∈ C and w k ∈ C with | w k | = 1 .

Original: A proof of this result can be found in any textbook on complex analysis, e.g., [1].
Revised: A proof of this result can be found in any textbook on complex analysis, e.g., [1].

Original: With this result we can prove Theorem 3.
Revised: With this result we can prove Theorem 3.

Original: Proof of Theorem 3.
Revised: Proof of Theorem 3.

Original: Assume there are two probability distribution P 1 ∈ P conf2 and P 2 ∈ P conf with densities p 1 and p 2 supported on ﬁnite rectangles R 1 and R 2 and a conformal map f : R 1 → Rsuch that f ∗ P 1 = P 2 .
Revised: Assume there are two probability distribution P 1 ∈ P conf2 and P 2 ∈ P conf with densities p 1 and p 2 supported on ﬁnite rectangles R 1 and R 2 and a conformal map f : R 1 → Rsuch that f ∗ P 1 = P 2 .

Original: Let h : D → R 1 be a conformal map which by the Riemann mapping theorem exists and by Theorem 8 can be expressed as in (77), i.e.,
Revised: Let h : D → R 1 be a conformal map which by the Riemann mapping theorem exists and by Theorem 8 can be expressed as in (77), i.e.,

Original: [Equation]
Revised: [Equation]

Original: where | w k | = 1 and we used that all angles in a rectangle are π/ 2 .
Revised: where | w k | = 1 and we used that all angles in a rectangle are π/ 2 .

Original: The points w k are the preimages of the corners of R 1 .
Revised: The points w k are the preimages of the corners of R 1 .

Original: Then the derivative of h can be written as
Revised: Then the derivative of h can be written as

Original: [Equation]
Revised: [Equation]

Original: Note that h (cid:48) is bounded above and below away from the points w k and | h (cid:48)
Revised: Note that h (cid:48) is bounded above and below away from the points w k and | h (cid:48)

Original: ( x ) | → ∞ as x → w k .
Revised: ( x ) | → ∞ as x → w k .

Original: We can write f = ( f ◦ h ) ◦ h − 

Revised: We can write f = ( f ◦ h ) ◦ h −

Original: where g = f ◦ h is a conformal map from D to R 2 .
Revised: where g = f ◦ h is a conformal map from D to R 2 .

Original: This can be written again as in (77) with points w (cid:48) 1 , . . .
Revised: This can be written again as in (77) with points w (cid:48) 1 , . . .

Original: , w (cid:48) 4 which are the preimages of the corners of R 2 , i.e.,
Revised: , w (cid:48) 4 which are the preimages of the corners of R 2 , i.e.,

Original: [Equation]
Revised: [Equation]

Original: The condition f ∗ P 1 = P 2 implies that
Revised: The condition f ∗ P 1 = P 2 implies that

Original: [Equation]
Revised: [Equation]

Original: (using the relation (76) between complex derivative and maps from R 2 → R 2 ).
Revised: (using the relation (76) between complex derivative and maps from R 2 → R 2 ).

Original: Now we infer
Revised: Now we infer

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where y = h − 1 ( x ) .
Revised: where y = h − 1 ( x ) .

Original: Applying this with y → w k we get h (cid:48) ( y ) → ∞ .
Revised: Applying this with y → w k we get h (cid:48) ( y ) → ∞ .

Original: Using the assumption that the densities of P 1 and P 2 are bounded above and below we infer from (81) and (82)
Revised: Using the assumption that the densities of P 1 and P 2 are bounded above and below we infer from (81) and (82)

Original: that then
Revised: that then

Original: | g (cid:48) ( y ) | → ∞ from which we conclude y → w (cid:48) k (cid:48) for some k (cid:48) .
Revised: | g (cid:48) ( y ) | → ∞ from which we conclude y → w (cid:48) k (cid:48) for some k (cid:48) .

Original: This implies w k = w (cid:48) k (cid:48) .
Revised: This implies w k = w (cid:48) k (cid:48) .

Original: After renaming the corners, we get w k = w (cid:48) k for all k .
Revised: After renaming the corners, we get w k = w (cid:48) k for all k .

Original: From (78) and (80) we see that
Revised: From (78) and (80) we see that

Original: [Equation]

Original: ( g

Original: [Equation]
Revised: [Equation]

Original: z
Revised: g

Original: [Equation]
Revised: [Equation]

Original: Thus g ( z ) =
Revised: Thus g ( z ) =

Original: C (cid:48) h ( z ) /C
Revised: C (cid:48) h ( z ) /C

Original: − KC (cid:48) /C
Revised: − KC (cid:48) /C

Original: + K (cid:48)
Revised: + K (cid:48)

Original: so we conclude that
Revised: so we conclude that

Original: [Equation]
Revised: [Equation]

Original: In particular there are constants A, B ∈ C such that f ( x )
Revised: In particular there are constants A, B ∈ C such that f ( x )

Original: =
Revised: =

Original: ( g ◦ h − 1 )( x ) =
Revised: ( g ◦ h − 1 )( x ) =

Original: Ax + B .
Revised: Ax + B .

Original: From f ( R 1 ) =
Revised: From f ( R 1 ) =

Original: R 

Revised: R

Original: we infer that A ∈ R ∪ i R .
Revised: we infer that A ∈ R ∪ i R .

Original: This ends the proof.
Revised: This ends the proof.

Original: Let us ﬁnally remark that the identiﬁability of conformal maps for distributions with full support in d = 2 follows just as in d ≥ 3 because every bijective conformal map of the Riemann sphere to itself is a Moebius transformation so we can apply Lemma 3.
Revised: Let us ﬁnally remark that the identiﬁability of conformal maps for distributions with full support in d = 2 follows just as in d ≥ 3 because every bijective conformal map of the Riemann sphere to itself is a Moebius transformation so we can apply Lemma 3.

Original: We expect that the result can be extended to more general densities using the same strategy and a more careful analysis of the density close to the boundary.
Revised: We expect that the result can be extended to more general densities using the same strategy and a more careful analysis of the density close to the boundary.

Original: F Proofs for the results on OCTs
Revised: F Proofs for the results on OCTs

Original: In this section we collect the missing proofs for Section 4.
Revised: In this section we collect the missing proofs for Section 4.

Original: F.1 Proof of Proposition
Revised: F.1 Proof of Proposition

Original: First we prove Proposition 1.
Revised: First we prove Proposition 1.

Original: We refer to Appendix C for a general review and characterisation of spurious solutions.
Revised: We refer to Appendix C for a general review and characterization of spurious solutions.

Original: Proof of Proposition 1.
Revised: Proof of Proposition 1.

Original: The proof essentially shows that polar coordinates are an example of such a function.
Revised: The proof essentially shows that polar coordinates are an example of such a function.

Original: In dimension d they are deﬁned by Φ : ( a, b ) × [0 , π ] × [0 , π/ 2] d − 2 → R d with x 1 = r sin( ϕ ) sin( θ 1 ) . . 
Revised: In dimension d they are deﬁned by Φ : ( a, b ) × [0 , π ] × [0 , π/ 2] d − 2 → R d with x 1 = r sin( ϕ ) sin( θ 1 ) . . .

Original: [Equation]
Revised: [Equation]

Original: The following lemma is well known.
Revised: The following lemma is well known.

Original: Lemma 4.
Revised: Lemma 4.

Original: The polar coordinates Φ deﬁned above satisfy:
Revised: The polar coordinates Φ deﬁned above satisfy:

Original: The transformation satisﬁes Φ ∈ F OCT . 2.
Revised: The transformation satisﬁes Φ ∈ F OCT . 2.

Original: The determinant of the Jacobian is given by
Revised: The determinant of the Jacobian is given by

Original: [Equation]
Revised: [Equation]

Original: The image of Φ d is, up to a set of measure zero, the annulus { x ∈ R n : a < | x | < b } .
Revised: The image of Φ d is, up to a set of measure zero, the annulus { x ∈ R n : a < | x | < b } .

Original: Proof.
Revised: Proof.

Original: For the ﬁrst part we only need to show that D Φ has orthogonal columns which can formally be done by induction noting that
Revised: For the ﬁrst part we only need to show that D Φ has orthogonal columns which can formally be done by induction noting that

Original: [Equation]
Revised: [Equation]

Original: The determinant and the image can also be derived from this recursion.
Revised: The determinant and the image can also be derived from this recursion.

Original: We now deﬁne for k ∈ N the functions g k : (0 , π/ 2) → R
Revised: We now deﬁne for k ∈ N the functions g k : (0 , π/ 2) → R

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Clearly g (cid:48) k
Revised: Clearly g (cid:48) k

Original: ( θ ) =
Revised: ( θ ) =

Original: sin( θ ) k .
Revised: sin( θ ) k .

Original: They are strictly increasing functions with positive derivative, i.e., diffeomorphisms, so we can deﬁne their inverses on an open interval h k : I k
Revised: They are strictly increasing functions with positive derivative, i.e., diffeomorphisms, so we can deﬁne their inverses on an open interval h k : I k

Original: → (0 , π/ 2) which are also differentiable functions.
Revised: → (0 , π/ 2) which are also differentiable functions.

Original: Note that
Revised: Note that

Original: [Equation]
Revised: [Equation]

Original: Denote the density of P by p .
Revised: Denote the density of P by p .

Original: By assumption P is invariant under rotations so we can write with a slight abuse of notation p ( s )
Revised: By assumption P is invariant under rotations so we can write with a slight abuse of notation p ( s )

Original: = p ( | s | ) .
Revised: = p ( | s | ) .

Original: We assume that p ( | s | ) is positive on a, possibly unbounded, interval ( a, b ) .
Revised: We assume that p ( | s | ) is positive on a, possibly unbounded, interval ( a, b ) .

Original: We consider q : R + → R given by
Revised: We consider q : R + → R given by

Original: [Equation]
Revised: [Equation]

Original: where ω d is the volume of the unit ball in dimension d and these constants ensure that q is a probability density.
Revised: where ω d is the volume of the unit ball in dimension d and these constants ensure that q is a probability density.

Original: Deﬁne F q : (0 , ∞ ) → [0 , 1] as the cdf for the probability density q , i.e.,
Revised: Deﬁne F q : (0 , ∞ ) → [0 , 1] as the cdf for the probability density q , i.e.,

Original: [Equation]
Revised: [Equation]

Original: Since q ( z ) > 0 iff t ∈ ( a, b ) we conclude that F q restricted to ( a, b ) is a continuous and strictly increasing from 0 to 1 and has a positive derivative.
Revised: Since q ( z ) > 0 iff t ∈ ( a, b ) we conclude that F q restricted to ( a, b ) is a continuous and strictly increasing from 0 to 1 and has a positive derivative.

Original: Hence, we can deﬁne ψ : (0 , 1) →
Revised: Hence, we can deﬁne ψ : (0 , 1) →

Original: ( a, b ) by ψ ( t ) =
Revised: ( a, b ) by ψ ( t ) =

Original: F − 1 q ( t ) and ψ is differentiable with
Revised: F − 1 q ( t ) and ψ is differentiable with

Original: [Equation]
Revised: [Equation]

Original: We deﬁne the domains D d = (0 , 1) × (0 , 2 π ) ×
Revised: We deﬁne the domains D d = (0 , 1) × (0 , 2 π ) ×

Original: I 1 × . . .
Revised: I 1 × . . .

Original: I d − 

Revised: I d −

Original: Now we consider the map h :
Revised: Now we consider the map h :

Original: D
Revised: D

Original: d → ( a, b ) × (0 , 2 π ) × (0 , π ) d − 2 given by
Revised: d → ( a, b ) × (0 , 2 π ) × (0 , π ) d − 2 given by

Original: [Equation]
Revised: [Equation]

Original: Note that h is a coordinate-wise transformation and the determinant of its Jacobian is given by
Revised: Note that h is a coordinate-wise transformation and the determinant of its Jacobian is given by

Original: [Equation]
Revised: [Equation]

Original: Moreover Lemma 4 gives
Revised: Moreover Lemma 4 gives

Original: [Equation]
Revised: [Equation]

Original: and thus
Revised: and thus

Original: [Equation]
Revised: [Equation]

Original: We now consider the probability measure Q =
Revised: We now consider the probability measure Q =

Original: U (0 , 1) ⊗U
Revised: U (0 , 1) ⊗U

Original: ((0 , 2 π ))
Revised: ((0 , 2 π ))

Original: ⊗U ( I 1 ) ⊗ . . .
Revised: ⊗U ( I 1 ) ⊗ . . .

Original: ⊗U ( I n − 2 ) where
Revised: ⊗U ( I n − 2 ) where

Original: U denotes the uniform distribution on an interval.
Revised: U denotes the uniform distribution on an interval.

Original: Denote by λ : R d → R d the scaling function λ ( s ) =
Revised: Denote by λ : R d → R d the scaling function λ ( s ) =

Original: ( s 1 , 2 πs 2 , | I 1 | s 3 , . .
Revised: ( s 1 , 2 πs 2 , | I 1 | s 3 , . .

Original: , | I d − 2 | s d ) .
Revised: , | I d − 2 | s d ) .

Original: Clearly λ maps C d to (0 , 1) × (0 , 2 π ) × I 1 × . . .
Revised: Clearly λ maps C d to (0 , 1) × (0 , 2 π ) × I 1 × . . .

Original: ×
Revised: ×

Original: I d −and λ ∗ ν = Q .
Revised: I d −and λ ∗ ν = Q .

Original: We now consider the measure
Revised: We now consider the measure

Original: [Equation]
Revised: [Equation]

Original: We denote its density by ˜ p and obtain using (13), (96), and that the density of Q is constant
Revised: We denote its density by ˜ p and obtain using (13), (96), and that the density of Q is constant

Original: [Equation]
Revised: [Equation]

Original: Note that by deﬁnition of Φ we have Φ − 1 ( x ) 1 = | x | and h acts coordinatewise where the action on the ﬁrst coordinate is ψ so that
Revised: Note that by deﬁnition of Φ we have Φ − 1 ( x ) 1 = | x | and h acts coordinatewise where the action on the ﬁrst coordinate is ψ so that

Original: [Equation]
Revised: [Equation]

Original: We conclude, using (90) and the last display, that
Revised: We conclude, using (90) and the last display, that

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: This shows
Revised: This shows

Original: [Equation]
Revised: [Equation]

Original: It remains to be shown that Φ ◦ h ◦ λ ∈ F OCT .
Revised: It remains to be shown that Φ ◦ h ◦ λ ∈ F OCT .

Original: But we have seen in Lemma 4 that Φ ∈ F OCT and h ◦ λ is an invertible coordinate-wise transformation so that h ◦ λ ∈ F R OCT (see (28)) implying Φ ◦ h ◦ λ ∈ F OCT .
Revised: But we have seen in Lemma 4 that Φ ∈ F OCT and h ◦ λ is an invertible coordinate-wise transformation so that h ◦ λ ∈ F R OCT (see (28)) implying Φ ◦ h ◦ λ ∈ F OCT .

Original: Note that we here essentially use the fact that precomposition with a function that acts on each coordinate separately preserves orthogonality of the columns of the Jacobian.
Revised: Note that we here essentially use the fact that precomposition with a function that acts on each coordinate separately preserves orthogonality of the columns of the Jacobian.

Original: F.2 Proof of Theorems
Revised: F.2 Proof of Theorems

Original: We now consider smooth deformations of a data generating mechanism x = f ( s ) .
Revised: We now consider smooth deformations of a data generating mechanism x = f ( s ) .

Original: For this it is helpful to phrase these as ﬂows generated by vector ﬁelds.
Revised: For this it is helpful to phrase these as ﬂows generated by vector ﬁelds.

Original: For a brief review of these notions we refer to Appendix A and for an extensive introduction we refer to any textbook on differential geometry.
Revised: For a brief review of these notions we refer to Appendix A and for an extensive introduction we refer to any textbook on differential geometry.

Original: We now give a complete proof of Theorem 4.
Revised: We now give a complete proof of Theorem 4.

Original: Proof of Theorem 4.
Revised: Proof of Theorem 4.

Original: To simplify the notation we ﬁrst focus on the case where M = R d .
Revised: To simplify the notation we ﬁrst focus on the case where M = R d .

Original: The necessary modiﬁcations for the manifold case are indicated afterwards.
Revised: The necessary modiﬁcations for the manifold case are indicated afterwards.

Original: We deﬁne
Revised: We deﬁne

Original: [Equation]
Revised: [Equation]

Original: so that Ψ t : R d → R d is a diffeomorphism for every t and Ψ 0 ( s )
Revised: so that Ψ t : R d → R d is a diffeomorphism for every t and Ψ 0 ( s )

Original: =
Revised: =

Original: s by assumption.
Revised: s by assumption.

Original: We denote the vector ﬁeld that generates Ψ t by X : (0 , 1) d → R d , i.e., X satisﬁes
Revised: We denote the vector ﬁeld that generates Ψ t by X : (0 , 1) d → R d , i.e., X satisﬁes

Original: [Equation]
Revised: [Equation]

Original: The assumption (Φ t ) ∗ ν = f ∗ ν implies ν = (Ψ t ) ∗ f ∗ ν .
Revised: The assumption (Φ t ) ∗ ν = f ∗ ν implies ν = (Ψ t ) ∗ f ∗ ν .

Original: Then the continuity equation (16) implies that Div( X t ) =
Revised: Then the continuity equation (16) implies that Div( X t ) =

Original: on (0 , 1) d .
Revised: on (0 , 1) d .

Original: By assumption, Φ t ∈ F OCT , which means that
Revised: By assumption, Φ t ∈ F OCT , which means that

Original: [Equation]
Revised: [Equation]

Original: where Λ t : (0 , 1) d → Diag( d ) maps to diagonal matrices.
Revised: where Λ t : (0 , 1) d → Diag( d ) maps to diagonal matrices.

Original: Similarly f t ∈ F OCT implies that there is a function Ω t : (0 , 1) d → Diag( d ) such that
Revised: Similarly f t ∈ F OCT implies that there is a function Ω t : (0 , 1) d → Diag( d ) such that

Original: [Equation]
Revised: [Equation]

Original: We now evaluate ∂ t Ω t in terms of Ψ t and Φ t .
Revised: We now evaluate ∂ t Ω t in terms of Ψ t and Φ t .

Original: To evaluate the time derivative it is convenient to write Ψ( t, s ) instead of Ψ t ( s ) .
Revised: To evaluate the time derivative it is convenient to write Ψ( t, s ) instead of Ψ t ( s ) .

Original: We get, using f t ( s ) =
Revised: We get, using f t ( s ) =

Original: Φ( t, Ψ( t, s )) ,
Revised: Φ( t, Ψ( t, s )) ,

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Note that also
Revised: Note that also

Original: [Equation]
Revised: [Equation]

Original: Combining this with the last display we get (dropping the positional argument s for conciseness)
Revised: Combining this with the last display we get (dropping the positional argument s for conciseness)

Original: [Equation]
Revised: [Equation]

Original: We now consider t = 0 and drop the t argument from the notation.
Revised: We now consider t = 0 and drop the t argument from the notation.

Original: By assumption Ψ 

Revised: By assumption Ψ

Original: = Id
Revised: = Id

Original: so the equation simpliﬁes to
Revised: so the equation simpliﬁes to

Original: [Equation]
Revised: [Equation]

Original: Now we use that Λ t and Ω t map to diagonal matrices for all t , in particular ∂ t Λ and ∂ t Ω are diagonal matrices.
Revised: Now we use that Λ t and Ω t map to diagonal matrices for all t , in particular ∂ t Λ and ∂ t Ω are diagonal matrices.

Original: We conclude that for i (cid:54) =
Revised: We conclude that for i (cid:54) =

Original: j the equation
Revised: j the equation

Original: [Equation]
Revised: [Equation]

Original: holds.
Revised: holds.

Original: Thus, we obtain a system of ﬁrst order Partial Differential Equations (PDE) for X t =0 .
Revised: Thus, we obtain a system of ﬁrst order Partial Differential Equations (PDE) for X t =0 .

Original: We now write Λ j = Λ jj .
Revised: We now write Λ j = Λ jj .

Original: We also ﬁx an i ∈ { 1 , . . .
Revised: We also ﬁx an i ∈ { 1 , . . .

Original: , d } in the following.
Revised: , d } in the following.

Original: Then we can rewrite (110) concisely as
Revised: Then we can rewrite (110) concisely as

Original: [Equation]
Revised: [Equation]

Original: We divide equation (111) by Λ j apply ∂ j and sum over j (cid:54)
Revised: We divide equation (111) by Λ j apply ∂ j and sum over j (cid:54)

Original: =
Revised: =

Original: i to obtain
Revised: i to obtain

Original: [Equation]
Revised: [Equation]

Original: This implies that X i satisﬁes the wave equation
Revised: This implies that X i satisﬁes the wave equation

Original: [Equation]
Revised: [Equation]

Original: 113) where a j = Λ i / Λ j .
Revised: 113) where a j = Λ i / Λ j .