Original: Consistent Interpolating Ensembles via the Manifold-Hilbert Kernel
Revised: Consistent Interpolating Ensembles via the Manifold-Hilbert Kernel

Original: Abstract
Revised: Abstract

Original: Recent research in the theory of overparametrized learning has sought to establish generalization guarantees in the interpolating regime.
Revised: Recent research in the theory of overparametrized learning has sought to establish generalization guarantees in the interpolating regime.

Original: Such results have been established for a few common classes of methods, but so far not for ensemble methods.
Revised: Such results have been established for a few common classes of methods, but so far not for ensemble methods.

Original: We devise an ensemble classification method that simultaneously interpolates the training data, and is consistent for a broad class of data distributions.
Revised: We devise an ensemble classification method that simultaneously interpolates the training data, and is consistent for a broad class of data distributions.

Original: To this end, we define the manifold-Hilbert kernel for data distributed on a Riemannian manifold.
Revised: To this end, we define the manifold-Hilbert kernel for data distributed on a Riemannian manifold.

Original: We prove that kernel smoothing regression and classification using the manifold-Hilbert kernel are weakly consistent in the setting of Devroye et al.
Revised: We prove that kernel smoothing regression and classification using the manifold-Hilbert kernel are weakly consistent in the setting of Devroye et al.

Original: For the sphere, we show that the manifold-Hilbert kernel can be realized as a weighted random partition kernel, which arises as an infinite ensemble of partition-based classifiers.
Revised: For the sphere, we show that the manifold-Hilbert kernel can be realized as a weighted random partition kernel, which arises as an infinite ensemble of partition-based classifiers.

Original:  Introduction
Revised:  Introduction

Original: Ensemble methods are among the most often applied learning algorithms, yet their theoretical properties have not been fully understood [12].
Revised: Ensemble methods are among the most often applied learning algorithms, yet their theoretical properties have not been fully understood [12].

Original: Based on empirical evidence, Wyner et al.
Revised: Based on empirical evidence, Wyner et al.

Original: [42] conjectured that interpolation of the training data plays a key role in explaining the success of
Revised: [42] conjectured that interpolation of the training data plays a key role in explaining the success of

Original: AdaBoost and random forests.
Revised: AdaBoost and random forests.

Original: However, while a few classes of learning methods have been analyzed in the interpolating regime [6, 4], ensembles have not.
Revised: However, while a few classes of learning methods have been analyzed in the interpolating regime [6, 4], ensembles have not.

Original: Towards developing the theory of interpolating ensembles, we examine an ensemble classification method for data distributed on the sphere and show that this classifier interpolates the training data and is consistent for a broad class of data distributions.
Revised: Towards developing the theory of interpolating ensembles we examine an ensemble classification method for data distributed on the sphere and show that this classifier interpolates the training data and is consistent for a broad class of data distributions.

Original: To show this result, we develop two additional contributions that may be of independent interest.
Revised: To show this result, we develop two additional contributions that may be of independent interest.

Original: First, for data distributed on a Riemannian manifold
Revised: First, for data distributed on a Riemannian manifold

Original: M , we introduce the manifold-Hilbert kernel K H M , a manifold extension of the Hilbert kernel [39].
Revised: M , we introduce the manifold-Hilbert kernel K H M , a manifold extension of the Hilbert kernel [39].

Original: Under the same setting as Devroye et al.
Revised: Under the same setting as Devroye et al.

Original: [21], we prove that kernel smoothing regression with K H M is weakly consistent while interpolating the training data.
Revised: [22], we prove that kernel smoothing regression with K H M is weakly consistent while interpolating the training data.

Original: Consequently, the classifier obtained by taking the sign of the kernel smoothing estimate has zero training error and is consistent.
Revised: Consequently, the classifier obtained by taking the sign of the kernel smoothing estimate has zero training error and is consistent.

Original: Second, we introduce a class of kernels called weighted random partition kernels.
Revised: Second, we introduce a class of kernels called weighted random partition kernels.

Original: These are kernels that can be realized as an infinite, weighted ensemble of partition-based histogram classifiers.
Revised: These are kernels that can be realized as an infinite, weighted ensemble of partition-based histogram classifiers.

Original: Our main result is established by showing that when M = S d , the d -dimensional sphere, the manifoldHilbert kernel is a weighted random partition kernel.
Revised: Our main result is established by showing that when M = S d , the d -dimensional sphere, the manifoldHilbert kernel is a weighted random partition kernel.

Original: In particular, we show that on the sphere, the manifold-Hilbert kernel is a weighted ensemble based on random hyperplane arrangements.
Revised: In particular, we show that on the sphere, the manifold-Hilbert kernel is a weighted ensemble based on random hyperplane arrangements.

Original: This implies that the kernel smoothing classifier is a consistent, interpolating ensemble on S d .
Revised: This implies that the kernel smoothing classifier is a consistent, interpolating ensemble on S d .

Original: To our knowledge, this is the first demonstration of an interpolating ensemble method that is consistent for a broad class of distributions in arbitrary dimensions.
Revised: To our knowledge, this is the first demonstration of an interpolating ensemble method that is consistent for a broad class of distributions in arbitrary dimensions.

Original: 1.1 Problem statement
Revised: 1.1 Problem statement

Original: Consider the problem of binary classification on a Riemannian manifold M .
Revised: Consider the problem of binary classification on a Riemannian manifold M .

Original: Let ( X, Y ) be random variables jointly distributed on M × {± 1 } .
Revised: Let ( X, Y ) be random variables jointly distributed on M × {± 1 } .

Original: Let D n := { ( X i , Y i ) } ni =1 be the (random) training data consisting of n i.i.d copies of X, Y .
Revised: Let D n := { ( X i , Y i ) } ni =1 be the (random) training data consisting of n i.i.d copies of X, Y .

Original: A classifier , i.e., a mapping from D n to a function (cid:98) f ( •∥ D n ) :
Revised: A classifier , i.e., a mapping from D n to a function (cid:98) f ( •∥ D n ) :

Original: M → {± 1 } , has the interpolating-consistent property if, when X has a continuous distribution, both of the following hold: 1) (cid:98) f
Revised: M → {± 1 } , has the interpolating-consistent property if, when X has a continuous distribution, both of the following hold: 1) (cid:98) f

Original: ( X i ∥ D n ) =
Revised: ( X i ∥ D n ) =

Original: Y i , for all i ∈ { 1 , . . . , n } , and 2)
Revised: Y i , for all i ∈ { 1 , . . . , n } , and 2)

Original: [Equation]
Revised: [Equation]

Original: Our goal is to find an interpolating-consistent ensemble of histogram classifiers , to be defined below.
Revised: Our goal is to find an interpolating-consistent ensemble of histogram classifiers , to be defined below.

Original: A partition on M , denoted by P , is a set of subsets of M such that P ∩ P ′ = ∅ for all P, P ′ ∈ P and M = (cid:83)
Revised: A partition on M , denoted by P , is a set of subsets of M such that P ∩ P ′ = ∅ for all P, P ′ ∈ P and M = (cid:83)

Original: P ∈P P .
Revised: P ∈P P .

Original: Given x ∈ M , let P [ x ] denote the unique element P ∈ P such that x ∈ P .
Revised: Given x ∈ M , let P [ x ] denote the unique element P ∈ P such that x ∈ P .

Original: The set of all partitions on a space M is denoted Part ( M ) .
Revised: The set of all partitions on a space M is denoted Part ( M ) .

Original: The histogram classifier with respect to D n over
Revised: The histogram classifier with respect to D n over

Original: P is the sign of the function (cid:98) h ( •∥ D n , P ) :
Revised: P is the sign of the function (cid:98) h ( •∥ D n , P ) :

Original: M → R given by
Revised: M → R given by

Original: [Equation]
Revised: [Equation]

Original: where I is the indicator function.
Revised: where I is the indicator function.

Original: See Figure 1-left panels.
Revised: See Figure 1-left panels.

Original: Definition 1.1.
Revised: Definition 1.1.

Original: A weighted random partition (WRP) over M is a 3-tuple (Θ , P , α ) consisting of (i) parameter space of partitions : a set Θ where P θ ∈ Part ( M ) for each θ ∈ Θ , (ii) random partitions : a probability measure P on Θ , and (iii) weights : a nonnegative function α : Θ → R ≥ 0 that is integrable with respect to the measure P .
Revised: A weighted random partition (WRP) over M is a 3-tuple (Θ , P , α ) consisting of (i) parameter space of partitions : a set Θ where P θ ∈ Part ( M ) for each θ ∈ Θ , (ii) random partitions : a probability measure P on Θ , and (iii) weights : a nonnegative function α : Θ → R ≥ 0 that is integrable with respect to the measure P .

Original: Example 1.2 (Regular partition of the d -cube) .
Revised: Example 1.2 (Regular partition of the d -cube) .

Original: Let M =
Revised: Let M =

Original: [0 , 1] d and Θ = { 1 , 2 . . .
Revised: [0 , 1] d and Θ = { 1 , 2 . . .

Original: =: N + .
Revised: =: N + .

Original: For each n ∈ N + , denote by P n the regular partition of M into n d d -cubes of side length 1 /n .
Revised: For each n ∈ N + , denote by P n the regular partition of M into n d d -cubes of side length 1 /n .

Original: For any probability mass function P on N + and weights α : N + → R ≥ 0 , the 3-tuple (Θ , P , α ) is a WRP.
Revised: For any probability mass function P on N + and weights α : N + → R ≥ 0 , the 3-tuple (Θ , P , α ) is a WRP.

Original: Below, WRPs will be denoted with 2-letter names in the sans-serif font, e.g., “ rp ” for a generic WRP, and “ ha ” for the weighted hyperplane arrangement random partition (Definition 5.1).
Revised: Below, WRPs will be denoted with 2-letter names in the sans-serif font, e.g., “ rp ” for a generic WRP, and “ ha ” for the weighted hyperplane arrangement random partition (Definition 5.1).

Original: The weighted random partition kernel associated to rp =
Revised: The weighted random partition kernel associated to rp =

Original: (Θ , P , α ) is defined as
Revised: (Θ , P , α ) is defined as

Original: [Equation]
Revised: [Equation]

Original: When α ≡ 1 , we recover the notion of unweighted random partition kernel introduced in [20].
Revised: When α ≡ 1 , we recover the notion of unweighted random partition kernel introduced in [21].

Original: Note that the kernel is symmetric since I { x ∈ P θ [ z ] }
Revised: Note that the kernel is symmetric since I { x ∈ P θ [ z ] }

Original: =
Revised: =

Original: I { z ∈ P θ [ x ] } .
Revised: I { z ∈ P θ [ x ] } .

Original: If K rp M < ∞ , then K rp M is a positive definite (PD) kernel.
Revised: If K rp M < ∞ , then K rp M is a positive definite (PD) kernel.

Original: When K rp
Revised: When K rp

Original: M can evaluate to ∞ , the definition of a PD kernel is not applicable since the positive definite property is defined only for to kernels taking finite values [10].
Revised: M can evaluate to ∞ , the definition of a PD kernel is not applicable since the positive definite property is defined only for to kernels taking finite values [10].

Original: Let sgn : R ∪{±∞} → {± 1 } be the sign function.
Revised: Let sgn : R ∪{±∞} → {± 1 } be the sign function.

Original: For a WRP, define the weighted infinite-ensemble
Revised: For a WRP, define the weighted infinite-ensemble

Original: [Equation]
Revised: [Equation]

Original: Note that the equality on the right follows immediately from linearity of the expectation and the definition of (cid:98) h ( •∥ D n , P θ ) in Equation (2).
Revised: Note that the equality on the right follows immediately from linearity of the expectation and the definition of (cid:98) h ( •∥ D n , P θ ) in Equation (2).

Original: See Figure 1-right panel.
Revised: See Figure 1-right panel.

Original: Main problem .
Revised: Main problem .

Original: Find a WRP such that sgn ( (cid:98) u ( •∥ D n , K rp M )) has the interpolating-consistent property.
Revised: Find a WRP such that sgn ( (cid:98) u ( •∥ D n , K rp M )) has the interpolating-consistent property.

Original: 1.2 Outline of approach and contributions
Revised: 1.2 Outline of approach and contributions

Original: In the regression setting, we have ( X, Y ) jointly distributed on M × R .
Revised: In the regression setting, we have ( X, Y ) jointly distributed on M × R .

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

Original: E [ Y | X = x ] .
Revised: E [ Y | X = x ] .

Original: Recall from Belkin et al.
Revised: Recall from Belkin et al.

Original: [8, Equation (7)] the definition of the kernel smoothing estimator with a so-called singular 1 kernel K : M × M → [0 , + ∞ ] :
Revised: [8, Equation (7)] the definition of the kernel smoothing estimator with a so-called singular 1 kernel K : M × M → [0 , + ∞ ] :

Original: [Equation]
Revised: [Equation]

Original: We note that Equation (5) is referred as the Nadaraya-Watson estimate in [8].
Revised: We note that Equation (5) is referred as the Nadaraya-Watson estimate in [8].

Original: Now, we simply write (cid:98) m n ( x ) instead of (cid:98) m ( x ∥ D n , K ) when there is no ambiguity.
Revised: Now, we simply write (cid:98) m n ( x ) instead of (cid:98) m ( x ∥ D n , K ) when there is no ambiguity.

Original: Similarly, we write (cid:98) u n ( x ) instead of (cid:98) u ( x ∥ D n , K ) from earlier.
Revised: Similarly, we write (cid:98) u n ( x ) instead of (cid:98) u ( x ∥ D n , K ) from earlier.

Original: Note that sgn ( (cid:98) m n ( x )) =
Revised: Note that sgn ( (cid:98) m n ( x )) =

Original: sgn ( (cid:98) u n ( x ))
Revised: sgn ( (cid:98) u n ( x ))

Original: if (cid:80) nj
Revised: if (cid:80) nj

Original: [Equation]
Revised: [Equation]

Original: K ( x, X j ) > 0 .
Revised: K ( x, X j ) > 0 .

Original: Observe that (cid:98) m n is interpolating by construction.
Revised: Observe that (cid:98) m n is interpolating by construction.

Original: Let µ X denote the marginal distribution of X .
Revised: Let µ X denote the marginal distribution of X .

Original: The
Revised: The

Original: L 1 -error of (cid:98) m n in approximating m is J n :=
Revised: L 1 -error of (cid:98) m n in approximating m is J n :=

Original: (cid:82) M | (cid:98) m n ( x ) − m ( x ) | µ X ( dx ) .
Revised: (cid:82) M | (cid:98) m n ( x ) − m ( x ) | µ X ( dx ) .

Original: For M = R d and the
Revised: For M = R d and the

Original: Hilbert kernel defined by K H R d ( x, z ) :
Revised: Hilbert kernel defined by K H R d ( x, z ) :

Original: =
Revised: =

Original: ∥ x − z
Revised: ∥ x − z

Original: ∥ − d , Devroye et al.
Revised: ∥ − d , Devroye et al.

Original: [21] proved L 1 -consistency for regression : J n → 0 in probability when Y is bounded and X is continuously distributed.
Revised: [22] proved L 1 -consistency for regression : J n → 0 in probability when Y is bounded and X is continuously distributed.

Original: Our contributions.
Revised: Our contributions.

Original: Our primary contribution is to demonstrate an ensemble method with the consistent-interpolating property.
Revised: Our primary contribution is to demonstrate an ensemble method with the consistent-interpolating property.

Original: Toward this end, in Section 3, we introduce the manifold-Hilbert kernel K H M on a Riemannian manifold M .
Revised: Toward this end, in Section 3, we introduce the manifold-Hilbert kernel K H M on a Riemannian manifold M .

Original: When show that when M is complete, connected, and smooth, kernel smoothing regression with K H M has the same consistency guarantee (Theorem 3.2) as
Revised: When show that when M is complete, connected, and smooth, kernel smoothing regression with K H M has the same consistency guarantee (Theorem 3.2) as

Original: K H R d mentioned in the preceding paragraph.
Revised: K H R d mentioned in the preceding paragraph.

Original: In Section 5, we consider the case when M = S d , and show that the manifold-Hilbert kernel K H S d is a weighted random partition kernel (Proposition 5.2).
Revised: In Section 5, we consider the case when M = S d , and show that the manifold-Hilbert kernel K H S d is a weighted random partition kernel (Proposition 5.2).

Original: Devroye et al.
Revised: Devroye et al.

Original: [21, Section 7] observed that the L 1 -consistency of (cid:98) m n for regression implies the consistency for classification of sgn ◦ (cid:98) u n .
Revised: [22, Section 7] observed that the L 1 -consistency of (cid:98) m n for regression implies the consistency for classification of sgn ◦ (cid:98) u n .

Original: Furthermore, (cid:98) m n is interpolating for regression implies that sgn ◦ (cid:98) u n is interpolating for classification.
Revised: Furthermore, (cid:98) m n is interpolating for regression implies that sgn ◦ (cid:98) u n is interpolating for classification.

Original: These observations together with our results demonstrate the existence of a weighted infinite-ensemble classifier with the interpolating-consistent property.
Revised: These observations together with our results demonstrate the existence of a weighted infinite-ensemble classifier with the interpolating-consistent property.

Original: 1.3 Related work
Revised: 1.3 Related work

Original: Kernel regression.
Revised: Kernel regression.

Original: Kernel smoothing regression, or simply kernel regression, is an interpolator when the kernel used is singular, a fact known to Shepard [39] in 1968.
Revised: Kernel smoothing regression, or simply kernel regression, is an interpolator when the kernel used is singular, a fact known to Shepard [39] in 1968.

Original: Devroye et al.
Revised: Devroye et al.

Original: [21] showed that kernel regression with the Hilbert kernel is interpolating and weakly consistent for data with a density and bounded labels.
Revised: [22] showed that kernel regression with the Hilbert kernel is interpolating and weakly consistent for data with a density and bounded labels.

Original: Using singular kernels with compact support, Belkin et al.
Revised: Using singular kernels with compact support, Belkin et al.

Original: [8] showed that minimax optimality can be achieved under additional distributional assumptions.
Revised: [8] showed that minimax optimality can be achieved under additional distributional assumptions.

Original: Random forests.
Revised: Random forests.

Original: Wyner et al.
Revised: Wyner et al.

Original: [42] proposed that interpolation may be a key mechanism for the success of random forests and gave a compelling intuitive rationale.
Revised: [42] proposed that interpolation may be a key mechanism for the success of random forests and gave a compelling intuitive rationale.

Original: Belkin et al.
Revised: Belkin et al.

Original: [6] studied empirically the double descent phenomenon in random forests by considering the generalization performance past the interpolation threshold.
Revised: [6] studied empirically the double descent phenomenon in random forests by considering the generalization performance past the interpolation threshold.

Original: The PERT variant of random forests, introduced by
Revised: The PERT variant of random forests, introduced by

Original: Cutler and Zhao [19], provably interpolates in 1-dimension.
Revised: Cutler and Zhao [20], provably interpolates in 1-dimension.

Original: Belkin et al.
Revised: Belkin et al.

Original: [7] pose as an interesting question whether the result of Cutler and Zhao [19] extends to higher dimension.
Revised: [7] pose as an interesting question whether the result of Cutler and Zhao [20] extends to higher dimension.

Original: Many work have established consistency of random forest and its variants under different settings [15, 11, 38].
Revised: Many work have established consistency of random forest and its variants under different settings [15, 11, 38].

Original: However, none of these work addressed interpolation.
Revised: However, none of these work addressed interpolation.

Original: Boosting.
Revised: Boosting.

Original: For classification under the noiseless setting (i.e., the Bayes error is zero), AdaBoost is interpolating and consistent (see Freund and Schapire [25, first paragraph of Chapter 12]).
Revised: For classification under the noiseless setting (i.e., the Bayes error is zero), AdaBoost is interpolating and consistent (see Freund and Schapire [26, first paragraph of Chapter 12]).

Original: However, this setting is too restrictive and the result does not answer if consistency is possible when fitting the noise.
Revised: However, this setting is too restrictive and the result does not answer if consistency is possible when fitting the noise.

Original: Bartlett and Traskin [5] proved that AdaBoost with early stopping is universally consistent, however without the interpolation guarantee.
Revised: Bartlett and Traskin [5] proved that AdaBoost with early stopping is universally consistent, however without the interpolation guarantee.

Original: To the best of our knowledge, whether AdaBoost or any other variant of boosting can be interpolating and consistent remains open.
Revised: To the best of our knowledge, whether AdaBoost or any other variant of boosting can be interpolating and consistent remains open.

Original: Random partition kernels.
Revised: Random partition kernels.

Original: Breiman [16] and Geurts et al.
Revised: Breiman [16] and Geurts et al.

Original: [27] studied infinite ensembles of simplified variants of random forest and connections to certain kernels.
Revised: [28] studied infinite ensembles of simplified variants of random forest and connections to certain kernels.

Original: Davies and Ghahramani [20] formalized this connection and coined the term random partition kernel .
Revised: Davies and Ghahramani [21] formalized this connection and coined the term random partition kernel .

Original: Scornet [37] further developed the theory of random forest kernels and obtained upper bounds on the rate of convergence.
Revised: Scornet [37] further developed the theory of random forest kernels and obtained upper bounds on the rate of convergence.

Original: However, it is not clear if these variants of random forests are interpolating.
Revised: However, it is not clear if these variants of random forests are interpolating.

Original: Previously defined (unweighted) random partition kernels are bounded, and thus cannot be singular.
Revised: Previously defined (unweighted) random partition kernels are bounded, and thus cannot be singular.

Original: On the other hand, the manifold-Hilbert kernel is always singular.
Revised: On the other hand, the manifold-Hilbert kernel is always singular.

Original: To bridge between ensemble methods and theory on interpolating kernel smoothing regression, we propose weighted random partitions (Definition 1.1), whose associated kernel (Equation 3) can be singular.
Revised: To bridge between ensemble methods and theory on interpolating kernel smoothing regression, we propose weighted random partitions (Definition 1.1), whose associated kernel (Equation 3) can be singular.

Original: Learning on Riemannian manifolds.
Revised: Learning on Riemannian manifolds.

Original: Strong consistency of a kernel-based classification method on manifolds has been established by Loubes and Pelletier [31].
Revised: Strong consistency of a kernel-based classification method on manifolds has been established by Loubes and Pelletier [32].

Original: However, the result requires the kernel to be bounded and thus the method is not guaranteed to be interpolating.
Revised: However, the result requires the kernel to be bounded and thus the method is not guaranteed to be interpolating.

Original: See Feragen and Hauberg [24] for a review of theoretical results regarding kernels on Riemannian manifolds.
Revised: See Feragen and Hauberg [25] for a review of theoretical results regarding kernels on Riemannian manifolds.

Original: Beyond kernel methods, other classical methods for Euclidean data have been extended to Riemannian manifolds, e.g., regression [40], classification [43], and dimensionality reduction and clustering [44][33].
Revised: Beyond kernel methods, other classical methods for Euclidean data have been extended to Riemannian manifolds, e.g., regression [40], classification [43], and dimensionality reduction and clustering [44][34].

Original: To the best of our knowledge, no previous works have demonstrated an interpolatingconsistent classifiers on manifolds other than R d .
Revised: To the best of our knowledge, no previous works have demonstrated an interpolatingconsistent classifiers on manifolds other than R d .

Original: In many applications, the data naturally belong to a Riemannian manifold.
Revised: In many applications, the data naturally belong to a Riemannian manifold.

Original: Spherical data arise from a range of disciplines in natural sciences.
Revised: Spherical data arise from a range of disciplines in natural sciences.

Original: See the influential textbook by Mardia and Jupp [32,
Revised: See the influential textbook by Mardia and Jupp [33,

Original: Ch.1§4].
Revised: Ch.1§4].

Original: For applications of the Grassmanian manifold in computer vision, see Jayasumana et al.
Revised: For applications of the Grassmanian manifold in computer vision, see Jayasumana et al.

Original: [29] and the references therein.
Revised: [30] and the references therein.

Original: Topological data analysis [41] presents another interesting setting of manifold-valued data in the form of persistence diagrams [3, 30].
Revised: Topological data analysis [41] presents another interesting setting of manifold-valued data in the form of persistence diagrams [3, 31].

Original:  Background on Riemannian Manifolds
Revised:  Background on Riemannian Manifolds

Original: We give an intuitive overview of the necessary concepts and results on Riemannian manifolds.
Revised: We give an intuitive overview of the necessary concepts and results on Riemannian manifolds.

Original: A longer, more precise version of this overview is in the Supplemental Materials Section A.1.
Revised: A longer, more precise version of this overview is in the Supplemental Materials Section A.1.

Original: A smooth d -dimensional manifold M is a topological space that is locally diffeomorphic 2 to open subsets of R d .
Revised: A smooth d -dimensional manifold M is a topological space that is locally diffeomorphic 2 to open subsets of R d .

Original: For simplicity, suppose that M is embedded in R N for some N ≥ d , e.g., S d ⊆ R d +1 .
Revised: For simplicity, suppose that M is embedded in R N for some N ≥ d , e.g., S d ⊆ R d +1 .

Original: Let x ∈ M be a point.
Revised: Let x ∈ M be a point.

Original: The tangent space at x , denoted T x M , is the set of vectors that is tangent to M at x .
Revised: The tangent space at x , denoted T x M , is the set of vectors that is tangent to M at x .

Original: Since linear combinations of tangent vectors are also tangent, the tangent space T x M is a vector space.
Revised: Since linear combinations of tangent vectors are also tangent, the tangent space T x M is a vector space.

Original: Tangent vectors can also be viewed as the time derivative of smooth curves.
Revised: Tangent vectors can also be viewed as the time derivative of smooth curves.

Original: In particular, let x ∈ M .
Revised: In particular, let x ∈ M .

Original: If ϵ > 0 is an open set and γ : ( − ϵ, ϵ ) → M is a smooth curve such that γ (0) =
Revised: If ϵ > 0 is an open set and γ : ( − ϵ, ϵ ) → M is a smooth curve such that γ (0) =

Original: x , then dγdt (0)
Revised: x , then dγdt (0)

Original: ∈
Revised: ∈

Original: T x M .
Revised: T x M .

Original: A Riemannian metric on M is a choice of inner product ⟨· , ·⟩ x on T x M for each x such that ⟨· , ·⟩ x varies smoothly with x .
Revised: A Riemannian metric on M is a choice of inner product ⟨· , ·⟩ x on T x M for each x such that ⟨· , ·⟩ x varies smoothly with x .

Original: Naturally,
Revised: Naturally,

Original: ∥ z
Revised: ∥ z

Original: ∥ x := (cid:112) ⟨ z, z ⟩ x defines a norm on T x M .
Revised: ∥ x := (cid:112) ⟨ z, z ⟩ x defines a norm on T x M .

Original: The length of a piecewise smooth curve γ : [ a, b ] → M is defined by len ( γ ) :
Revised: The length of a piecewise smooth curve γ : [ a, b ] → M is defined by len ( γ ) :

Original: = (cid:82) b a ∥ ˙
Revised: = (cid:82) b a ∥ ˙

Original: γ ( t ) ∥ γ ( t ) dt .
Revised: γ ( t ) ∥ γ ( t ) dt .

Original: Define dist M ( x, ξ ) := inf { len ( γ ) : γ is a piecewise smooth curve from x to ξ } , which is a metric on M in the sense of metric spaces (see Sakai [36, Proposition 1.1]).
Revised: Define dist M ( x, ξ ) := inf { len ( γ ) : γ is a piecewise smooth curve from x to ξ } , which is a metric on M in the sense of metric spaces (see Sakai [36, Proposition 1.1]).

Original: For x ∈ M and r ∈ (0 , ∞ ) , the open metric ball centered at x of radius r is denoted B x
Revised: For x ∈ M and r ∈ (0 , ∞ ) , the open metric ball centered at x of radius r is denoted B x

Original: ( r, M ) := { ξ ∈ M : dist M ( x, ξ ) < r } .
Revised: ( r, M ) := { ξ ∈ M : dist M ( x, ξ ) < r } .

Original: A curve γ : [ a, b ] → M is a geodesic if γ is locally distance minimizing and has constant speed, i.e., ∥ dγdt ( τ ) ∥ γ ( τ ) is constant.
Revised: A curve γ : [ a, b ] → M is a geodesic if γ is locally distance minimizing and has constant speed, i.e., ∥ dγdt ( τ ) ∥ γ ( τ ) is constant.

Original: Now, suppose x ∈ M and v ∈
Revised: Now, suppose x ∈ M and v ∈

Original: T x M are such that there exists a geodesic γ : [0 , 1] → M where γ (0)
Revised: T x M are such that there exists a geodesic γ : [0 , 1] → M where γ (0)

Original: =
Revised: =

Original: x and dγdt (0) = v .
Revised: x and dγdt (0) = v .

Original: Define exp x ( v ) :
Revised: Define exp x ( v ) :

Original: = γ (1) , the element reached by traveling along γ at time = 1 . See Figure 2 for the case when M = S 2 .
Revised: = γ (1) , the element reached by traveling along γ at time = 1 . See Figure 2 for the case when M = S 2 .

Original: For a fixed x ∈ M , the above function exp x , the exponential map , can be defined on an open subset of T x M containing the origin.
Revised: For a fixed x ∈ M , the above function exp x , the exponential map , can be defined on an open subset of T x M containing the origin.

Original: The Hopf-Rinow theorem ([22, Ch. 8, Theorem 2.8]) states that if
Revised: The Hopf-Rinow theorem ([23, Ch. 8, Theorem 2.8]) states that if

Original: M is connected and complete with respect to the metric dist M , then exp x can be defined on all of
Revised: M is connected and complete with respect to the metric dist M , then exp x can be defined on all of

Original: T x M .
Revised: T x M .

Original:  The Manifold-Hilbert kernel
Revised:  The Manifold-Hilbert kernel

Original: Throughout the remainder of this work, we assume that M is a complete, connected, and smooth
Revised: Throughout the remainder of this work, we assume that M is a complete, connected, and smooth

Original: Riemannian manifold of dimension d .
Revised: Riemannian manifold of dimension d .

Original: Definition 3.1.
Revised: Definition 3.1.

Original: We define the manifold-Hilbert kernel K H M : M × M → [0 , ∞ ] for each x, ξ ∈ M by K H M ( x, ξ ) :
Revised: We define the manifold-Hilbert kernel K H M : M × M → [0 , ∞ ] for each x, ξ ∈ M by K H M ( x, ξ ) :

Original: = dist M ( x, ξ )
Revised: = dist M ( x, ξ )

Original: − d
Revised: − d

Original: if x ̸ = ξ and K H M ( x, x ) := ∞ otherwise.
Revised: if x ̸ = ξ and K H M ( x, x ) := ∞ otherwise.

Original: [Figure]
Revised: [Figure]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

Original: [Figure]
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

Original: Let λ M be the Riemann–Lebesgue volume measure of M .
Revised: Let λ M be the Riemann–Lebesgue volume measure of M .

Original: Integration with respect to this measure is denoted (cid:82) M fdλ M for a function f : M → R .
Revised: Integration with respect to this measure is denoted (cid:82) M fdλ M for a function f : M → R .

Original: For details of the construction of λ M , see Amann and
Revised: For details of the construction of λ M , see Amann and

Original: Escher [1, Proposition 1.5].
Revised: Escher [1, Proposition 1.5].

Original: When M = R d , λ M is the ordinary Lebesgue measure and (cid:82) R d fdλ R d is the ordinary Lebesgue integral.
Revised: When M = R d , λ M is the ordinary Lebesgue measure and (cid:82) R d fdλ R d is the ordinary Lebesgue integral.

Original: For this case, we simply write λ instead of λ R d .
Revised: For this case, we simply write λ instead of λ R d .

Original: We now state our first main result, a manifold theory extension of Devroye et al.
Revised: We now state our first main result, a manifold theory extension of Devroye et al.

Original: [21, Theorem 1].
Revised: [22, Theorem 1].

Original: Theorem 3.2.
Revised: Theorem 3.2.

Original: Suppose that X has a density f
Revised: Suppose that X has a density f

Original: X with respect to λ M and that Y is bounded.
Revised: X with respect to λ M and that Y is bounded.

Original: Let
Revised: Let

Original: P Y | X be a conditional distribution of Y given X and m Y |
Revised: P Y | X be a conditional distribution of Y given X and m Y |

Original: X be its conditional expectation.
Revised: X be its conditional expectation.

Original: Let (cid:98) m n ( x ) :
Revised: Let (cid:98) m n ( x ) :

Original: = (cid:98) m ( x ∥ D n , K H M ) .
Revised: = (cid:98) m ( x ∥ D n , K H M ) .

Original: Then
Revised: Then

Original: at almost all x ∈ M with f X ( x ) > 0 , we have (cid:98) m n ( x ) → m Y | X
Revised: at almost all x ∈ M with f X ( x ) > 0 , we have (cid:98) m n ( x ) → m Y | X

Original: ( x ) in probability, 2.
Revised: ( x ) in probability, 2.

Original: J n := (cid:82) M | (cid:98) m n ( x ) − m Y | X ( x ) |
Revised: J n := (cid:82) M | (cid:98) m n ( x ) − m Y | X ( x ) |

Original: f X ( x ) dλ M ( x ) → 0 in probability.
Revised: f X ( x ) dλ M ( x ) → 0 in probability.

Original: In words, the kernel smoothing regression estimate (cid:98) m n based on the manifold-Hilbert kernel is consistent and interpolates the training data, provided X has a density and Y is bounded.
Revised: In words, the kernel smoothing regression estimate (cid:98) m n based on the manifold-Hilbert kernel is consistent and interpolates the training data, provided X has a density and Y is bounded.

Original: As a consequence, following the same logic as in Devroye et al.
Revised: As a consequence, following the same logic as in Devroye et al.

Original: [21], the associated classifier sgn ◦ (cid:98) u n has the interpolating-consistent property.
Revised: [22], the associated classifier sgn ◦ (cid:98) u n has the interpolating-consistent property.

Original: Before proving Theorem 3.2, we first review key concepts in probability theory on Riemannian manifolds.
Revised: Before proving Theorem 3.2, we first review key concepts in probability theory on Riemannian manifolds.

Original: 3.1 Probability on Riemannian manifolds
Revised: 3.1 Probability on Riemannian manifolds

Original: Let B M be the Borel σ -algebra of M , i.e., the smallest σ -algebra containing all open subsets of M .
Revised: Let B M be the Borel σ -algebra of M , i.e., the smallest σ -algebra containing all open subsets of M .

Original: We recall the definition of M -valued random variables, following Pennec [34, Definition 2]:
Revised: We recall the definition of M -valued random variables, following Pennec [35, Definition 2]:

Original: Definition 3.3.
Revised: Definition 3.3.

Original: Let (Ω , P , A ) be a probability space with measure P and σ -algebra A .
Revised: Let (Ω , P , A ) be a probability space with measure P and σ -algebra A .

Original: A M -valued random variable
Revised: A M -valued random variable

Original: X is a Borel-measurable function Ω → M , i.e., X − 1 ( B ) ∈
Revised: X is a Borel-measurable function Ω → M , i.e., X − 1 ( B ) ∈

Original: A for all B ∈ B M .
Revised: A for all B ∈ B M .

Original: Definition 3.4 (Density) .
Revised: Definition 3.4 (Density) .

Original: A random variable X taking values in M has a density if there exists a nonnegative Borel-measurable function f : M → [0 , ∞ ] such that for all Borel sets B in M , we have
Revised: A random variable X taking values in M has a density if there exists a nonnegative Borel-measurable function f : M → [0 , ∞ ] such that for all Borel sets B in M , we have

Original: Pr( X ∈ B ) = (cid:82) B fdλ M .
Revised: Pr( X ∈ B ) = (cid:82) B fdλ M .

Original: The function f is said to be a probability density function (PDF) of X .
Revised: The function f is said to be a probability density function (PDF) of X .

Original: Next, we recall the definition of conditional distributions, following Dudley [23, Ch. 10 §2]:
Revised: Next, we recall the definition of conditional distributions, following Dudley [24, Ch. 10 §2]:

Original: Definition 3.5 (Conditional distribution 3 ) .
Revised: Definition 3.5 (Conditional distribution 3 ) .

Original: Let ( X, Y ) be a random variable jointly distributed on
Revised: Let ( X, Y ) be a random variable jointly distributed on

Original: M × R .
Revised: M × R .

Original: Let P X ( · ) be the probability measure corresponding to the marginal distribution of X .
Revised: Let P X ( · ) be the probability measure corresponding to the marginal distribution of X .

Original: A conditional distribution for Y given X is a collection of probability measures P Y | X ( ·| x ) on R indexed by x ∈ M satisfying the following:
Revised: A conditional distribution for Y given X is a collection of probability measures P Y | X ( ·| x ) on R indexed by x ∈ M satisfying the following:

Original: For all Borel sets A ⊆ R , the function M ∋ x
Revised: For all Borel sets A ⊆ R , the function M ∋ x

Original: (cid:55)→ P Y | X ( A | x ) ∈ [0 , 1] is Borel-measurable.
Revised: (cid:55)→ P Y | X ( A | x ) ∈ [0 , 1] is Borel-measurable.

Original: For all A ⊆ R and B ⊆ M Borel sets, Pr( Y ∈ A, X ∈ B ) =
Revised: For all A ⊆ R and B ⊆ M Borel sets, Pr( Y ∈ A, X ∈ B ) =

Original: (cid:82) B P Y | X ( A | x ) P X ( dx ) .
Revised: (cid:82) B P Y | X ( A | x ) P X ( dx ) .

Original: The conditional expectation 4 is defined as
Revised: The conditional expectation 4 is defined as

Original: [Equation]
Revised: [Equation]

Original: The existence of a conditional probability for a joint distribution ( X, Y ) is guaranteed by Dudley [23, Theorem 10.2.2].
Revised: The existence of a conditional probability for a joint distribution ( X, Y ) is guaranteed by Dudley [24, Theorem 10.2.2].

Original: When ( X, Y ) has a joint density f XY and marginal density f X , the above definition gives the classical formula P Y | X
Revised: When ( X, Y ) has a joint density f XY and marginal density f X , the above definition gives the classical formula P Y | X

Original: ( A | x ) = (cid:82) A f XY ( x, y ) /f
Revised: ( A | x ) = (cid:82) A f XY ( x, y ) /f

Original: X ( x ) dy when ∞ > f X ( x ) > 0 .
Revised: X ( x ) dy when ∞ > f X ( x ) > 0 .

Original: See the first example in Dudley [23, Ch. 10 §2].
Revised: See the first example in Dudley [24, Ch. 10 §2].

Original: 3.2 Lebesgue points on manifolds
Revised: 3.2 Lebesgue points on manifolds

Original: Devroye et al.
Revised: Devroye et al.

Original: [21] proved Theorem 3.2 when M = R d and, moreover, that part 1 holds for the so-called Lebesgue points , whose definition we now recall.
Revised: [22] proved Theorem 3.2 when M = R d and, moreover, that part 1 holds for the so-called Lebesgue points , whose definition we now recall.

Original: Definition 3.6.
Revised: Definition 3.6.

Original: Let f : M → R be an absolutely integrable function and x ∈ M .
Revised: Let f : M → R be an absolutely integrable function and x ∈ M .

Original: We say that x is a
Revised: We say that x is a

Original: Lebesgue point of f if f ( x ) =
Revised: Lebesgue point of f if f ( x ) =

Original: lim r →
Revised: lim r →

Original: [Equation]
Revised: [Equation]

Original: For an integrable function, the following result states that almost all points are its Lebesgue points.
Revised: For an integrable function, the following result states that almost all points are its Lebesgue points.

Original: For the proof, see Fukuoka [26, Remark 2.4].
Revised: For the proof, see Fukuoka [27, Remark 2.4].

Original: Theorem 3.7 (Lebesgue differentation) .
Revised: Theorem 3.7 (Lebesgue differentation) .

Original: Let f : M → R be an absolutely integrable function.
Revised: Let f : M → R be an absolutely integrable function.

Original: Then there exists a set A ⊆ M such that λ M ( A ) = 0 and every x ∈ M \ A is a Lebesgue point of f .
Revised: Then there exists a set A ⊆ M such that λ M ( A ) = 0 and every x ∈ M \ A is a Lebesgue point of f .

Original: Next, for the reader’s convenience, we restate Devroye et al.
Revised: Next, for the reader’s convenience, we restate Devroye et al.

Original: [21, Theorem 1], emphasizing the connection to Lebesgue points.
Revised: [22, Theorem 1], emphasizing the connection to Lebesgue points.

Original: The result will be used in our proof of Theorem 3.2 in the next section.
Revised: The result will be used in our proof of Theorem 3.2 in the next section.

Original: Theorem 3.8 (Devroye et al.
Revised: Theorem 3.8 (Devroye et al.

Original: Let M
Revised: Let M

Original: =
Revised: =

Original: R
Revised: R

Original: d be the flat Euclidean space.
Revised: d be the flat Euclidean space.

Original: Then Theorem 3. holds.
Revised: Then Theorem 3. holds.

Original: Moreover, Part 1 holds for all x that is a Lebesgue point to both f X and m Y | X · f X .
Revised: Moreover, Part 1 holds for all x that is a Lebesgue point to both f X and m Y | X · f X .

Original:  Proof of Theorem 3. 
Revised:  Proof of Theorem 3.

Original: The focal point of the first subsection is Lemma 4.1 which shows the Borel measurability of extensions of the so-called Riemannian logarithm.
Revised: The focal point of the first subsection is Lemma 4.1 which shows the Borel measurability of extensions of the so-called Riemannian logarithm.

Original: The second subsection contains two key results regarding densities of M -valued random variables transformed by the Riemannian logarithm.
Revised: The second subsection contains two key results regarding densities of M -valued random variables transformed by the Riemannian logarithm.

Original: The final subsection proves Theorem 3.2 leveraging results from the preceding two subsections.
Revised: The final subsection proves Theorem 3.2 leveraging results from the preceding two subsections.

Original: 4.1 The Riemannian logarithm
Revised: 4.1 The Riemannian logarithm

Original: Throughout, x is assumed to be an arbitrary point of M .
Revised: Throughout, x is assumed to be an arbitrary point of M .

Original: Let U x M = { v ∈ T x M : ∥ v ∥ x = 1 } ⊆
Revised: Let U x M = { v ∈ T x M : ∥ v ∥ x = 1 } ⊆

Original: T x M denote the set of unit tangent vectors.
Revised: T x M denote the set of unit tangent vectors.

Original: Define a function τ x : U x M → (0 , ∞ ] as follows 5 :
Revised: Define a function τ x : U x M → (0 , ∞ ] as follows 5 :

Original: [Equation]
Revised: [Equation]

Original: The tangent cut locus is the set
Revised: The tangent cut locus is the set

Original: ˜ C x ⊆ T x M defined by ˜
Revised: ˜ C x ⊆ T x M defined by ˜

Original: C x := { τ x ( u ) u : u ∈ U x M, τ x ( u ) < ∞} .
Revised: C x := { τ x ( u ) u : u ∈ U x M, τ x ( u ) < ∞} .

Original: Note that it is possible for τ x ( u ) = ∞ for all u ∈ U x M in which case ˜ C x is empty.
Revised: Note that it is possible for τ x ( u ) = ∞ for all u ∈ U x M in which case ˜ C x is empty.

Original: The cut locus is the set C x := exp x ( ˜ C x ) ⊆ M .
Revised: The cut locus is the set C x := exp x ( ˜ C x ) ⊆ M .

Original: The tangent interior set is ˜ I x := { tu : 0 ≤ t < τ x ( u ) , u ∈ U x M } and the interior set is the set
Revised: The tangent interior set is ˜ I x := { tu : 0 ≤ t < τ x ( u ) , u ∈ U x M } and the interior set is the set

Original: I x := exp x (˜ I x ) .
Revised: I x := exp x (˜ I x ) .

Original: Finally, define ˜ D
Revised: Finally, define ˜ D

Original: x := ˜
Revised: x := ˜

Original: I x ∪ ˜ C x .
Revised: I x ∪ ˜ C x .

Original: Note that for each z = tu ∈
Revised: Note that for each z = tu ∈

Original: ˜
Revised: ˜

Original: I x , we have
Revised: I x , we have

Original: [Equation]
Revised: [Equation]

Original: Consider the example where M = S 2 as in Figure 2.
Revised: Consider the example where M = S 2 as in Figure 2.

Original: Then τ x ( u ) =
Revised: Then τ x ( u ) =

Original: π for all u ∈ U x M .
Revised: π for all u ∈ U x M .

Original: Thus, the tangent interior set ˜ I
Revised: Thus, the tangent interior set ˜ I

Original: x = B 0 ( π, R 2 ) , the open disc of radius π centered at the origin.
Revised: x = B 0 ( π, R 2 ) , the open disc of radius π centered at the origin.

Original: When restricted to ˜ I x , the exponential map exp x | ˜ I x : ˜ I x
Revised: When restricted to ˜ I x , the exponential map exp x | ˜ I x : ˜ I x

Original: → I x is a diffeomorphism.
Revised: → I x is a diffeomorphism.

Original: Its functional inverse, denoted by log x | I x , is called the Riemannian Logarithm [9, 45].
Revised: Its functional inverse, denoted by log x | I x , is called the Riemannian Logarithm [9, 45].

Original: In previous works, log x | I x is only defined from I x to ˜ I x .
Revised: In previous works, log x | I x is only defined from I x to ˜ I x .

Original: The next result shows that the domain of log x | I x : I x → ˜ I x can be extended to log x : M → ˜ D x while remaining Borel-measurable.
Revised: The next result shows that the domain of log x | I x : I x → ˜ I x can be extended to log x : M → ˜ D x while remaining Borel-measurable.

Original: Lemma 4.1.
Revised: Lemma 4.1.

Original: For all x ∈ M , there exists a Borel measurable map log x : M → T x M such that log x ( M ) ⊆
Revised: For all x ∈ M , there exists a Borel measurable map log x : M → T x M such that log x ( M ) ⊆

Original: ˜ D x and exp x ◦ log x is the identity on M .
Revised: ˜ D x and exp x ◦ log x is the identity on M .

Original: Furthermore, for all x, ξ ∈ M , we have dist M ( x, ξ ) =
Revised: Furthermore, for all x, ξ ∈ M , we have dist M ( x, ξ ) =

Original: ∥ log x ( ξ ) ∥ x .
Revised: ∥ log x ( ξ ) ∥ x .

Original: Proof sketch.
Revised: Proof sketch.

Original: The full proof of the lemma is provided in Section A.2 of the Supplemental Materials 6 .
Revised: The full proof of the lemma is provided in Section A.2 of the Supplemental Materials 6 .

Original: Below, we illustrate the idea of the proof using the example when M = S 2 as in Figure 2.
Revised: Below, we illustrate the idea of the proof using the example when M = S 2 as in Figure 2.

Original: Let x ∈ S 2 be the “northpole” (the blue point).
Revised: Let x ∈ S 2 be the “northpole” (the blue point).

Original: The tangent cut locus ˜ C x is the dashed circle in the left panel of Figure 2.
Revised: The tangent cut locus ˜ C x is the dashed circle in the left panel of Figure 2.

Original: The exponential map exp x is one-to-one on ˜ D
Revised: The exponential map exp x is one-to-one on ˜ D

Original: x
Revised: x

Original: except on the dashed circle, which all gets mapped to − x , the “southpole” (the orange point).
Revised: except on the dashed circle, which all gets mapped to − x , the “southpole” (the orange point).

Original: A consequence of the measurable selection theorem 7 is that log x can be extended to be a Borel-measurable right inverse of exp x by selecting z point on ˜ C x such that log x ( − x ) =
Revised: A consequence of the measurable selection theorem 7 is that log x can be extended to be a Borel-measurable right inverse of exp x by selecting z point on ˜ C x such that log x ( − x ) =

Original: z .
Revised: z .

Original: Thus, we’ve shown that log x : M → T x M is Borel-measurable.
Revised: Thus, we’ve shown that log x : M → T x M is Borel-measurable.

Original: Now, recall that T x M is equipped with the inner product ⟨· , ·⟩ x , i.e., the Riemannian metric.
Revised: Now, recall that T x M is equipped with the inner product ⟨· , ·⟩ x , i.e., the Riemannian metric.

Original: Below, for each x ∈ M choose an orthonormal basis on T x M with respect to ⟨· , ·⟩ .
Revised: Below, for each x ∈ M choose an orthonormal basis on T x M with respect to ⟨· , ·⟩ .

Original: Then T x M is isomorphic as an inner product space to R d with the usual dot product.
Revised: Then T x M is isomorphic as an inner product space to R d with the usual dot product.

Original: Our next two results are “change-of-variables formulas” for computing the densities/conditional distributions of M -valued random variables after the log x transform.
Revised: Our next two results are “change-of-variables formulas” for computing the densities/conditional distributions of M -valued random variables after the log x transform.

Original: Recall that λ M is the RiemannLebesgue measure on M and λ is the ordinary Lebesgue measure on R d = T x M .
Revised: Recall that λ M is the RiemannLebesgue measure on M and λ is the ordinary Lebesgue measure on R d = T x M .

Original: Proposition 4.2.
Revised: Proposition 4.2.

Original: Let x ∈ M be fixed.
Revised: Let x ∈ M be fixed.

Original: There exists a Borel measurable function ν x : M → R with the following properties:
Revised: There exists a Borel measurable function ν x : M → R with the following properties:

Original: (i) Let X be a random variable on M with density f X and let Z :=
Revised: (i) Let X be a random variable on M with density f X and let Z :=

Original: log x ( X ) .
Revised: log x ( X ) .

Original: Then Z is a random variable on T x M with density f Z ( z ) :
Revised: Then Z is a random variable on T x M with density f Z ( z ) :

Original: = f X (exp x ( z )) ·
Revised: = f X (exp x ( z )) ·

Original: ν x (exp x ( z )) .
Revised: ν x (exp x ( z )) .

Original: (ii) Let f : M → R be an absolutely integrable function such that x is a Lebesgue point of f .
Revised: (ii) Let f : M → R be an absolutely integrable function such that x is a Lebesgue point of f .

Original: Define f : T x M → R by h ( z ) :
Revised: Define f : T x M → R by h ( z ) :

Original: = f
Revised: = f

Original: (exp x ( z )) · ν x
Revised: (exp x ( z )) · ν x

Original: (exp x ( z )) .
Revised: (exp x ( z )) .

Original: Then 0 ∈ T x M is a Lebesgue point for h .
Revised: Then 0 ∈ T x M is a Lebesgue point for h .

Original: Proposition 4.3.
Revised: Proposition 4.3.

Original: Let ( X, Y ) have a joint distribution on M × R such that the marginal of X has a density f X on M .
Revised: Let ( X, Y ) have a joint distribution on M × R such that the marginal of X has a density f X on M .

Original: Let P Y | X ( ·|· ) be a conditional distribution for Y given X .
Revised: Let P Y | X ( ·|· ) be a conditional distribution for Y given X .

Original: Let x ∈ M .
Revised: Let x ∈ M .

Original: Define Z := log x ( X ) and consider the joint distribution ( Z, Y ) on T p M × R .
Revised: Define Z := log x ( X ) and consider the joint distribution ( Z, Y ) on T p M × R .

Original: Then P Y | Z ( ·|· ) :
Revised: Then P Y | Z ( ·|· ) :

Original: =
Revised: =

Original: P Y | X ( ·| exp x ( · )) is a conditional distribution for Y given Z .
Revised: P Y | X ( ·| exp x ( · )) is a conditional distribution for Y given Z .

Original: Consequently, m Y | X ◦ exp x
Revised: Consequently, m Y | X ◦ exp x

Original: = m Y
Revised: = m Y

Original: | Z .
Revised: | Z .

Original: The above propositions are straightforward manifold-theoretic extensions of well-known results on
Revised: The above propositions are straightforward manifold-theoretic extensions of well-known results on

Original: Euclidean spaces.
Revised: Euclidean spaces.

Original: For completeness, the full proofs are in Supplemental Materials Section A.4.
Revised: For completeness, the full proofs are in Supplemental Materials Section A.4.

Original: An anonymous reviewer brought to our attention that Proposition 4.2 is the consequence of a well-known formula from geometric measure theory, called the area formula [2, p. 44-45].
Revised: An anonymous reviewer brought to our attention that Proposition 4.2 is the consequence of a well-known formula from geometric measure theory, called the area formula [2, p. 44-45].

Original: 4. Proof of Theorem 3. 
Revised: 4. Proof of Theorem 3.

Original: Fix x ∈ M such that x is a Lebesgue point of f X and m Y | X · f X .
Revised: Fix x ∈ M such that x is a Lebesgue point of f X and m Y | X · f X .

Original: Note that by Theorem 3.7, almost all x ∈ M has this property.
Revised: Note that by Theorem 3.7, almost all x ∈ M has this property.

Original: Next, let Z = log x ( X ) and f Z be as in Proposition 4.2-(i).
Revised: Next, let Z = log x ( X ) and f Z be as in Proposition 4.2-(i).

Original: Then
Revised: Then

Original: f
Revised: f

Original: Z
Revised: Z

Original: =
Revised: =

Original: ( f X ◦ exp x ) · ( ν x ◦ exp x ) , and 2.
Revised: ( f X ◦ exp x ) · ( ν x ◦ exp x ) , and 2.

Original: ( m Y | X ◦ exp x ) · f Z = ( m Y | X ◦ exp x ) · ( f X ◦ exp x ) · ( ν x ◦ exp x ) .
Revised: ( m Y | X ◦ exp x ) · f Z = ( m Y | X ◦ exp x ) · ( f X ◦ exp x ) · ( ν x ◦ exp x ) .

Original: Now, proposition 4.2-(ii) implies that 0 is a Lebesgue point of both f Z and ( m Y | X ◦ exp x ) · f Z .
Revised: Now, proposition 4.2-(ii) implies that 0 is a Lebesgue point of both f Z and ( m Y | X ◦ exp x ) · f Z .

Original: Furthermore, by Proposition 4.3, we have m Y | X ◦ exp x
Revised: Furthermore, by Proposition 4.3, we have m Y | X ◦ exp x

Original: = m Y
Revised: = m Y

Original: | Z .
Revised: | Z .

Original: Thus, 0 is a Lebesgue point of f Z and m Y | Z · f Z .
Revised: Thus, 0 is a Lebesgue point of f Z and m Y | Z · f Z .

Original: Now, let D n := { ( X i , Y i ) } i ∈ [ n ] .
Revised: Now, let D n := { ( X i , Y i ) } i ∈ [ n ] .

Original: Define Z i :=
Revised: Define Z i :=

Original: log x
Revised: log x

Original: ( X i ) , which are i.i.d copies of the random variable
Revised: ( X i ) , which are i.i.d copies of the random variable

Original: Z :=
Revised: Z :=

Original: log x ( X ) , and let ˜ D n := { ( Z i , Y i ) } i ∈ [ n ] .
Revised: log x ( X ) , and let ˜ D n := { ( Z i , Y i ) } i ∈ [ n ] .

Original: Then we have
Revised: Then we have

Original: [Equation]
Revised: [Equation]

Original: where equations marked by (a) and (d) follow from Equation (5), (b) from Lemma 4.1, and (c) from the fact that the inner product space T x M with ⟨· , ·⟩ x is isomorphic to R d with the usual dot product.
Revised: where equations marked by (a) and (d) follow from Equation (5), (b) from Lemma 4.1, and (c) from the fact that the inner product space T x M with ⟨· , ·⟩ x is isomorphic to R d with the usual dot product.

Original: By Theorem 3.8, we have (cid:98) m (0 ∥ ˜ D n , K H R d ) →
Revised: By Theorem 3.8, we have (cid:98) m (0 ∥ ˜ D n , K H R d ) →

Original: m Y | Z (0) in probability.
Revised: m Y | Z (0) in probability.

Original: In other words, for all ϵ > 0 , lim n
Revised: In other words, for all ϵ > 0 , lim n

Original: [Equation]
Revised: [Equation]

Original: |
Revised: |

Original: [Equation]
Revised: [Equation]

Original: By Proposition 4.3, we have m Y | Z (0) =
Revised: By Proposition 4.3, we have m Y | Z (0) =

Original: m Y | Z (exp x (0))
Revised: m Y | Z (exp x (0))

Original: =
Revised: =

Original: m Y | Z ( x ) .
Revised: m Y | Z ( x ) .

Original: Therefore,
Revised: Therefore,

Original: [Equation]
Revised: [Equation]

Original: as events.
Revised: as events.

Original: Thus, (cid:98) m ( x ∥ D n , K H M )
Revised: Thus, (cid:98) m ( x ∥ D n , K H M )

Original: → m Y | X ( x ) converges in probability, proving Theorem 3.2 part 1.
Revised: → m Y | X ( x ) converges in probability, proving Theorem 3.2 part 1.

Original: As noted in Devroye et al.
Revised: As noted in Devroye et al.

Original: [21, §2], part 2 of Theorem 3.2 is an immediate consequence of part 1.
Revised: [22, §2], part 2 of Theorem 3.2 is an immediate consequence of part 1.

Original:  Application to the d -Sphere
Revised:  Application to the d -Sphere

Original: The d -dimensional round sphere is S d := { x ∈ R d +1 : x 21 + · · · + x 2 d +1 = 1 } .
Revised: The d -dimensional round sphere is S d := { x ∈ R d +1 : x 21 + · · · + x 2 d +1 = 1 } .

Original: Here, a round sphere assumes that S d has the arc-length metric : dist S d ( x
Revised: Here, a round sphere assumes that S d has the arc-length metric : dist S d ( x

Original: [Equation]
Revised: [Equation]

Original: z
Revised: z

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: Let S be a set and σ : M → S be a function.
Revised: Let S be a set and σ : M → S be a function.

Original: The partition induced by σ is defined by { σ − 1 ( s ) : s ∈
Revised: The partition induced by σ is defined by { σ − 1 ( s ) : s ∈

Original: Range ( σ ) } .
Revised: Range ( σ ) } .

Original: For example, when M = S d and W ∈ R ( d +1) × h , then the function σ W : S d → {± 1 } h defined by σ W ( x ) = sgn ( W ⊤ x ) induces a hyperplane arrangement partition.
Revised: For example, when M = S d and W ∈ R ( d +1) × h , then the function σ W : S d → {± 1 } h defined by σ W ( x ) = sgn ( W ⊤ x ) induces a hyperplane arrangement partition.

Original: Let N = { 1 , 2 , . . . }
Revised: Let N = { 1 , 2 , . . . }

Original: and N 0 = N ∪ { 0 } denote the positive and non-negative integers.
Revised: and N 0 = N ∪ { 0 } denote the positive and non-negative integers.

Original: Definition 5.1 (Random hyperplane arrangement partition) .
Revised: Definition 5.1 (Random hyperplane arrangement partition) .

Original: Let d ∈ N and M = S d .
Revised: Let d ∈ N and M = S d .

Original: Let q < 0 be a negative number, and let H be a random variable with probability mass function p H : N 0 → [0 , 1] such that p H ( h ) > 0 for all h .
Revised: Let q < 0 be a negative number, and let H be a random variable with probability mass function p H : N 0 → [0 , 1] such that p H ( h ) > 0 for all h .

Original: Define the following weighted random partition ha :=
Revised: Define the following weighted random partition ha :=

Original: (Θ , P , α ) :
Revised: (Θ , P , α ) :

Original: The parameter space Θ = (cid:70) ∞
Revised: The parameter space Θ = (cid:70) ∞

Original: h =

Revised: h =

Original: R ( d +1)
Revised: R ( d +1)

Original: × h is the disjoint union of all ( d + 1) × h matrices.
Revised: × h is the disjoint union of all ( d + 1) × h matrices.

Original: Element of Θ are matrices θ = W ∈ R ( d +1)
Revised: Element of Θ are matrices θ = W ∈ R ( d +1)

Original: × h where the number of columns h ∈ { 0 , 1 , 2 , . . .
Revised: × h where the number of columns h ∈ { 0 , 1 , 2 , . . .

Original: } varies.
Revised: } varies.

Original: By convention, if h = 0 , the partition P θ =
Revised: By convention, if h = 0 , the partition P θ =

Original: P W is the trivial partition { S d } .
Revised: P W is the trivial partition { S d } .

Original: If h > 0 , P W is the partition induced by x (cid:55)→ sgn ( W ⊤ x ) .
Revised: If h > 0 , P W is the partition induced by x (cid:55)→ sgn ( W ⊤ x ) .

Original: The probability P is constructed by the procedure where we first sample h ∼ p H ( h ) , then sample the entries of W ∈ R d × h i.i.d according to Gaussian (0 , 1) .
Revised: The probability P is constructed by the procedure where we first sample h ∼ p H ( h ) , then sample the entries of W ∈ R d × h i.i.d according to Gaussian (0 , 1) .

Original: For θ ∈ Θ , define α ( θ ) :
Revised: For θ ∈ Θ , define α ( θ ) :

Original: = π q p H ( h ) − 1 ( − 1) h (cid:0) qh (cid:1) , where (cid:0) qh (cid:1) :
Revised: = π q p H ( h ) − 1 ( − 1) h (cid:0) qh (cid:1) , where (cid:0) qh (cid:1) :

Original: = 1 h !
Revised: = 1 h !

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

Original: h − 1 j =0 ( q − j ) .
Revised: h − 1 j =0 ( q − j ) .

Original: Note that ( − 1) h (cid:0) qh (cid:1)
Revised: Note that ( − 1) h (cid:0) qh (cid:1)

Original: = 1 h !
Revised: = 1 h !

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

Original: h − 1 j =0 (q −  j ) > 0 when q < 0 .
Revised: h − 1 j =0 ( − q + j ) > 0 when q < 0 .

Original: Theorem 5.2.
Revised: Theorem 5.2.

Original: Let ha =
Revised: Let ha =

Original: (Θ , P , α ) be as in Definition 5.1.
Revised: (Θ , P , α ) be as in Definition 5.1.

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original:  ) q : ∠ ( x, z ) ̸ =+ ∞ : otherwise.
Revised: z ) q : ∠ ( x, z ) ̸ =+ ∞ : otherwise.

Original: When q = − d , we have K ha S d = K H S d where the right hand side is the manifold-Hilbert kernel.
Revised: When q = − d , we have K ha S d = K H S d where the right hand side is the manifold-Hilbert kernel.

Original: Proof of Theorem 5.2.
Revised: Proof of Theorem 5.2.

Original: Before proceeding, we have the following useful lemma:
Revised: Before proceeding, we have the following useful lemma:

Original: Lemma 5.3.
Revised: Lemma 5.3.

Original: Let rp =
Revised: Let rp =

Original: (Θ , P , α ) be a WRP.
Revised: (Θ , P , α ) be a WRP.

Original: Let H be a random variable.
Revised: Let H be a random variable.

Original: Let θ ∼ P .
Revised: Let θ ∼ P .

Original: Suppose that for all x, z ∈ M , the random variables α ( θ ) and I { x ∈ P θ [ z ] } are conditionally independent given H .
Revised: Suppose that for all x, z ∈ M , the random variables α ( θ ) and I { x ∈ P θ [ z ] } are conditionally independent given H .

Original: Then we have K rp M ( x, z ) =
Revised: Then we have K rp M ( x, z ) =

Original: E H (cid:104) α
Revised: E H (cid:104) α

Original: ( H ) ·
Revised: ( H ) ·

Original: E θ ∼ P [ I { x ∈ P θ [ z ] }| H ] (cid:105) where α ( h ) :
Revised: E θ ∼ P [ I { x ∈ P θ [ z ] }| H ] (cid:105) where α ( h ) :

Original: =
Revised: =

Original: E θ ∈ P [ α ( θ ) | H
Revised: E θ ∈ P [ α ( θ ) | H

Original: = h ] for a realization h of H .
Revised: = h ] for a realization h of H .

Original: The lemma follows immediately from the Definition of K rp M ( x, z ) in Equation 3 and the conditional independence assumption.
Revised: The lemma follows immediately from the Definition of K rp M ( x, z ) in Equation 3 and the conditional independence assumption.

Original: Now, we proceed with the proof of Theorem 5.2.
Revised: Now, we proceed with the proof of Theorem 5.2.

Original: Let ϕ :
Revised: Let ϕ :

Original: = ∠ ( x, z ) /π .
Revised: = ∠ ( x, z ) /π .

Original: Let H ∼ p H and θ ∼ P be the random variables in Definition 5.1.
Revised: Let H ∼ p H and θ ∼ P be the random variables in Definition 5.1.

Original: Note that by construction, the following condition is satisfied: for all x, z ∈ M , the random variables α ( θ ) and I { x ∈ P θ [ z ] } are conditionally independent given H .
Revised: Note that by construction, the following condition is satisfied: for all x, z ∈ M , the random variables α ( θ ) and I { x ∈ P θ [ z ] } are conditionally independent given H .

Original: In fact, α ( θ ) =
Revised: In fact, α ( θ ) =

Original: π q p H ( h ) − 1 ( − 1) h (cid:0) qh (cid:1) is constant given H = h .
Revised: π q p H ( h ) − 1 ( − 1) h (cid:0) qh (cid:1) is constant given H = h .

Original: Hence, applying Lemma 5.3, we have
Revised: Hence, applying Lemma 5.3, we have

Original: [Equation]
Revised: [Equation]

Original: Next, we claim that Pr { x ∈ P θ [ z ] | H = h } =
Revised: Next, we claim that Pr { x ∈ P θ [ z ] | H = h } =

Original: (1 − ϕ ) h .
Revised: (1 − ϕ ) h .

Original: When h = 0 , x ∈ P θ [ z ] is always true since
Revised: When h = 0 , x ∈ P θ [ z ] is always true since

Original: P θ = { S d } is the trivial partition.
Revised: P θ = { S d } is the trivial partition.

Original: In this case, we have Pr { x ∈ P θ
Revised: In this case, we have Pr { x ∈ P θ

Original: [ z ] | H = h } = 

Revised: [ z ] | H = h } =

Original: = (1 − ϕ ) 0 .
Revised: = (1 − ϕ ) 0 .

Original: When h > 0 , we recall a result of Pinelis [35]:
Revised: When h > 0 , we recall an identity involving the cosine angle:

Original: Lemma 5.4.
Revised: Lemma 5.4 (Charikar [19]) .

Original: Let x, z ∈ S d .
Revised: Let x, z ∈ S d .

Original: Let w ∈ R
Revised: Let w ∈ R

Original: d
Revised: d

Original: +1 be a random vector whose entries are sampled i.i.d according to Gaussian (0 , 1) .
Revised: +1 be a random vector whose entries are sampled i.i.d according to Gaussian (0 , 1) .

Original: Then Pr { sgn ( w ⊤ x ) = sgn ( w ⊤ z ) } = 1 − ( ∠ ( x, z ) /π ) .
Revised: Then Pr { sgn ( w ⊤ x ) = sgn ( w ⊤ z ) } = 1 − ( ∠ ( x, z ) /π ) .

Original: Let W =
Revised: Let W =

Original: [ w 1 , . . .
Revised: [ w 1 , . . .

Original: , w h ] be as in Definition 5.1 where w j denotes the j -th column of W .
Revised: , w h ] be as in Definition 5.1 where w j denotes the j -th column of W .

Original: Then by construction, w j is distributed identically as w in Lemma 5.4.
Revised: Then by construction, w j is distributed identically as w in Lemma 5.4.

Original: Furthermore, w j and w j ′ are independent for j, j ′ ∈
Revised: Furthermore, w j and w j ′ are independent for j, j ′ ∈

Original: [ h ] where j ̸ = j ′ .
Revised: [ h ] where j ̸ = j ′ .

Original: Thus, the claim follows from
Revised: Thus, the claim follows from

Original: Pr { x ∈P θ
Revised: Pr { x ∈

Original: [ z ] | H =

Original: [Equation]
Revised: [Equation]

Original: where equality (a) follows from Definition 5.1, (b) from W ∈ R ( d +1)
Revised: where equality (a) follows from Definition 5.1, (b) from W ∈ R ( d +1)

Original: × h having i.i.d standard
Revised: × h having i.i.d standard

Original: Gaussian entries given H = h , and (c) from Lemma 5.4.
Revised: Gaussian entries given H = h , and (c) from Lemma 5.4.

Original: Putting it all together, we have
Revised: Putting it all together, we have

Original: [Equation]
Revised: [Equation]

Original: For the last step, we used the fact that for all q ∈ R the binomial series (1 + t ) q = (cid:80) ∞
Revised: For the last step, we used the fact that for all q ∈ R the binomial series (1 + t ) q = (cid:80) ∞

Original: h =0 (cid:0) qh (cid:1)
Revised: h =0 (cid:0) qh (cid:1)

Original: t h converges absolutely for | t | < 1 (when ϕ ∈ (0 , 1] ) and diverges to + ∞ for t = − 1 (when ϕ = 0 ).
Revised: t h converges absolutely for | t | < 1 (when ϕ ∈ (0 , 1] ) and diverges to + ∞ for t = − 1 (when ϕ = 0 ).

Original: Corollary 5.5.
Revised: Corollary 5.5.

Original: Let q := − d
Revised: Let q := − d

Original: and K ha S d be as in Theorem 5.2.
Revised: and K ha S d be as in Theorem 5.2.

Original: The infinite-ensemble classifier sgn ( (cid:98) u ( •∥ D n , K ha S d ))
Revised: The infinite-ensemble classifier sgn ( (cid:98) u ( •∥ D n , K ha S d ))

Original: (see Equation 4 for definition) has the interpolating-consistent property.
Revised: (see Equation 4 for definition) has the interpolating-consistent property.

Original: Proof.
Revised: Proof.

Original: As observed in Devroye et al.
Revised: As observed in Devroye et al.

Original: [21, Section 7], for an arbitrary kernel K , the L 1 -consistency of (cid:98) m ( •∥ D n , K ) for regression implies the consistency for classification of sgn ( (cid:98) u ( •∥ D n , K )) .
Revised: [22, Section 7], for an arbitrary kernel K , the L 1 -consistency of (cid:98) m ( •∥ D n , K ) for regression implies the consistency for classification of sgn ( (cid:98) u ( •∥ D n , K )) .

Original: Furthermore, (cid:98) m ( •∥ D n , K ) is interpolating for regression implies that sgn ( (cid:98) u ( •∥ D n , K )) is interpolating for classification.
Revised: Furthermore, (cid:98) m ( •∥ D n , K ) is interpolating for regression implies that sgn ( (cid:98) u ( •∥ D n , K )) is interpolating for classification.

Original: While the argument there is presented in the R d case, the argument holds in the more general manifold case mutatis mutandis .
Revised: While the argument there is presented in the R d case, the argument holds in the more general manifold case mutatis mutandis .

Original: Thus, by Theorem 3.2, we have sgn ( (cid:98) u ( •∥ D n , K H S d )) is consistent for classification, i.e., Equation (1) holds.
Revised: Thus, by Theorem 3.2, we have sgn ( (cid:98) u ( •∥ D n , K H S d )) is consistent for classification, i.e., Equation (1) holds.

Original: It is also interpolating since (cid:98) m ( •∥ D n , K ) is interpolating.
Revised: It is also interpolating since (cid:98) m ( •∥ D n , K ) is interpolating.

Original: By Proposition 5.2, we have
Revised: By Proposition 5.2, we have

Original: K ha S d = K H S d .
Revised: K ha S d = K H S d .

Original: Thus sgn ( (cid:98) u ( •∥ D n , K ha S d )) is an ensemble method having the interpolating-consistent property.
Revised: Thus sgn ( (cid:98) u ( •∥ D n , K ha S d )) is an ensemble method having the interpolating-consistent property.

Original:  Discussion
Revised:  Discussion

Original: We have shown that using the manifold-Hilbert kernel in kernel smoothing regression, also known as
Revised: We have shown that using the manifold-Hilbert kernel in kernel smoothing regression, also known as

Original: Nadaraya-Watson regression, results in a consistent estimator that interpolates the training data on a
Revised: Nadaraya-Watson regression, results in a consistent estimator that interpolates the training data on a

Original: Riemannian manifold M .
Revised: Riemannian manifold M .

Original: We proposed weighted random partition kernels , a generalization of the unweighted analogous definition by Davies and Ghahramani [20] which provided a framework for analyzing ensemble methods such as random forest via kernels.
Revised: We proposed weighted random partition kernels , a generalization of the unweighted analogous definition by Davies and Ghahramani [21] which provided a framework for analyzing ensemble methods such as random forest via kernels.

Original: When M = S d is the sphere, we showed that the manifold-Hilbert kernel is a weighted random partition kernel, where the random partitions are induced by random hyperplane arrangements.
Revised: When M = S d is the sphere, we showed that the manifold-Hilbert kernel is a weighted random partition kernel, where the random partitions are induced by random hyperplane arrangements.

Original: This demonstrates an ensemble method that has the interpolating-consistent property.
Revised: This demonstrates an ensemble method that has the interpolating-consistent property.

Original: One limitation of this work is the lack of rate of convergence of the ensemble methods.
Revised: One limitation of this work is the lack of rate of convergence of the ensemble methods.

Original: The analogous result for the Nadaraya-Watson regression have been obtained by Belkin et al.
Revised: The analogous result for the Nadaraya-Watson regression have been obtained by Belkin et al.

Original: However, it is not clear if the kernels used in [8] are weighted random partition kernels.
Revised: However, it is not clear if the kernels used in [8] are weighted random partition kernels.

Original: Resolving this is an interesting future direction.
Revised: Resolving this is an interesting future direction.

Original: Similar to PERT [19] another limitation of this work is that the base classifiers in the ensemble are data-independent.
Revised: Similar to PERT [20] another limitation of this work is that the base classifiers in the ensemble are data-independent.

Original: Such ensemble methods in these line of work (including ours) are easier to analyze than the data- dependent ensemble methods used in practice.
Revised: Such ensemble methods in these line of work (including ours) are easier to analyze than the data- dependent ensemble methods used in practice.

Original: See Biau and Scornet [12] and [13] for an in-depth discussion.
Revised: See Biau and Scornet [12] and [13] for an in-depth discussion.

Original: We believe our work offers one theoretical basis towards understanding generalization in the interpolation regime of ensembles of histogram classifiers over data-dependent partitions, e.g., decision trees à la CART [17].
Revised: We believe our work offers one theoretical basis towards understanding generalization in the interpolation regime of ensembles of histogram classifiers over data-dependent partitions, e.g., decision trees à la CART [17].

Original: Acknowledgements
Revised: Acknowledgements

Original: The authors were supported in part by the National Science Foundation under awards 1838179 and 2008074.
Revised: The authors were supported in part by the National Science Foundation under awards 1838179 and 2008074.

Original: The authors would like to thank Vidya Muthukumar for helpful discussions, as well as an anonymous reviewer for bringing to our attention facts from geometric measure theory that provided context for the ad-hoc technical result Proposition 4.2.
Revised: The authors would like to thank Vidya Muthukumar for helpful discussions, as well as an anonymous reviewer for bringing to our attention facts from geometric measure theory that provided context for the ad-hoc technical result Proposition 4.2.

Original: Checklist
Revised: Checklist

Original: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?

Original: [Yes] (b) Did you describe the limitations of your work?
Revised: [Yes] (b) Did you describe the limitations of your work?

Original: [Yes] See the Section 6.
Revised: [Yes] See the Section 6.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: [N/A] (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 2.
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] This is the opening sentence of our first main technical section, Section 3.
Revised: [Yes] This is the opening sentence of our first main technical section, Section 3.

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] 3.
Revised: [Yes] 3.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [N/A] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
Revised: [N/A] (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [N/A] (c) Did you report error bars (e
Revised: [N/A] (c) Did you report error bars (e

Original: g., with respect to the random seed after running experiments multiple times)?
Revised: g., with respect to the random seed after running experiments multiple times)?

Original: [N/A] (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
Revised: [N/A] (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [N/A] 4.
Revised: [N/A] 4.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [N/A] (b) Did you mention the license of the assets?
Revised: [N/A] (b) Did you mention the license of the assets?

Original: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: [N/A] (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [N/A] (d) Did you discuss whether
Revised: [N/A] (d) Did you discuss whether

Original: and how consent was obtained from people whose data you’re using/curating?
Revised: and how consent was obtained from people whose data you’re using/curating?

Original: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
Revised: [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?

Original: [N/A] 5.
Revised: [N/A] 5.

Original: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?[N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] via the Manifold-
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Original: Hilbert Kernel A.1 Basics of Riemannian Manifolds

Original: In this section, we review the main concepts from Riemannian manifold theory essential to this work.

Original: Our main references are Sakai [36] and Do Carmo [22].

Original: Throughout, d ∈ N denotes the dimension.

Original: We use the word smooth to mean infinitely differentiable.

Original: Manifolds.

Original: A smooth manifold M of dimension d is a Hausdorff, second countable topological space together with an atlas : a set Atlas :

Original: = { ( U α , φ α ) } α ∈

Original: A where 1).

Original: { U α } α ∈

Original: A is an open cover of M , 2).

Original: for each α ∈ A , φ α : U α → φ α ( U α ) ⊆

Original: R d is a homeomorphism onto its image, and 3).

Original: φ α ◦ φ − 1 β : φ β ( U α ∩ U β ) → φ − 1 α ( U α ∩ U β ) is smooth for each pair α, β ∈ A .

Original: An element ( U, φ ) of Atlas is called a chart .

Original: Smooth maps.

Original: A real-valued function f : M → R is a smooth function if f ◦ φ − 1 is smooth (in the elementary calculus sense) for all charts ( U, φ ) .

Original: The set of all smooth functions is denoted

Original: Fn ( M ) , which forms an R -vectorspace.

Original: Let N be another smooth manifold with atlas B .

Original: A function Φ : M → N is a smooth map if g ◦ Φ ∈ Fn

Original: ( M ) for all g ∈ Fn ( N ) .

Original: Tangent space.

Original: Let x ∈ M .

Original: A derivation at x is a linear function v : Fn ( M )

Original: → R satisfying the product rule : v [ fg ] = f

Original: ( x ) v [ f ] + g ( x ) v [ g ] for all f, g ∈ Fn ( M ) .

Original: The tangent space at x , denoted T x M , is the vector space of all derivations at x .

Original: Elements of T x M are referred to as tangent vectors at x .

Original: For a given chart ( U, φ ) where x ∈ U , define a derivation at x , denoted ∂ i | x , by f (cid:55)→ d ( f ◦ φ − 1 )

Original: dz i ( φ ( x )) where ddz i is the i -th partial derivative in ordinary calculus.

Original: It is a fact that { ∂ i | x : i = 1 , . . .

Original: , d } is a basis for T x M .

Original: Although the above definition of a tangent vector is abstract, it can be concretely interpreted in terms of derivative along a curve.

Original: Let a < t 0 < b be real numbers.

Original: A curve through x is a smooth map γ : ( a, b ) → M such that γ ( t 0 )

Original: =

Original: x .

Original: Then Fn ( M ) ∋ f

Original: (cid:55)→ ddt f ( γ ( t )) | t = t 0 ∈ R defines a derivation at x .

Original: Oftentimes, this derivation is denoted ˙ γ ( t 0 )

Original: ∈

Original: T x M

Original: Riemannian metric.

Original: The tangent bundle is the set TM :

Original: = (cid:83) x T x M , which itself is a smooth manifold of dimension 2 d .

Original: A vector field on M is a smooth map V : M → TM such that

Original: V ( x ) ∈ T x M for all x ∈ M .

Original: The set of all vectors fields on M is denoted Vf ( M ) .

Original: A Riemannian metric on M is a choice of an inner product ⟨· , ·⟩ x (and thus, a norm ∥ · ∥ x ) on T x M for each x ∈ M such that the function M → R given by x (cid:55)→ ⟨ V ( x ) , U ( x ) ⟩ x is smooth for all V ,

Original: U ∈ Vf ( M ) .

Original: As shorthands, when x is clear from context, we drop the subscripts and simply write ⟨· , ·⟩ and ∥ · ∥ instead.

Original: Choosing an orthonormal basis for T x M with respect to ⟨· , ·⟩ x for each x , we can identify T x M with R d with the ordinary dot inner product.

Original: Let x ∈ M and ( U, φ ) be a chart such that x ∈ U .

Original: Define g ij ( x ) = ⟨ ∂ i | x , ∂ j | x ⟩ x .

Original: Denote by G ( x ) the d × d positive definite matrix [ g ij ( x )] ij .

Original: Below, we will refer to the function G : U → R d × d as the coordinate representation of the Riemannian metric.

Original: Define g ij ( x ) :=

Original: [ G ( x ) − 1 ] ij .

Original: The Christoffel symbols with respect to ( U, φ ) are defined by Γ kij :

Original: = 12 (cid:80) dℓ

Original: =1 g kℓ ( ∂ i | x g jℓ + ∂ j | x g iℓ − ∂ ℓ | x g ij ) .

Original: Note that g kℓ , g kℓ , G , Γ kij , and ∂ i | x g jℓ are all functions with domain U .

Original: Geodesics.

Original: Fix a chart ( U, φ ) .

Original: Consider a smooth curve γ : [ a, b ] → U .

Original: Let ζ i ( t ) :

Original: =

Original: [ φ ( γ ( t ))]

Original: i be the i -th component functions.

Original: The curve γ is a geodesic if ζ is a solution to the following system of second order ordinary differential equations (ODEs): d 2 ζ

Original: i dt 2 + (cid:80) dj,ℓ

Original: =

Original: 1 Γ ijℓ ◦ γ dζ j dt dζ ℓ dt = 0 for all i = 1 , . . .

Original: , d at all time t ∈ [ a, b ] .

Original: Geodesics are minimizers of the so-called energy functional E ( γ ) =

Original: 12 (cid:82) b a ∥ ˙

Original: γ ( t ) ∥ 2 γ ( t ) dt .

Original: The above system of ODEs are the analog of the “first derivative test” for local minimizers of E .

Original: Thus, geodesics are defined independently of the choice of the chart.

Original: Exponential map.

Original: For x ∈ M and v ∈ T x M , there exists ϵ > 0 and a unique geodesic curve γ v : [ − ϵ, ϵ ]

Original: → M such that γ v (0) =

Original: x and ˙ γ v (0) = v .

Original: This follows from the existence and uniqueness of the solution to an ODE given initial conditions where the ODE is as discussed above.

Original: A

Original: Note that although geodesics are previously defined in U where ( U, φ ) is a chart, they can be extended outside of U using additional charts.

Original: Let x ∈ M and v ∈ T x

Original: M be fixed and let γ v : [ − ϵ, ϵ ]

Original: → M be as in the preceding paragraph.

Original: If ∥ v ∥ x ≤ ϵ , then define exp x ( v ) :

Original: = γ v (1) .

Original: A fundamental fact is that exp x , known as the exponential map at x , can be defined on an open set of T x M containing the origin.

Original: Distance function.

Original: Let x, ξ ∈ M and a < b be real numbers.

Original: A piecewise smooth curve from x to ξ is a piecewise smooth map γ : [ a, b ] → M

Original: such that γ ( a ) = x and γ ( b ) = ξ .

Original: Assume that

Original: M is connected.

Original: Then for all x, ξ ∈ M , there exists a piecewise smooth curve from x to ξ .

Original: The length of γ is defined as len ( γ ) :

Original: = (cid:82) b a ∥ ˙

Original: γ ( t ) ∥ γ ( t ) dt .

Original: Define dist M ( x, ξ ) := inf { len ( γ ) : γ is a piecewise smooth curve from x to ξ } , which is a metric on M in the sense of metric spaces (see [36,

Original: Proposition 1.1]).

Original: For x ∈ M and r ∈ (0 , ∞ ) , the open ball centered at x of radius r is denoted

Original: B x ( r, M ) :

Original: = { z ∈ M : dist M ( x, z ) < r } .

Original: Complete Riemannian manifolds.

Original: A Riemannian manifold is complete if it is a complete metric space under the metric dist M .

Original: The Hopf-Rinow theorem ([22, Ch. 8, Theorem 2.8]) states that if M is connected and complete, then the exponential exp x can be defined on the entire T x M .

Original: A.2 Proof of Lemma 4.

Original: This section uses definitions and notations introduced in Section 4.1.

Original: In particular, recall the cut locus C x , the tangent cut locus ˜ C x , the interior set I x and the tangent interior set ˜ I x .

Original: The proof of

Original: Lemma 4.1 is presented towards the end of the section.

Original: At this point, we compile some facts from various sources about the cut locus.

Original: Lemma A.1.

Original: For all x ∈ M , we have

Original: C x is a closed subset of M (Hebda [28, Proposition 1.2]).

Original: I x ∩ C x = ∅

Original: and I x ∪ C x = M

Original: (Sakai [36, Ch II, Lemma 4.4 (1)]) 3.

Original: I x is an open subset of M (immediate from 1 and 2 above) 4.

Original: exp x : ˜ I x

Original: → I x is a diffeomorphism ([36, Ch II, Lemma 4.4 (2)]) 5.

Original: λ M ( C x ) = 0 , where λ M is the Riemann-Lebesgue measure ([36, Lemma 4.4 (3)]) 6.

Original: τ x is continuous and inf u ∈

Original: U x M τ x ( u ) > 0 ([36, Ch II, Propositions 4.1 (2) and 4.13 (1)])

Original: While the following lemma is elementary, we provide a proof since we could not find one in the literature.

Original: Lemma A.2.

Original: For all x ∈ M , the (topological) closure of ˜ I x in T x M is ˜ D x .

Original: Furthermore, for all x ∈ M , we have exp x ( ˜ D x ) =

Original: M .

Original: Proof of Lemma A.2.

Original: Take a convergent sequence { t i u i } i ∈ N ⊆

Original: ˜

Original: I x where u i ∈

Original: U x M and 0 ≤ t

Original: i <

Original: τ x ( u i ) .

Original: Let v ∗ =

Original: lim i t i u i .

Original: Our goal is to show that v ∗ ∈

Original: ˜ D x = ˜

Original: I x ∪ ˜ C x .

Original: Since U x M is compact, we may assume that u ∗ :=

Original: lim i u

Original: i exists after passing to a subsequence if necessary.

Original: Furthermore, ∥ t i u i ∥ x

Original: = t i implies that t ∗ :

Original: = lim i t i exists as well (i.e., t ∗ < ∞ ).

Original: Hence, v ∗ =

Original: t ∗ u ∗ .

Original: Consider the case that τ x ( u ∗ )

Original: = ∞ .

Original: Then 0 ≤ t ∗ < τ x ( u ∗ ) implies that v ∗ =

Original: t ∗ u ∗ ∈ ˜

Original: I x .

Original: For the other case that t ( u ) < ∞ , we first note that t

Original: i u i ∈ ˜

Original: I x implies that t i <

Original: τ x ( u i ) .

Original: Taking the limit of both sides, we have t ∗ =

Original: lim

Original: i t i ≤ lim

Original: i τ x ( u i ) =

Original: τ x ( u ∗ ) .

Original: Note that the last limit can be exchanged since τ x is continuous (Lemma A.1 part 6).

Original: Thus, either t ∗ <

Original: τ x ( u ∗ ) in which case v ∗ ∈ ˜

Original: I x , or t ∗ =

Original: τ x ( u ∗ ) in which case v ∗ =

Original: τ x ( u ∗ ) u ∗ ∈ ˜

Original: C x .

Original: For the “furthermore” part, note that

Original: [Equation]

Original: where the last equality is Lemma A.1 part 2.

Original: Proof of Lemma 4.1.

Original: Denote by cl( T x M ) the set of closed subsets of T x M . Define ψ : M → cl( T x M ) by ψ ( ξ ) : = { x ∈ ˜ D x : exp x ( x ) = ξ } = exp − 1 x ( ξ ) ∩ ˜
Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

Original: D x .

Original: Note that ψ ( ξ ) is a closed set by

Original: Lemma A.2.

Original: A

Original: We claim that ψ is weakly-measurable , i.e., for every open set ˜ U ⊆ T x M , the subset of M defined by { ξ ∈ M : ψ ( ξ ) ∩ ˜ U ̸ = ∅} is Borel.

Original: To see this, note that

Original: [Equation]

Original: As inner product spaces, T x M and R d are isomorphic (see Section A.1-Riemannian metric).

Original: Since,

Original: T x M and R d are homeomorphic as topological spaces, R d being locally compact implies T x M is locally compact as well.

Original: Thus, we can write ˜ U

Original: = (cid:83) i ∈ N ˜ K i as a countable union of compact sets ˜ K

Original: i ⊆

Original: T x M .

Original: Furthermore, ˜ D x ∩ ˜ U

Original: =

Original: (cid:83) i ∈ N ˜ D x ∩ ˜ K

Original: i and so exp x ( ˜ D x ∩ ˜ U )

Original: =

Original: (cid:83) i ∈ N exp x

Original: ( ˜ D x ∩ ˜ K i ) .

Original: Since exp x is continuous, exp x ( ˜ D x ∩ ˜ K i ) is a compact subset of M , and hence closed and bounded by the Hopf-Rinow theorem ([22, Ch. 8, Theorem 2.8]).

Original: Thus, exp x ( ˜ D x ∩ ˜ U ) =

Original: (cid:83) i ∈ N exp x

Original: ( ˜ D x ∩ ˜ K i ) is a countable union of closed sets, which is Borel.

Original: This proves the claim that ψ is weakly Borel measurable.

Original: By the Kuratowski–Ryll-Nardzewski measurable selection theorem (see [14, Theorem 6.9.3]), there exists a Borel measurable function M → T x M , which we denote by log x , such that log x ( ξ ) ∈ ψ ( ξ ) =

Original: exp −x ( ξ ) for all ξ ∈ M , as desired.

Original: By construction, log x ( ξ ) ∈ exp − 1 x ( ξ ) for all ξ ∈ M , and so exp x (log x ( ξ )) =

Original: ξ is immediate.

Original: For the “furthermore” part, let ξ ∈ M be arbitrary and let z := log x ( ξ ) ∈

Original: ˜ D x . Let { z i } ⊆

Original: ˜

Original: I x be a sequence such that lim

Original: i z i

Original: = z .

Original: By Equation (6), we have dist M ( x, exp x

Original: ( z i ))

Original: =

Original: ∥

Original: z i ∥ x .

Original: By continuity of dist M and exp x , we have dist M ( x, ξ ) =

Original: dist M ( x, exp x ( z )) =

Original: lim i dist M ( x, exp x

Original: ( z i )) .

Original: To conclude, we have lim

Original: i dist M ( x, exp x

Original: ( z i ))

Original: =

Original: lim i ∥

Original: z i ∥ x = ∥

Original: z ∥ x = ∥ log x ( ξ ) ∥ x , as desired.

Original: A.3 Proof of Proposition 4.

Original: Recall from Section A.1-Riemannian metric, given a chart ( U, φ ) , one can define the matrix-valued function G : U → R d × d referred to earlier as the coordinate representation of the Riemannian metric.

Original: Now, Lemma A.1 part 3 states that I x is an open neighborhood of x .

Original: Furthermore, ˜ I x is an open subset of T x M , which is identified with R d using an orthonormal basis (see Section A.1-Riemannian metric).

Original: Hence, { ( I x , log x | I x ) } x ∈ M is an atlas of M (see Section A.1-Manifolds).

Original: Definition A.3.

Original: The chart ( I x , log x | I x ) is called a normal coordinate system at x .

Original: Let G : I x →

Original: R d × d be the coordinate representation of the Riemannian metric for this chart.

Original: To emphasize the dependency on x , we write G x := G .

Original: Denote by G ⊥ x : M → R d × d the zero extension of G x to the rest of M , i.e., G ⊥ x ( ξ ) =

Original: G x ( ξ ) for ξ ∈ I x and G ⊥ x ( ξ ) is the zero matrix for ξ ̸∈ I x .

Original: The normal coordinate system has the property that G x ( x ) =

Original: G ⊥ x

Original: ( x ) is the identity matrix.

Original: This is the result of Sakai [36, Ch.

Original: II §2 Exercise 4].

Original: Lemma A.4 (Change-of-Variables) .

Original: Let x ∈ M be fixed.

Original: Define the function ν x : M → R by ν x ( ξ ) =

Original: (cid:112) | det G ⊥ x ( ξ ) | where G ⊥ x is as in Definition A.3.

Original: Then ν x is Borel-measurable.

Original: Furthermore, ν x satisfies the following property: Let f : M

Original: → R be an absolutely integrable function.

Original: Define the function h : T x M → R by h ( z ) :

Original: = f

Original: (exp x ( z )) · ν x (exp x ( z )) .

Original: Then (i) h (0) =

Original: f

Original: ( x ) and (ii) for all Borel set ˜ B

Original: ⊆ T x M we have (cid:82) B fdλ M = (cid:82)

Original: ˜ B hdλ where

Original: B := exp x ( ˜ B ∩ ˜ I x ) .

Original: Proof of Lemma A.4.

Original: We first show that ν x is Borel-measurable.

Original: Recall that G ⊥ x : M → R d × d is the zero extension of G x : I x

Original: → R , which is by definition smooth (see Section A.1-Riemannian metric).

Original: In particular, G x : I x → R is continuous and so (cid:112) det( G x ( • )) is Borel-measurable.

Original: Now, note that (cid:112) det( G ⊥ x ( • )) is the zero extension of (cid:112) det( G x ( • )) from I x to M .

Original: Hence, (cid:112) det( G ⊥ x ( • )) , which is ν x by definition, is Borel-measurable.

Original: A

Original: Next, we prove the “Furthermore” part (i).

Original: Note that exp x (0) =

Original: x .

Original: Moreover, G ⊥ x ( x ) =

Original: G x ( x ) is the identity matrix as asserted after Definition A.3 (see Sakai [36, Ch.

Original: II §2 Exercise 4]).

Original: Thus, h (0) =

Original: f (exp x (0)) (cid:112) | det G ⊥ x (exp x (0))

Original: | = f ( x ) √ 1 = f ( x ) , as desired.

Original: For the “Furthermore” part (ii), we first note that ˜ B

Original: = ( ˜ B ∩ ˜ I x ) ∪ ( ˜ B ∩ ˜

Original: C x ) expresses ˜ B as a disjoint union.

Original: Thus, B = exp x ( ˜ B ) =

Original: exp x ( ˜ B ∩ ˜ I x ) ∪ exp( ˜ B ∩ ˜ C x ) expresses B as a disjoint union as well.

Original: Moreover, exp( ˜ B ∩ ˜ C x ) ⊆

Original: exp( ˜ C x ) =

Original: C x , which has λ M -measure zero (Lemma A.1 part 5).

Original: Recall that λ is the shorthand for the ordinary Lebesgue measure λ R d

Original: (see paragraph right after

Original: Definition 3.1).

Original: Now, we directly compute to obtain the formula

Original: [Equation]

Original: Amann and Escher [1, Ch XII, Thm 1.10]

Original: [Equation]

Original: as desired.

Original: Proposition A.5.

Original: Let x ∈ M be fixed.

Original: Let X be a random variable on M with density f X where the underlying probability space is (Ω , P , A ) (see Definition 3.3).

Original: Define Z :=

Original: log x ( X ) .

Original: Then Z is a random variable on T x M such that for all events E ∈ A and Borel sets

Original: ˜ B ⊆ T x M we have

Original: Pr( E ∩ { Z ∈ ˜ B } ) =

Original: Pr( E ∩ { X ∈ exp x

Original: ( ˜ B ∩ ˜ I x ) } ) ,

Original: Proof of Proposition A.5.

Original: To start with, we have

Original: [Equation]

Original: Since exp x : ˜ I x → I x is a diffeomorphism (Lemma A.1-part 4) with inverse log x , we have

Original: [Equation]

Original: as sets.

Original: On the other hand,

Original: [Equation]

Original: Finally, Pr( X ∈ C x ) =

Original: (cid:82) C x

Original: f X dλ M = 0 since C x has λ M -measure zero (Lemma A.1-part 5).

Original: Proof of Proposition 4.2 part (i).

Original: Recall that λ is the shorthand for the ordinary Lebesgue measure λ R d

Original: (see paragraph right after Definition 3.1).

Original: Let E = Ω in Proposition A.5.

Original: Then we have

Original: [Equation]

Original: is the density of X

Original: [Equation]

Original: Lemma A.

Original: [Equation]

Original: ∵ Definition of f Z

Original: A

Original: By assumption, f X is Borel-measurable.

Original: By Lemma A.4, ν x is Borel-measurable.

Original: Since exp x is continuous, we have that both f X ◦ exp x

Original: and ν x ◦ exp x are Borel-measurable.

Original: This proves that f Z is Borel-measurable.

Original: Hence, the integrand is Borel-measurable and a density function for Z .

Original: Proof of Proposition 4.2 part (ii).

Original: Recall that λ is the shorthand for the ordinary Lebesgue measure λ R d

Original: (see paragraph right after Definition 3.1).

Original: By Lemma A.1 part 6, we have τ ∗ x :=

Original: inf u ∈

Original: U x M τ x ( u ) > 0 .

Original: Now, let r ∈

Original: (0 , τ ∗ x ) .

Original: By the definition of r , we have B x

Original: ( r, M ) ⊆

Original: ˜

Original: I x .

Original: Hence letting z = log x ( ξ ) for ξ ∈ B x ( r, M ) , by Equation (6) we have

Original: [Equation]

Original: Thus, log

Original: [Equation]

Original: 9) and

Original: B x (

Original: [Equation]

Original: [Equation]

Original: Thus, by Lemma A.4, we have

Original: [Equation]

Original: Before proceeding, we need the following lemma:

Original: Lemma A.6.

Original: For all x ∈ M , we have lim r → 0 λ M ( B x ( r,M )) λ

Original: ( B 0 ( r,T x M ))

Original: = 1 .

Original: Proof of Lemma A.6.

Original: Let ω d := π d/ 2 / Γ( d 2 + 1) be the volume of the unit ball in R d where Γ is the gamma function.

Original: Then λ ( B 0 ( r, T x M )) =

Original: ω d r d .

Original: [-Next, [36, Ch II.
-]
Revised: [-Next, [36, Ch II.
-]

Original: Exercise 3] states that

Original: [Equation]

Original: where S x ∈ R is a constant that depends only on x (it is the scalar curvature of M at x ).

Original: By simple algebra, the above yields

Original: [Equation]
Revised: [N/A]

Original: [-In particular, we have lim r → 
-]
Revised: [-In particular, we have lim r → 
-]

Original: [Equation]

Original: r

Original: [Equation]

Original: d = 0 , as desired.

Original: Now we continue with the proof of Proof of Proposition 4.2 part (ii).

Original: We observe that

Original: [Equation]

Original: ∵ x is a Lebesgue point of f

Original: [Equation]

Original: ∵ definition of h and equation (11)

Original: [Equation]

Original: Since f ( x ) = h (0) (Lemma A.4) , we’ve shown that

Original: [Equation]

Original: Thus, 0 is a Lebesgue point of h , as desired.

Original: A

Original: A.4 Proof of Proposition 4.

Original: Recall that λ is the shorthand for the ordinary Lebesgue measure λ R d

Original: (see paragraph right after

Original: Definition 3.1).

Original: Let A ⊆ R and ˜ B ⊆ T x M be Borel subsets.

Original: Then

Original: [Equation]

Original: ∵ Definition of P Y | Z = z

Original: [Equation]

Original: This proves that P Y | Z ( ·|· ) is a conditional probability for Y given Z .

Original: A