Original: F UNCTIONAL VS . PARAMETRIC EQUIVALENCE OF R E LU NETWORKS
Revised: F UNCTIONAL VS . PARAMETRIC EQUIVALENCE OF R E LU NETWORKS

Original: Anonymous authors

Original: Paper under double-blind review

Original: We address the following question: How redundant is the parameterisation of ReLU net- works?
Revised: We address the following question: How redundant is the parameterisation of ReLU net- works?

Original: Speciﬁcally, we consider transformations of the weight space which leave the function implemented by the network intact.
Revised: Speciﬁcally, we consider transformations of the weight space which leave the function implemented by the network intact.

Original: Two such transformations are known for feed- forward architectures: permutation of neurons within a layer, and positive scaling of all incoming weights of a neuron coupled with inverse scaling of its outgoing weights.
Revised: Two such transformations are known for feed- forward architectures: permutation of neurons within a layer, and positive scaling of all incoming weights of a neuron coupled with inverse scaling of its outgoing weights.

Original: In this work, we show for architectures with non-increasing widths that permutation and scaling are in fact the only function-preserving weight transformations.
Revised: In this work, we show for architectures with non-increasing widths that permutation and scaling are in fact the only function-preserving weight transformations.

Original: For any eligible architec- ture we give an explicit construction of a neural network such that any other network that implements the same function can be obtained from the original one by the application of permutations and rescaling.
Revised: For any eligible architec- ture we give an explicit construction of a neural network such that any other network that implements the same function can be obtained from the original one by the application of permutations and rescaling.

Original: The proof relies on a geometric understanding of boundaries between linear regions of ReLU networks and we hope the developed mathematical tools are of independent interest.
Revised: The proof relies on a geometric understanding of boundaries between linear regions of ReLU networks, and we hope the developed mathematical tools are of independent interest.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Ever since its early successes, deep learning has been a puzzle for machine learning theorists.
Revised: Ever since its early successes deep learning has been a puzzle for machine learning theorists.

Original: Multiple aspects of deep learning seem at ﬁrst sight to contradict common sense: single-hidden-layer networks sufﬁce to approximate any continuous function (Cybenko, 1989;
Revised: Multiple aspects of deep learning seem at ﬁrst sight to contradict common sense: single-hidden-layer networks sufﬁce to approximate any continuous function (Cybenko, 1989;

Original: Hornik et al., 1989), yet in practice deeper is better; the loss surface is highly non-convex, yet it can be minimised by ﬁrst-order methods; the capacity of the model class is immense, yet deep networks tend not to overﬁt (Zhang et al., 2017).
Revised: Hornik et al., 1989) yet in practice deeper is better; the loss surface is highly non-convex, yet it can be minimised by ﬁrst-order methods; the capacity of the model class is immense yet deep networks tend not to overﬁt (Zhang et al., 2017)

Original: Recent investigations into these and other questions have emphasised the role of overparameterisation, or highly redundant function representation.
Revised: Recent investigations into these and other questions have emphasised the role of overparameterisation, or highly redundant function representation

Original: It is now known that overparameterised networks enjoy both easier training (Allen-Zhu et al., 2019; Du et al., 2019; Frankle & Carbin, 2019), and better generalisation (Belkin et al., 2019; Neyshabur et al., 2019; Novak et al., 2018).
Revised: It is now known that overparameterised networks enjoy both easier training (Allen-Zhu et al., 2019; Du et al., 2019; Frankle & Carbin, 2019), and better generalisation (Belkin et al., 2019; Neyshabur et al., 2019; Novak et al., 2018).

Original: However, the speciﬁc mechanism by which over-parameterisation operates is still largely a mystery.
Revised: However, the speciﬁc mechanism by which over-parameterisation operates is still largely a mystery.

Original: In this work, we study one particular aspect of over-parameterisation, namely the ability of neural networks to represent a target function in many different ways.
Revised: In this work, we study one particular aspect of over-parameterisation, namely the ability of neural networks to represent a target function in many different ways.

Original: In other words, we ask whether many different parameter conﬁgurations can give rise to the same function.
Revised: In other words, we ask whether many different parameter conﬁgurations can give rise to the same function.

Original: Such a notion of parameterisation redundancy has so far remained unexplored, despite its potential connections to the structure of the loss landscape, as well as to the literature on neural network capacity in general.
Revised: Such a notion of parameterisation redundancy has so far remained unexplored, despite its potential connections to the structure of the loss landscape, as well as to the literature on neural network capacity in general.

Original: Speciﬁcally, we consider feed-forward ReLU networks, with weight matrices W 1 , . . .
Revised: Speciﬁcally, we consider feed-forward ReLU networks, with weight matrices W 1 , . . .

Original: , W L , and biases b 1 , . . .
Revised: , W L , and biases b 1 , . . .

Original: , b L , .
Revised: , b L , .

Original: We study parameter transformations which preserve the output behaviour of the network
Revised: We study parameter transformations which preserve the output behaviour of the network

Original: h ( z ) =
Revised: h ( z ) =

Original: [Equation]

Original:  L
Revised: W L

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: L −

Original: [Equation]
Revised: [Equation]

Original: W 1 z + b 1 . . . )

Original: +
Revised: +

Original: [Equation]
Revised: [Equation]

Original: L − 1 )

Original: +
Revised: +

Original: [Equation]
Revised: [Equation]

Original: L for all inputs z in some domain Z .
Revised: L for all inputs z in some domain Z .

Original: Two such transformations are known for feed-forward ReLU architectures:
Revised: Two such transformations are known for feed-forward ReLU architectures:

Original: Permutation of units (neurons) within a layer,
Revised: Permutation of units (neurons) within a layer,

Original: i
Revised: i

Original: e. for some permutation matrix P 
Revised: e. for some permutation matrix P ,

Original: [Equation]
Revised: [Equation]

Original: Positive scaling of all incoming weights of a unit coupled with inverse scaling of its outgoing weights.
Revised: Positive scaling of all incoming weights of a unit coupled with inverse scaling of its outgoing weights.

Original: Applied to a whole layer, with potentially different scaling factors arranged into a diagonal matrix M , this can be written as
Revised: Applied to a whole layer, with potentially different scaling factors arranged into a diagonal matrix M , this can be written as

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Our main theorem applies to architectures with non-increasing widths, and shows that there are no other function-preserving parameter transformations besides permutation and scaling .
Revised: Our main theorem applies to architectures with non-increasing widths, and shows that there are no other function-preserving parameter transformations besides permutation and scaling .

Original: Stated formally:
Revised: Stated formally:

Original: Theorem 1.
Revised: Theorem 1.

Original: Consider a bounded open nonempty domain Z ⊆
Revised: Consider a bounded open nonempty domain Z ⊆

Original: R
Revised: R

Original: d 0 and any architecture ( d 0 , . . .
Revised: d 0 and any architecture ( d 0 , . . .

Original: , d L ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 , d L = 1 .
Revised: , d L ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 , d L = 1 .

Original: For this architecture, there exists a
Revised: For this architecture, there exists a

Original: ReLU network h θ : Z → R , or equivalently a setting of the weights θ (cid:44) ( W 1 , b 1 , . . .
Revised: ReLU network h θ : Z → R , or equivalently a setting of the weights θ (cid:44) ( W 1 , b 1 , . . .

Original: , W L , b L ) , such that for any ‘general’ ReLU network h
Revised: , W L , b L ) , such that for any ‘general’ ReLU network h

Original: η : Z → R (with the same architecture) satisfying h θ ( z ) =
Revised: η : Z → R (with the same architecture) satisfying h θ ( z ) =

Original: h η ( z ) for all z ∈ Z , there exist permutation matrices P 1 , . . .
Revised: h η ( z ) for all z ∈ Z , there exist permutation matrices P 1 , . . .

Original: P L − 1 , and positive diagonal matrices M 1 , . .
Revised: P L − 1 , and positive diagonal matrices M 1 , . .

Original: , M L − 1 , such that
Revised: , M L − 1 , such that

Original: [Equation]
Revised: [Equation]

Original: where η (cid:44) ( W (cid:48) 1 , b (cid:48) 1 , . .
Revised: where η (cid:44) ( W (cid:48) 1 , b (cid:48) 1 , . .

Original: , W (cid:48)
Revised: , W (cid:48)

Original: L , b (cid:48) L ) are the parameters of h η .
Revised: L , b (cid:48) L ) are the parameters of h η .

Original: In the above, ‘general’ networks is a class of networks meant to exclude degenerate cases.
Revised: In the above, ‘general’ networks is a class of networks meant to exclude degenerate cases.

Original: We give a more precise deﬁnition in Section 3; for now it sufﬁces to note that almost all networks are general.
Revised: We give a more precise deﬁnition in Section 3; for now it sufﬁces to note that almost all networks are general.

Original: The proof of the result relies on a geometric understanding of prediction surfaces of ReLU networks.
Revised: The proof of the result relies on a geometric understanding of prediction surfaces of ReLU networks.

Original: These surfaces are piece-wise linear functions, with non-differentiabilities or ‘folds’ between linear regions.
Revised: These surfaces are piece-wise linear functions, with non-differentiabilities or ‘folds’ between linear regions.

Original: It turns out that folds carry a lot of information about the parameters of a network, so much in fact, that some networks are uniquely identiﬁed (up to permutation and scaling) by the function they implement.
Revised: It turns out that folds carry a lot of information about the parameters of a network, so much in fact, that some networks are uniquely identiﬁed (up to permutation and scaling) by the function they implement.

Original: This is the main insight of the theorem.
Revised: This is the main insight of the theorem.

Original: In the following sections, we introduce in more detail the concept of a fold-set, and describe its geometric structure for a subclass of ReLU networks.
Revised: In the following sections, we introduce in more detail the concept of a fold-set, and describe its geometric structure for a subclass of ReLU networks.

Original: The paper culminates in a proof sketch of the main result.
Revised: The paper culminates in a proof sketch of the main result.

Original: The full proof, including proofs of intermediate results, is included in the Appendix.
Revised: The full proof, including proofs of intermediate results, is included in the Appendix.

Original:  R ELATED WORK
Revised:  R ELATED WORK

Original: The functional equivalence of neural networks is a well-researched topic in classical connectionist literature.
Revised: The functional equivalence of neural networks is a well-researched topic in classical connectionist literature.

Original: The problem was ﬁrst posed by Hecht-Nielsen (1990), and soon resolved for feed-forward networks with the tanh activation function by Chen et al.
Revised: The problem was ﬁrst posed by Hecht-Nielsen (1990), and soon resolved for feed-forward networks with the tanh activation function by Chen et al.

Original: (1993), who showed that any smooth transformation of the weight space that preserves the function of all neural networks is necessarily a composition of permutations and sign ﬂips.
Revised: (1993), who showed that any smooth transformation of the weight space that preserves the function of all neural networks is necessarily a composition of permutations and sign ﬂips.

Original: For the same class of networks, Fefferman & Markel (1994) showed a somewhat stronger result: knowledge of the input-output mapping of a neural network determines both its architecture and its weights, up to permutations and sign ﬂips.
Revised: For the same class of networks, Fefferman & Markel (1994) showed a somewhat stronger result: knowledge of the input-output mapping of a neural network determines both its architecture and its weights, up to permutations and sign ﬂips.

Original: Similar results have been proven for single-layer networks with a saturating activation function such as sigmoid or RBF (K ˚ urkov ´ a & Kainen, 1994), as well as single-layer recurrent networks with a smooth activation function (Albertini & Sontag, 1993a;b).
Revised: Similar results have been proven for single-layer networks with a saturating activation function such as sigmoid or RBF (K ˚ urkov ´ a & Kainen, 1994), as well as single-layer recurrent networks with a smooth activation function (Albertini & Sontag, 1993a;b).

Original: To the best of our knowledge, no such theoretical results exist for networks with the ReLU activation, which is non-saturating, asymmetric and non-smooth.
Revised: To the best of our knowledge, no such theoretical results exist for networks with the ReLU activation, which is non-saturating, asymmetric and non-smooth.

Original: Broadly related is the recent work by Petersen et al.
Revised: Broadly related is the recent work by Petersen et al.

Original: and Berner et al.
Revised: and Berner et al.

Original: (2019) who study whether two neural networks (ReLU or otherwise) that are close in the functional space have parameterisations that are close in the weight space.
Revised: (2019) who study whether two neural networks (ReLU or otherwise) that are close in the functional space have parameterisations that are close in the weight space.

Original: This is called inverse stability.
Revised: This is called inverse stability.

Original: In contrast, we are interested in ReLU networks that are functionally identical, and ask about all their possible parameterisations.
Revised: In contrast, we are interested in ReLU networks that are functionally identical, and ask about all their possible parameterisations.

Original: In terms of proof technique, our approach is based on the geometry of piece-wise linear functions, speciﬁcally the boundaries between linear regions.
Revised: In terms of proof technique, our approach is based on the geometry of piece-wise linear functions, speciﬁcally the boundaries between linear regions.

Original: The intuition for this kind of analysis has previously been presented by Raghu et al.
Revised: The intuition for this kind of analysis has previously been presented by Raghu et al.

Original: (2017) and Serra et al.
Revised: (2017) and Serra et al.

Original: (2018), and somewhat similar proof techniques to ours have been used by Hanin & Rolnick (2019) in the context of counting the number of linear decision regions.
Revised: (2018), and somewhat similar proof techniques to ours have been used by Hanin & Rolnick (2019) in the context of counting the number of linear decision regions.

Original: Finally, the sets of equivalent parametrisations can be viewed as symmetries in the weight space, with implications for optimisation.
Revised: Finally, the sets of equivalent parametrisations can be viewed as symmetries in the weight space, with implications for optimisation.

Original: Multiple authors, including e.g. Neyshabur et al. (2015); Badrinarayanan et al. (2016); Stock et al.
Revised: Multiple authors, including e.g. Neyshabur et al. (2015); Badrinarayanan et al. (2016); Stock et al.

Original: (2019), have observed that the naive loss gradient is sensitive to reparametrisation by scaling, and proposed alternative, scaling-invariant optimisation procedures.
Revised: (2019), have observed that the naive loss gradient is sensitive to reparametrisation by scaling, and proposed alternative, scaling-invariant optimisation procedures.

Original:  R E LU NETWORKS
Revised:  R E LU NETWORKS

Original: This section introduces notation and two important classes of ReLU networks that we refer to throughout the manuscript.
Revised: This section introduces notation and two important classes of ReLU networks that we refer to throughout the manuscript.

Original: We denote by σ the ReLU function: σ ( u ) i = max { 0 , u i } for i ∈ [dim( u )] ; the subscript index denotes the corresponding vector element.
Revised: We denote by σ the ReLU function: σ ( u ) i = max { 0 , u i } for i ∈ [dim( u )] ; the subscript index denotes the corresponding vector element.

Original: ReLU network.
Revised: ReLU network.

Original: Let Z ⊆
Revised: Let Z ⊆

Original: R d 0 with d 0 ≥ 2 be a nonempty open set, and let θ (cid:44) ( W 1 , b 1 , . .
Revised: R d 0 with d 0 ≥ 2 be a nonempty open set, and let θ (cid:44) ( W 1 , b 1 , . .

Original: , W L , b L ) be the network’s parameters, with W l ∈ R d l × d l − 1 , b l ∈ R d l , and
Revised: , W L , b L ) be the network’s parameters, with W l ∈ R d l × d l − 1 , b l ∈ R d l , and

Original: d L = 1 .
Revised: d L = 1 .

Original: We denote the corresponding ReLU network by h θ : Z → R , where
Revised: We denote the corresponding ReLU network by h θ : Z → R , where

Original: [Equation]
Revised: [Equation]

Original: and h l θ ( z ) =
Revised: and h l θ ( z ) =

Original: W l · z + b l .
Revised: W l · z + b l .

Original: For 1 ≤ l ≤ k ≤ L , we also introduce notation for truncated networks,
Revised: For 1 ≤ l ≤ k ≤ L , we also introduce notation for truncated networks,

Original: [Equation]
Revised: [Equation]

Original: We will omit the subscript θ when it is clear from the context.
Revised: We will omit the subscript θ when it is clear from the context.

Original: General ReLU network.
Revised: General ReLU network.

Original: In this work, we restrict our attention to so-called general ReLU networks.
Revised: In this work, we restrict our attention to so-called general ReLU networks.

Original: Intuitively, a general network is one that satisﬁes a number of non-degeneracy properties, such as all weight matrices having non-zero entries and full rank, no two network units exactly cancelling each other out, etc.
Revised: Intuitively, a general network is one that satisﬁes a number of non-degeneracy properties, such as all weight matrices having non-zero entries and full rank, no two network units exactly cancelling each other out, etc.

Original: It can be shown 1 that almost all ReLU networks are general.
Revised: It can be shown 1 that almost all ReLU networks are general.

Original: In other words, a sufﬁcient condition for a ReLU network to be general with probability one is that its weights are sampled from a distribution with a density.
Revised: In other words, a sufﬁcient condition for a ReLU network to be general with probability one is that its weights are sampled from a distribution with a density.

Original: More formally, a general ReLU network is one that satisﬁes the following three conditions.
Revised: More formally, a general ReLU network is one that satisﬁes the following three conditions.

Original: For any unit ( l, i ) , the local optima of h 1: l i do not have value exactly zero.
Revised: For any unit ( l, i ) , the local optima of h 1: l i do not have value exactly zero.

Original: For all k ≤ l and all diagonal matrices ( I k , . . .
Revised: For all k ≤ l and all diagonal matrices ( I k , . . .

Original: I l ) with entries in { 0 , 1 } , rank( I l W l I l − 1 · · · I k W k ) =
Revised: I l ) with entries in { 0 , 1 } , rank( I l W l I l − 1 · · · I k W k ) =

Original: min { d k − 1 , rank( I k ) , . .
Revised: min { d k − 1 , rank( I k ) , . .

Original: , rank( I l − 1 ) , rank( I l ) } .
Revised: , rank( I l − 1 ) , rank( I l ) } .

Original: For any two units ( l, i ) , ( k, j ) , any linear region R 1 ⊆ Z of h 1: l i , and any linear region R 2 ⊆
Revised: For any two units ( l, i ) , ( k, j ) , any linear region R 1 ⊆ Z of h 1: l i , and any linear region R 2 ⊆

Original: Z of h 1: k j , the linear functions implemented by h 1: l i on R 1 and h 1: k j on R 2 are not multiples of each other.
Revised: Z of h 1: k j , the linear functions implemented by h 1: l i on R 1 and h 1: k j on R 2 are not multiples of each other.

Original: General networks are convenient to study, as they exclude many degenerate special cases.
Revised: General networks are convenient to study, as they exclude many degenerate special cases.

Original: The second important class of ReLU networks are so-called transparent networks.
Revised: The second important class of ReLU networks are so-called transparent networks.

Original: Their signiﬁcance as well as their name will become clear in the next section.
Revised: Their signiﬁcance as well as their name will become clear in the next section.

Original: For now, we state the deﬁnition.
Revised: For now, we state the deﬁnition.

Original: Transparent ReLU network.
Revised: Transparent ReLU network.

Original: A ReLU network h :
Revised: A ReLU network h :

Original: Z → R is called transparent if for all z ∈ Z and l ∈ [ L − 1] , there exists
Revised: Z → R is called transparent if for all z ∈ Z and l ∈ [ L − 1] , there exists

Original: i ∈
Revised: i ∈

Original: [ d l ] such that h 1: l i ( z ) ≥ 0 .
Revised: [ d l ] such that h 1: l i ( z ) ≥ 0 .

Original: In words, we require that for any input, at least one unit on each layer is active.
Revised: In words, we require that for any input, at least one unit on each layer is active.

Original:  F OLD - SETS
Revised:  F OLD - SETS

Original: In this section we introduce the concept of fold-sets , which is key to our understanding of ReLU networks and their prediction surfaces.
Revised: In this section we introduce the concept of fold-sets , which is key to our understanding of ReLU networks and their prediction surfaces.

Original: Since ReLU networks are piece-wise linear functions, a great deal about them is revealed by the boundaries between individual linear regions.
Revised: Since ReLU networks are piece-wise linear functions, a great deal about them is revealed by the boundaries between individual linear regions.

Original: A network’s fold-set is simply the union of all these boundaries.
Revised: A network’s fold-set is simply the union of all these boundaries.

Original: More formally, if Z is an open set, and f : Z → R is any continuous, piece-wise linear function, we deﬁne the fold-set of f , denoted by F ( f ) , as the set of all points at which f is non-differentiable.
Revised: More formally, if Z is an open set, and f : Z → R is any continuous, piece-wise linear function, we deﬁne the fold-set of f , denoted by F ( f ) , as the set of all points at which f is non-differentiable.

Original: It turns out there is a class of networks whose fold-sets are especially easy to understand; these are the ones we have termed transparent .
Revised: It turns out there is a class of networks whose fold-sets are especially easy to understand; these are the ones we have termed transparent .

Original: For transparent networks, we have the following characterisation of the fold-set (which also motivates the name ‘transparent’).
Revised: For transparent networks, we have the following characterisation of the fold-set (which also motivates the name ‘transparent’).

Original: Lemma 1.
Revised: Lemma 1.

Original: If h : Z → R is a general and transparent ReLU network, then
Revised: If h : Z → R is a general and transparent ReLU network, then

Original: [Equation]
Revised: [Equation]

Original: To appreciate the signiﬁcance of the lemma, suppose we are given some transparent ReLU network function h and we want to infer its parameters.
Revised: To appreciate the signiﬁcance of the lemma, suppose we are given some transparent ReLU network function h and we want to infer its parameters.

Original: This lemma shows that the knowledge of the endto-end mapping h (cid:44) h 1: L in fact gives us information about the network’s hidden units h 1: l i (hence ‘transparent’).
Revised: This lemma shows that the knowledge of the endto-end mapping h (cid:44) h 1: L in fact gives us information about the network’s hidden units h 1: l i (hence ‘transparent’).

Original: Moreover, this information is very explicit: we observe the units’ zero-level sets, which in the case of a linear unit on a full-dimensional space already determines the unit’s parameters up to scaling 2 .
Revised: Moreover, this information is very explicit: we observe the units’ zero-level sets, which in the case of a linear unit on a full-dimensional space already determines the unit’s parameters up to scaling 2 .

Original: Of course, dealing with piece-wise linearity and disambiguating the union into its constituent zero-level sets remains a challenge for upcoming sections.
Revised: Of course, dealing with piece-wise linearity and disambiguating the union into its constituent zero-level sets remains a challenge for upcoming sections.

Original:  P IECE - WISE LINEAR SURFACES
Revised:  P IECE - WISE LINEAR SURFACES

Original: In this section, we provide a geometric description of fold-sets of transparent networks.
Revised: In this section, we provide a geometric description of fold-sets of transparent networks.

Original: Intuitively, the fold-sets look like the sets shown in Figure 1.
Revised: Intuitively, the fold-sets look like the sets shown in Figure 1.

Original: The ﬁrst-layer units of a network are linear, so the component (cid:83) i (cid:8) z | h 1:

Revised: The ﬁrst-layer units of a network are linear, so the component (cid:83) i (cid:8) z | h 1:

Original: i ( z ) = 0 (cid:9) of the fold-set (9) is a union of hyperplanes, illustrated by the blue lines in Figure 1.
Revised: i ( z ) = 0 (cid:9) of the fold-set (9) is a union of hyperplanes, illustrated by the blue lines in Figure 1.

Original: These hyperplanes partition the input space into a number of regions that each correspond to a different activation pattern.
Revised: These hyperplanes partition the input space into a number of regions that each correspond to a different activation pattern.

Original: For a ﬁxed activation pattern, or equivalently on each region, the second-layer units are linear, so their zero-level sets (cid:83) i (cid:8) z | h 1:2 i ( z ) = 0 (cid:9) are composed of piece-wise hyperplanes on the partition induced by the ﬁrst-layer units.
Revised: For a ﬁxed activation pattern, or equivalently on each region, the second-layer units are linear, so their zero-level sets (cid:83) i (cid:8) z | h 1:2 i ( z ) = 0 (cid:9) are composed of piece-wise hyperplanes on the partition induced by the ﬁrst-layer units.

Original: This is shown by the orange lines in Figure 1.
Revised: This is shown by the orange lines in Figure 1.

Original: More generally, the l th -layer zero-level sets (cid:83) i (cid:8) z | h 1: l i ( z ) = 0 (cid:9) consist of piece-wise hyperplanes on the partition induced by all lower-layer units.
Revised: More generally, the l th -layer zero-level sets (cid:83) i (cid:8) z | h 1: l i ( z ) = 0 (cid:9) consist of piece-wise hyperplanes on the partition induced by all lower-layer units.

Original: This yields a fold-set that looks like the set in the right pane of Figure 1, but potentially much more complicated.
Revised: This yields a fold-set that looks like the set in the right pane of Figure 1, but potentially much more complicated.

Original: We now deﬁne these concepts more precisely.
Revised: We now deﬁne these concepts more precisely.

Original: Piece-wise hyperplane.
Revised: Piece-wise hyperplane.

Original: Let P be a partition of Z .
Revised: Let P be a partition of Z .

Original: We say H ⊆
Revised: We say H ⊆

Original: Z is a piece-wise hyperplane with respect to partition P , if H is nonempty and there exist ( w , b ) (cid:54) =
Revised: Z is a piece-wise hyperplane with respect to partition P , if H is nonempty and there exist ( w , b ) (cid:54) =

Original: ( 0 , 0) and P ∈ P
Revised: ( 0 , 0) and P ∈ P

Original: such that H = { z ∈ P | w (cid:124)
Revised: such that H = { z ∈ P | w (cid:124)

Original: z
Revised: z

Original: + b = 0 } .
Revised: + b = 0 } .

Original: Piece-wise linear surface.
Revised: Piece-wise linear surface.

Original: A set S ⊆
Revised: A set S ⊆

Original: Z is called a piece-wise linear surface on Z of order κ if it has a representation of the form S =
Revised: Z is called a piece-wise linear surface on Z of order κ if it has a representation of the form S =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li , where each H li is a piece-wise hyperplane with respect to the partition induced by (cid:83) k ∈ [ l − 1] ,j ∈ [ n k ] H kj , and no number smaller than κ admits such a representation.
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li , where each H li is a piece-wise hyperplane with respect to the partition induced by (cid:83) k ∈ [ l − 1] ,j ∈ [ n k ] H kj , and no number smaller than κ admits such a representation.

Original: Using these deﬁnitions, the following lemma formalises the intuition behind Figure 1.
Revised: Using these deﬁnitions, the following lemma formalises the intuition behind Figure 1.

Original: Lemma 2.
Revised: Lemma 2.

Original: If h is a general and transparent ReLU network, then its fold-set is a piece-wise linear surface of order at most L − 1 .
Revised: If h is a general and transparent ReLU network, then its fold-set is a piece-wise linear surface of order at most L − 1 .

Original: The ﬁnal ingredient we will need to be able to reason about the parameterisation of ReLU networks is a more precise characterisation of the fold-set, in particular, the dependence structure between individual piece-wise hyperplanes.
Revised: The ﬁnal ingredient we will need to be able to reason about the parameterisation of ReLU networks is a more precise characterisation of the fold-set, in particular, the dependence structure between individual piece-wise hyperplanes.

Original: For example, consider the piece-wise linear surface in Figureand compare it to the one in Figure 2.
Revised: For example, consider the piece-wise linear surface in Figureand compare it to the one in Figure 2.

Original: Suppose as before that the blue hyperplanes come from ﬁrst-layer units, the orange hyperplanes come from second-layer units, and the black hyperplanes come from third-layer units.
Revised: Suppose as before that the blue hyperplanes come from ﬁrst-layer units, the orange hyperplanes come from second-layer units, and the black hyperplanes come from third-layer units.

Original: The difference between Figure 1 and Figure 2 is that if we observe only the fold-set, i.e. only the union of the zero-level sets over all layers (as shown in the right pane of
Revised: The difference between Figure 1 and Figure 2 is that if we observe only the fold-set, i.e. only the union of the zero-level sets over all layers (as shown in the right pane of

Original: Figure 2), then in the case of Figure 2, it is impossible to know which folds come from which layers.
Revised: Figure 2), then in the case of Figure 2, it is impossible to know which folds come from which layers.

Original: For instance, the blue folds and the orange folds could be assigned to the ﬁrst and second layer almost arbitrarily; there is not enough information (i.e. intersection) in the fold-set to tell which is which.
Revised: For instance, the blue folds and the orange folds could be assigned to the ﬁrst and second layer almost arbitrarily; there is not enough information (i.e. intersection) in the fold-set to tell which is which.

Original: In contrast, the piece-wise linear surface in the right pane of Figure 1 could in principle be disambiguated into ﬁrst-, second- and third- layer folds by the following procedure:
Revised: In contrast, the piece-wise linear surface in the right pane of Figure 1 could in principle be disambiguated into ﬁrst-, second- and third- layer folds by the following procedure:

Original: Take the largest possible union of hyperplanes that is a subset of the fold-set, and assign the hyperplanes to layer one. 2.
Revised: Take the largest possible union of hyperplanes that is a subset of the fold-set, and assign the hyperplanes to layer one. 2.

Original: Take all piece-wise hyperplanes with respect to the partition induced by the ﬁrst-layer folds, and assign them to layer two. 3.
Revised: Take all piece-wise hyperplanes with respect to the partition induced by the ﬁrst-layer folds, and assign them to layer two. 3.

Original: Take all piece-wise hyperplanes with respect to the partition induced by the ﬁrst- and second- layer folds, and assign them to layer three.
Revised: Take all piece-wise hyperplanes with respect to the partition induced by the ﬁrst- and second- layer folds, and assign them to layer three.

Original: This procedure is not guaranteed to assign all folds to their original layers because it ignores how piece-wise hyperplanes are connected; for example for the piece-wise linear surface in Figure 1, the procedure yields the layer assignment shown in Figure 3.
Revised: This procedure is not guaranteed to assign all folds to their original layers because it ignores how piece-wise hyperplanes are connected; for example for the piece-wise linear surface in Figure 1, the procedure yields the layer assignment shown in Figure 3.

Original: However, it is sufﬁcient for our purposes, and it is easier to work with mathematically.
Revised: However, it is sufﬁcient for our purposes, and it is easier to work with mathematically.

Original: Formally, for a piece-wise linear surface S , we denote
Revised: Formally, for a piece-wise linear surface S , we denote

Original: [Equation]
Revised: [Equation]

Original: wise linear surface
Revised: linear surface

Original: [Equation]
Revised: [Equation]

Original: order
Revised: order

Original: [Equation]
Revised: [Equation]

Original: One can show 4 that (cid:3) k S is itself a piece-wise linear surface of order at most k , so one can think of (cid:3) k S as the ‘largest possible’ subset of S that is a piece-wise linear surface of order at most k .
Revised: One can show 4 that (cid:3) k S is itself a piece-wise linear surface of order at most k , so one can think of (cid:3) k S as the ‘largest possible’ subset of S that is a piece-wise linear surface of order at most k .

Original: For the piece-wise linear surface in Figure 3, the set (cid:3) 1 S consists of the blue hyperplanes, (cid:3) 2 S consists of the blue and the orange (piece-wise) hyperplanes, and (cid:3) 3 S = S .
Revised: For the piece-wise linear surface in Figure 3, the set (cid:3) 1 S consists of the blue hyperplanes, (cid:3) 2 S consists of the blue and the orange (piece-wise) hyperplanes, and (cid:3) 3 S = S .

Original: This deﬁnition allows us to uniquely decompose S into its piece-wise hyperplanes.
Revised: This deﬁnition allows us to uniquely decompose S into its piece-wise hyperplanes.

Original: Let S =
Revised: Let S =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li be any representation of S in terms of its piece-wise hyperplanes.
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li be any representation of S in terms of its piece-wise hyperplanes.

Original: We say the representation is canonical if each H li is distinct and (cid:83) l ∈
Revised: We say the representation is canonical if each H li is distinct and (cid:83) l ∈

Original: [ k ] ,i ∈
Revised: [ k ] ,i ∈

Original: [ n l ] H li = (cid:3) k S for all k ∈ [ κ ] .
Revised: [ n l ] H li = (cid:3) k S for all k ∈ [ κ ] .

Original: One can show 5 that such a representation exists and is unique up to subscript indexing.
Revised: One can show 5 that such a representation exists and is unique up to subscript indexing.

Original: Importantly, it assigns a unique ‘layer’ to each piece-wise hyperplane, its superscript.
Revised: Importantly, it assigns a unique ‘layer’ to each piece-wise hyperplane, its superscript.

Original: See
Revised: See

Original: H
Revised: H

Original: H H
Revised: H H

Original: H H
Revised: H H

Original: H H
Revised: H H

Original: H
Revised: H

Original: The dependency graph (see also Figure 4) is a way to formally describe the dependencies between piece-wise hyperplanes.
Revised: The dependency graph (see also Figure 4) is a way to formally describe the dependencies between piece-wise hyperplanes.

Original: Dependency graph.
Revised: Dependency graph.

Original: Let S =
Revised: Let S =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li be the canonical representation of S .
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li be the canonical representation of S .

Original: The dependency graph of S is the directed graph that has the piece-wise hyperplanes (cid:8) H li (cid:9) l,i as vertices, and has an edge H li → H kj iff l < k and relint H li ∩ cl H kj (cid:54) = ∅ .
Revised: The dependency graph of S is the directed graph that has the piece-wise hyperplanes (cid:8) H li (cid:9) l,i as vertices, and has an edge H li → H kj iff l < k and relint H li ∩ cl H kj (cid:54) = ∅ .

Original: That is, there is and edge H li
Revised: That is, there is and edge H li

Original: → H kj if
Revised: → H kj if

Original: H kj ‘depends on’ or ‘bends at’ H li .
Revised: H kj ‘depends on’ or ‘bends at’ H li .

Original:  M AIN RESULT
Revised:  M AIN RESULT

Original: With all the necessary concepts in place, we now put the pieces together and explain the proof idea behind the main result.
Revised: With all the necessary concepts in place, we now put the pieces together and explain the proof idea behind the main result.

Original: We restate the theorem here for the reader’s convenience.
Revised: We restate the theorem here for the reader’s convenience.

Original: Theorem 1.
Revised: Theorem 1.

Original: Consider a bounded open nonempty domain Z ⊆
Revised: Consider a bounded open nonempty domain Z ⊆

Original: R
Revised: R

Original: d 0 and any architecture ( d 0 , . . .
Revised: d 0 and any architecture ( d 0 , . . .

Original: , d L ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 , d L = 1 .
Revised: , d L ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 , d L = 1 .

Original: For this architecture, there exists a
Revised: For this architecture, there exists a

Original: ReLU network h θ : Z → R such that for any general ReLU network h
Revised: ReLU network h θ : Z → R such that for any general ReLU network h

Original: η : Z → R (with the same architecture) satisfying h θ ( z ) =
Revised: η : Z → R (with the same architecture) satisfying h θ ( z ) =

Original: h η ( z ) for all z ∈ Z , there exist permutation matrices P 1 , . . .
Revised: h η ( z ) for all z ∈ Z , there exist permutation matrices P 1 , . . .

Original: P L − 1 , and positive diagonal matrices M 1 , . .
Revised: P L − 1 , and positive diagonal matrices M 1 , . .

Original: , M L − 1 , such that
Revised: , M L − 1 , such that

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

Original: L = W (cid:48)
Revised: = [Equation]

Original: L P − 1 L − 1 M − 1 L − 1 , b L = b (cid:48) 
Revised: − [Equation] (cid:48) [Equation]

Original: L , where ( W 1 , b 1 , . .
Revised: where ( W 1 , b 1 , . .

Original: , W L , b L ) are the parameters of h θ , and ( W (cid:48) 1 , b (cid:48) 1 , . . .
Revised: , W L , b L ) are the parameters of h θ , and ( W (cid:48) 1 , b (cid:48) 1 , . . .

Original: , W (cid:48)
Revised: , W (cid:48)

Original: L , b (cid:48) L ) are the parameters of h η .
Revised: L , b (cid:48) L ) are the parameters of h η .

Original: In other words, for architectures with non-increasing widths, there exists a ReLU network h such that knowledge of the input-output mapping h determines the network’s parameters uniquely up to permutation and scaling.
Revised: In other words, for architectures with non-increasing widths, there exists a ReLU network h such that knowledge of the input-output mapping h determines the network’s parameters uniquely up to permutation and scaling.

Original: The idea behind the proof is as follows.
Revised: The idea behind the proof is as follows.

Original: Suppose we are given the function h .
Revised: Suppose we are given the function h .

Original: Then we also know its fold-set F ( h ) , and if h is general and transparent, the fold-set is a piece-wise linear surface (by
Revised: Then we also know its fold-set F ( h ) , and if h is general and transparent, the fold-set is a piece-wise linear surface (by

Original: Lemma 2) of the form F ( h ) =
Revised: Lemma 2) of the form F ( h ) =

Original: (cid:83) l,i (cid:8) z | h 1: l i ( z ) = 0 (cid:9) .
Revised: (cid:83) l,i (cid:8) z | h 1: l i ( z ) = 0 (cid:9) .

Original: As we have mentioned earlier, this union of zero-level sets contains a lot of information about the network’s parameters, provided we can disambiguate the union to obtain the zero-level sets of individual units.
Revised: As we have mentioned earlier, this union of zero-level sets contains a lot of information about the network’s parameters, provided we can disambiguate the union to obtain the zero-level sets of individual units.

Original: This disambiguation of the union is crucial, but is impossible in general.
Revised: This disambiguation of the union is crucial, but is impossible in general.

Original: To see why, consider the ﬁrst-layer units: given F ( h ) , we want to identify (cid:83) i (cid:8) z |
Revised: To see why, consider the ﬁrst-layer units: given F ( h ) , we want to identify (cid:83) i (cid:8) z |

Original: h 1:1 i ( z ) = 0 (cid:9) .
Revised: h 1:1 i ( z ) = 0 (cid:9) .

Original: We know that (cid:83) i (cid:8) z |
Revised: We know that (cid:83) i (cid:8) z |

Original: h 1:

Revised: h 1:

Original: i ( z ) = 0 (cid:9) is a union of d 1 hyperplanes that is a subset of (cid:3) 1 F ( h ) , so if (cid:3) 1 F ( h ) is a union of d 1 hyperplanes, we are done.
Revised: i ( z ) = 0 (cid:9) is a union of d 1 hyperplanes that is a subset of (cid:3) 1 F ( h ) , so if (cid:3) 1 F ( h ) is a union of d 1 hyperplanes, we are done.

Original: In general however, F ( h ) may contain more than d 1 hyperplanes, such as for example in Figure 2.
Revised: In general however, F ( h ) may contain more than d 1 hyperplanes, such as for example in Figure 2.

Original: In such a setting it is impossible to tell which hyperplanes come from the ﬁrst layer.
Revised: In such a setting it is impossible to tell which hyperplanes come from the ﬁrst layer.

Original: The key insight here is the following: even though, say, a last-layer unit can create a fold that looks like a hyperplane, this hyperplane cannot have any dependencies, or descendants in the dependency graph.
Revised: The key insight here is the following: even though, say, a last-layer unit can create a fold that looks like a hyperplane, this hyperplane cannot have any dependencies, or descendants in the dependency graph.

Original: This follows from the fact that the layer is the last.
Revised: This follows from the fact that the layer is the last.

Original: More generally, if a (piece-wise) hyperplane has a chain of descendants of length m , it must come from a layer that is at least m layers below the last one.
Revised: More generally, if a (piece-wise) hyperplane has a chain of descendants of length m , it must come from a layer that is at least m layers below the last one.

Original: Formally, we have the following lemma.
Revised: Formally, we have the following lemma.

Original: Lemma 3.
Revised: Lemma 3.

Original: Let h : Z → R be a general ReLU network.
Revised: Let h : Z → R be a general ReLU network.

Original: Denote S :
Revised: Denote S :

Original: = (cid:83)
Revised: = (cid:83)

Original: [Equation]

Original: [Equation]

Original: [Equation]

Original: ] (cid:8)

Original: z

Original: [Equation]
Revised: [Equation]

Original: = 0 (cid:9) and let S = (cid:83) k ∈ [ κ ] ,j ∈
Revised: and let S = (cid:83) k ∈ [ κ ] ,j ∈

Original: [ n k ] H kj be the canonical representation of S .
Revised: [ n k ] H kj be the canonical representation of S .

Original: Then for all H kj there exists a unit ( l, i ) with l ≥ k such that H kj ⊆ (cid:8) z
Revised: Then for all H kj there exists a unit ( l, i ) with l ≥ k such that H kj ⊆ (cid:8) z

Original: | h 1: l i ( z ) = 0 (cid:9) .
Revised: | h 1: l i ( z ) = 0 (cid:9) .

Original: Moreover, if the dependency graph of S contains a directed path of length m starting at H kj , then l ≤ λ − m .
Revised: Moreover, if the dependency graph of S contains a directed path of length m starting at H kj , then l ≤ λ − m .

Original: Main proof idea.
Revised: Main proof idea.

Original: This lemma motivates the main idea of the proof.
Revised: This lemma motivates the main idea of the proof.

Original: We explicitly construct a network h such that the dependency graph of its fold-set is well connected.
Revised: We explicitly construct a network h such that the dependency graph of its fold-set is well connected.

Original: More precisely, we ensure that each of the hyperplanes corresponding to ﬁrst-layer units has a chain of descendants of length L − 2 .
Revised: More precisely, we ensure that each of the hyperplanes corresponding to ﬁrst-layer units has a chain of descendants of length L − 2 .

Original: This implies by Lemma 3 that the ﬁrst-layer hyperplanes can be identiﬁed as such, using only the information contained in the fold-set.
Revised: This implies by Lemma 3 that the ﬁrst-layer hyperplanes can be identiﬁed as such, using only the information contained in the fold-set.

Original: One can show that this is sufﬁcient to recover the parameters W 1 , b 1 , up to permutation and scaling.
Revised: One can show that this is sufﬁcient to recover the parameters W 1 , b 1 , up to permutation and scaling.

Original: To extend the argument to higher-layers, we then consider the truncated network h l : L .
Revised: To extend the argument to higher-layers, we then consider the truncated network h l : L .

Original: In h l : L , layer l becomes the ﬁrst layer, and we apply the same reasoning as above to recover W l , b l .
Revised: In h l : L , layer l becomes the ﬁrst layer, and we apply the same reasoning as above to recover W l , b l .

Original: The next lemma shows that a network with a ‘well connected’ dependency graph exists.
Revised: The next lemma shows that a network with a ‘well connected’ dependency graph exists.

Original: In what follows, f | A denotes the restriction of a function f to a domain A , and Z l θ (cid:44) (cid:8) σ ( h 1: l θ ( z ))
Revised: In what follows, f | A denotes the restriction of a function f to a domain A , and Z l θ (cid:44) (cid:8) σ ( h 1: l θ ( z ))

Original: | z ∈ Z (cid:9) is the set of all possible inputs to the truncated network h l : L .
Revised: | z ∈ Z (cid:9) is the set of all possible inputs to the truncated network h l : L .

Original: For notational convenience, we deﬁne
Revised: For notational convenience, we deﬁne

Original: Z 0 θ (cid:44) Z .
Revised: Z 0 θ (cid:44) Z .

Original: Lemma 4.
Revised: Lemma 4.

Original: For a bounded open nonempty domain Z and architecture ( d 0 , . . .
Revised: For a bounded open nonempty domain Z and architecture ( d 0 , . . .

Original: , d L ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 , d L = 1 , there exists a general transparent ReLU network h :
Revised: , d L ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 , d L = 1 , there exists a general transparent ReLU network h :

Original: Z → R such that for l ∈ [ L − 1] , the fold-set F ( h l : L | int Z l − 1 ) is a piece-wise linear surface whose dependency graph contains d l directed paths of length ( L − 1 − l ) with distinct starting vertices.
Revised: Z → R such that for l ∈ [ L − 1] , the fold-set F ( h l : L | int Z l − 1 ) is a piece-wise linear surface whose dependency graph contains d l directed paths of length ( L − 1 − l ) with distinct starting vertices.

Original: Theorem 1 then follows by the inductive argument outlined above.
Revised: Theorem 1 then follows by the inductive argument outlined above.

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

Original: Let h
Revised: Let h

Original: θ be the network from Lemma 4.
Revised: θ be the network from Lemma 4.

Original: One can show that if h θ is transparent, and h η ( z ) =
Revised: One can show that if h θ is transparent, and h η ( z ) =

Original: h θ ( z ) for all z ∈ Z , then also h η is transparent, and all the truncated networks h l : L θ , h l : L η are transparent.
Revised: h θ ( z ) for all z ∈ Z , then also h η is transparent, and all the truncated networks h l : L θ , h l : L η are transparent.

Original: We proceed by induction.
Revised: We proceed by induction.

Original: Let l = 1 .
Revised: Let l = 1 .

Original: Then we have
Revised: Then we have

Original: [Equation]
Revised: [Equation]

Original: (12) which implies F ( h l : L θ | int Z l − 1 θ )
Revised: (12) which implies F ( h l : L θ | int Z l − 1 θ )

Original: =
Revised: =

Original: F ( h l : L η | int Z l − 1 θ ) .
Revised: F ( h l : L η | int Z l − 1 θ ) .

Original: (For notational convenience, we will omit the domain restriction for now.)
Revised: (For notational convenience, we will omit the domain restriction for now.)

Original: Because both networks are general and transparent, the fold-sets are representable as unions of the respective zero-level sets, and we obtain 
Revised: Because both networks are general and transparent, the fold-sets are representable as unions of the respective zero-level sets, and we obtain (cid:91)

Original: [Equation]
Revised: [Equation]

Original: This is a piece-wise linear surface, whose dependency graph by Lemma 4 contains d l directed paths of length ( L − 1 − l ) with distinct starting vertices.
Revised: This is a piece-wise linear surface, whose dependency graph by Lemma 4 contains d l directed paths of length ( L − 1 − l ) with distinct starting vertices.

Original: Denote these vertices H 1 , . . .
Revised: Denote these vertices H 1 , . . .

Original: , H d l .
Revised: , H d l .

Original: By Lemma 3,
Revised: By Lemma 3,

Original: H i ⊆
Revised: H i ⊆

Original: (cid:8) z | h l : l − 1+ λ θ [ ι ]( z ) = 0 (cid:9) for some ( λ, ι ) with λ ≤ ( L − l ) − ( L − 1 − l ) = 1 .
Revised: (cid:8) z | h l : l − 1+ λ θ [ ι ]( z ) = 0 (cid:9) for some ( λ, ι ) with λ ≤ ( L − l ) − ( L − 1 − l ) = 1 .

Original: We thus obtain (cid:83) i ∈
Revised: We thus obtain (cid:83) i ∈

Original: [ d l ] H i ⊆
Revised: [ d l ] H i ⊆

Original: (cid:83)
Revised: (cid:83)

Original: i ∈
Revised: i ∈

Original: [ d l ] (cid:8) z | h l θ [ ι ]( z ) = 0 (cid:9) , where on the left-hand side we have a union of d l hyperplanes, and on the right-hand side we have a union of at most d l hyperplanes.
Revised: [ d l ] (cid:8) z | h l θ [ ι ]( z ) = 0 (cid:9) , where on the left-hand side we have a union of d l hyperplanes, and on the right-hand side we have a union of at most d l hyperplanes.

Original: It follows that the two sides are equal, and by applying the same argument to h η , we get
Revised: It follows that the two sides are equal, and by applying the same argument to h η , we get

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: Therefore there must exist a permutation π : [ d l ] → [ d l ] such that
Revised: Therefore there must exist a permutation π : [ d l ] → [ d l ] such that

Original: [Equation]
Revised: [Equation]

Original: for all i .
Revised: for all i .

Original: One can show 6 that this implies the existence of scalars m 1 , . . .
Revised: One can show 6 that this implies the existence of scalars m 1 , . . .

Original: m d l , such that
Revised: m d l , such that

Original: [Equation]
Revised: [Equation]

Original: We know that m i (cid:54)
Revised: We know that m i (cid:54)

Original: = 

Revised: =

Original: because the folds (cid:8) z | h l θ [ i ]( z ) = 0 (cid:9) , (cid:8) z | h l η [ i ]( z ) = 0 (cid:9) , are nonempty; otherwise (cid:83) i ∈
Revised: because the folds (cid:8) z | h l θ [ i ]( z ) = 0 (cid:9) , (cid:8) z | h l η [ i ]( z ) = 0 (cid:9) , are nonempty; otherwise (cid:83) i ∈

Original: [ d l ] H i could not be a union of d l hyperplanes.
Revised: [ d l ] H i could not be a union of d l hyperplanes.

Original: We have thus shown that there exists a permutation matrix P l ∈ R d l × d l and a nonzero-entry diagonal matrix M l ∈ R d
Revised: We have thus shown that there exists a permutation matrix P l ∈ R d l × d l and a nonzero-entry diagonal matrix M l ∈ R d

Original: l × d l such that
Revised: l × d l such that

Original: W l = M l P l W (cid:48) l and b l = M l P l b (cid:48) l .
Revised: W l = M l P l W (cid:48) l and b l = M l P l b (cid:48) l .

Original: One can also show that the scalars m i are positive.
Revised: One can also show that the scalars m i are positive.

Original: For the inductive step, let l ∈ { 2 , . . .
Revised: For the inductive step, let l ∈ { 2 , . . .

Original: , L − 1 } , and assume that there exist permutation matrices
Revised: , L − 1 } , and assume that there exist permutation matrices

Original: P 1 , . .
Revised: P 1 , . .

Original: , P l − 1 , and positive-entry diagonal matrices M 1 , . . .
Revised: , P l − 1 , and positive-entry diagonal matrices M 1 , . . .

Original: , M l − 1 , such that (65) holds up to layer l − 1 .
Revised: , M l − 1 , such that (65) holds up to layer l − 1 .

Original: Then h 1: l − 1 θ ≡ M
Revised: Then h 1: l − 1 θ ≡ M

Original: [Equation]
Revised: [Equation]

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original:  Since the end-to-end mappings are the same, h 1: L θ ≡ h 1: L η , it follows that the truncated mappings satisfy
Revised: η . Since the end-to-end mappings are the same, h 1: L θ ≡ h 1: L η , it follows that the truncated mappings satisfy

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: where ˜ η := ( W (cid:48)
Revised: where ˜ η := ( W (cid:48)

Original: l P − 1 l − 1 M − 1 l − 1 , b (cid:48) l , W (cid:48)
Revised: l P − 1 l − 1 M − 1 l − 1 , b (cid:48) l , W (cid:48)

Original: l +1 , b (cid:48)
Revised: l +1 , b (cid:48)

Original: l +1 , . . .
Revised: l +1 , . . .

Original: , W (cid:48)
Revised: , W (cid:48)

Original: L , b (cid:48) L ) .
Revised: L , b (cid:48) L ) .

Original: We therefore apply the same argument to h l : L θ | int Z l − 1 θ and h l : L ˜ η
Revised: We therefore apply the same argument to h l : L θ | int Z l − 1 θ and h l : L ˜ η

Original: | int Z l − 1 θ as we presented above for the case l = 1 .
Revised: | int Z l − 1 θ as we presented above for the case l = 1 .

Original: We obtain that there exists a permutation matrix P l ∈ R d l × d l and a positive-entry diagonal matrix M l ∈ R d
Revised: We obtain that there exists a permutation matrix P l ∈ R d l × d l and a positive-entry diagonal matrix M l ∈ R d

Original: l × d l such that
Revised: l × d l such that

Original: [Equation]
Revised: [Equation]

Original: Finally, consider the last layer.
Revised: Finally, consider the last layer.

Original: We know that h 1: L − 1 θ ≡ M L − 1 P L − 1 h 1: L − 1 η , which implies h L θ ≡ h L η ◦ P − 1 L − 1 M − 1 L − 1 , i.e. h L θ and h L η ◦ P − 1 L − 1 M − 1 L − 1 are identical linear functions supported on the full-dimensional domain Z L − 1 θ .
Revised: We know that h 1: L − 1 θ ≡ M L − 1 P L − 1 h 1: L − 1 η , which implies h L θ ≡ h L η ◦ P − 1 L − 1 M − 1 L − 1 , i.e. h L θ and h L η ◦ P − 1 L − 1 M − 1 L − 1 are identical linear functions supported on the full-dimensional domain Z L − 1 θ .

Original: It follows that W L = W (cid:48)
Revised: It follows that W L = W (cid:48)

Original: L P − 1 L − 1 M − 1 L − 1 and b L = b (cid:48)
Revised: L P − 1 L − 1 M − 1 L − 1 and b L = b (cid:48)

Original: L .
Revised: L .

Original: Discussion of assumptions.
Revised: Discussion of assumptions.

Original: Most of the theorem’s assumptions have their origin in Lemma 4.
Revised: Most of the theorem’s assumptions have their origin in Lemma 4.

Original: The reason we restrict the domain of h l : L to the interior of Z l − 1 is that we want h l : L to be deﬁned on an open set (otherwise fold-sets become unwieldy).
Revised: The reason we restrict the domain of h l : L to the interior of Z l − 1 is that we want h l : L to be deﬁned on an open set (otherwise fold-sets become unwieldy).

Original: For similar reasons, we study only architectures with non-increasing widths; otherwise int Z l − 1 may be empty.
Revised: For similar reasons, we study only architectures with non-increasing widths; otherwise int Z l − 1 may be empty.

Original: We conjecture that the theorem does not hold for more general architectures.
Revised: We conjecture that the theorem does not hold for more general architectures.

Original: If it does, the proof will likely go beyond fold-sets.
Revised: If it does, the proof will likely go beyond fold-sets.

Original: To guarantee transparency, our construction is such that for each input z ∈ Z and layer l ∈ [ L − 1] , either h 1: l 1 ( z ) > 0 or h 1: l 2 ( z ) > 0 .
Revised: To guarantee transparency, our construction is such that for each input z ∈ Z and layer l ∈ [ L − 1] , either h 1: l 1 ( z ) > 0 or h 1: l 2 ( z ) > 0 .

Original: Transparency could in principle be achieved with just a single unit, but it would have to be positive everywhere.
Revised: Transparency could in principle be achieved with just a single unit, but it would have to be positive everywhere.

Original: This is why we impose d l ≥ 2 .
Revised: This is why we impose d l ≥ 2 .

Original: Guaranteeing transparency for the ﬁrst layer (whose inputs are not constrained to the positive quadrant) also necessitates boundedness of Z .
Revised: Guaranteeing transparency for the ﬁrst layer (whose inputs are not constrained to the positive quadrant) also necessitates boundedness of Z .

Original: Boundedness can be lifted if we consider a slightly modiﬁed deﬁnition of transparency; proofs become more complicated though and we do not consider this crucial.
Revised: Boundedness can be lifted if we consider a slightly modiﬁed deﬁnition of transparency; proofs become more complicated though and we do not consider this crucial.

Original: Almost all of the proof carries over to the case of leaky ReLU activations (where σ is deﬁned as σ ( u ) i = max { αu i , u i } for some small α > 0 ).
Revised: Almost all of the proof carries over to the case of leaky ReLU activations (where σ is deﬁned as σ ( u ) i = max { αu i , u i } for some small α > 0 ).

Original: The part that does not carry over is our proof that
Revised: The part that does not carry over is our proof that

Original: M l has only positive entries on the diagonal:
Revised: M l has only positive entries on the diagonal:

Original: In this part, we compare the slope of h l : L θ for inputs on the positive and negative side of a given ReLU unit, and notice that the negative-side slope is ‘singular’ in the sense that some basis directions have zero magnitude.
Revised: In this part, we compare the slope of h l : L θ for inputs on the positive and negative side of a given ReLU unit, and notice that the negative-side slope is ‘singular’ in the sense that some basis directions have zero magnitude.

Original: This particular argument does not work for the leaky ReLU, though we cannot rule out that a simple workaround exists.
Revised: This particular argument does not work for the leaky ReLU, though we cannot rule out that a simple workaround exists.

Original:  D ISCUSSION & FUTURE WORK
Revised:  D ISCUSSION & FUTURE WORK

Original: In this work, we have shown that for architectures with non-increasing widths, certain ReLU networks are almost uniquely identiﬁed by the function they implement.
Revised: In this work, we have shown that for architectures with non-increasing widths, certain ReLU networks are almost uniquely identiﬁed by the function they implement.

Original: The result suggests that the function-equivalence classes of ReLU networks are surprisingly small, i.e. there may be only little redundancy in the way ReLU networks are parameterised, contrary to what is commonly believed.
Revised: The result suggests that the function-equivalence classes of ReLU networks are surprisingly small, i.e. there may be only little redundancy in the way ReLU networks are parameterised, contrary to what is commonly believed.

Original: This apparent contradiction could be explained in a number of ways:
Revised: This apparent contradiction could be explained in a number of ways:

Original: • It could be the case that even though exact equivalence classes are small, approximate equivalence is much easier to achieve.
Revised: • It could be the case that even though exact equivalence classes are small, approximate equivalence is much easier to achieve.

Original: That is, it could be that (cid:107) h θ − h η (cid:107) ≤ (cid:15) is satisﬁed
Revised: That is, it could be that (cid:107) h θ − h η (cid:107) ≤ (cid:15) is satisﬁed

Original: 6 See Appendix, Lemma A.19.See Appendix, Theorem A.1.
Revised: 6 See Appendix, Lemma A.19.See Appendix, Theorem A.1.

Original: by a disproportionately larger class of parameters η than (cid:107) h θ − h η (cid:107) = 0 .
Revised: by a disproportionately larger class of parameters η than (cid:107) h θ − h η (cid:107) = 0 .

Original: This issue is related to the so-called inverse stability of the realisation map of neural nets, which is not yet well understood.
Revised: This issue is related to the so-called inverse stability of the realisation map of neural nets, which is not yet well understood.

Original: • Another possibility is that the kind of networks we consider in this paper is not representative of networks typically encountered in practice,
Revised: • Another possibility is that the kind of networks we consider in this paper is not representative of networks typically encountered in practice,

Original: i
Revised: i

Original: e. it could be that ‘typical networks’ do not have well connected dependency graphs, and are therefore not easily identiﬁable.
Revised: e. it could be that ‘typical networks’ do not have well connected dependency graphs, and are therefore not easily identiﬁable.

Original: Finally, we have considered only architectures with non-increasing widths, whereas some previous theoretical work has assumed much wider intermediate layers compared to the input dimension.
Revised: Finally, we have considered only architectures with non-increasing widths, whereas some previous theoretical work has assumed much wider intermediate layers compared to the input dimension.

Original: It is possible that parameterisation redundancy is much larger in such a regime compared to ours.
Revised: It is possible that parameterisation redundancy is much larger in such a regime compared to ours.

Original: However, gains from over-parameterisation have also been observed in practical settings with architectures not unlike those considered here.
Revised: However, gains from over-parameterisation have also been observed in practical settings with architectures not unlike those considered here.

Original: We consider these questions important directions for further research.
Revised: We consider these questions important directions for further research.

Original: We also hypothesise that our analysis could be extended to convolutional and recurrent networks, and to other piece-wise linear activation functions such as leaky
Revised: We also hypothesise that our analysis could be extended to convolutional and recurrent networks, and to other piece-wise linear activation functions such as leaky

Original: ReLU
Revised: ReLU

Original: A A PPENDIX
Revised: A A PPENDIX

Original: Use the following look-up table to ﬁnd a particular lemma or theorem and its proof.
Revised: Use the following look-up table to ﬁnd a particular lemma or theorem and its proof.

Original: Lemma→
Revised: Lemma→

Original: Lemma A.
Revised: Lemma A.

Original: Lemma→
Revised: Lemma→

Original: Corollary A.
Revised: Corollary A.

Original: Lemma→
Revised: Lemma→

Original: Lemma A.
Revised: Lemma A.

Original: Lemma→
Revised: Lemma→

Original: Lemma A.
Revised: Lemma A.

Original: Theorem→
Revised: Theorem→

Original: Theorem A.
Revised: Theorem A.

Original: A. P IECE - WISE LINEAR SURFACES
Revised: A. P IECE - WISE LINEAR SURFACES

Original: Deﬁnition A.1 (Partition) .
Revised: Deﬁnition A.1 (Partition) .

Original: Let S ⊆
Revised: Let S ⊆

Original: Z .
Revised: Z .

Original: We deﬁne the partition of Z induced by S , denoted P Z ( S ) , as the set of connected components of Z \ S .
Revised: We deﬁne the partition of Z induced by S , denoted P Z ( S ) , as the set of connected components of Z \ S .

Original: Deﬁnition A.2 (Piece-wise hyperplane) .
Revised: Deﬁnition A.2 (Piece-wise hyperplane) .

Original: Let P be a partition of Z .
Revised: Let P be a partition of Z .

Original: We say H ⊆
Revised: We say H ⊆

Original: Z is a piece-wise hyperplane with respect to partition P , if H (cid:54) = ∅ and there exist ( w , b ) (cid:54) =
Revised: Z is a piece-wise hyperplane with respect to partition P , if H (cid:54) = ∅ and there exist ( w , b ) (cid:54) =

Original: ( 0 , 0) and P ∈ P
Revised: ( 0 , 0) and P ∈ P

Original: such that H = { z ∈ P | w (cid:124)
Revised: such that H = { z ∈ P | w (cid:124)

Original: z
Revised: z

Original: + b = 0 } .
Revised: + b = 0 } .

Original: Deﬁnition A.3 (Piece-wise linear surface / pwl. surface) .
Revised: Deﬁnition A.3 (Piece-wise linear surface / pwl. surface) .

Original: A set S ⊆
Revised: A set S ⊆

Original: Z is called a piece-wise linear surface on Z of order κ if it can be written as S
Revised: Z is called a piece-wise linear surface on Z of order κ if it can be written as S

Original: =
Revised: =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li , where each H li is a piece-wise hyperplane with respect to P Z ( (cid:83)
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li , where each H li is a piece-wise hyperplane with respect to P Z ( (cid:83)

Original: k ∈
Revised: k ∈

Original: [ l − 1] ,j ∈ [ n k ] H kj ) , and no number smaller than κ admits such a representation.
Revised: [ l − 1] ,j ∈ [ n k ] H kj ) , and no number smaller than κ admits such a representation.

Original: Lemma A.1.
Revised: Lemma A.1.

Original: If S 1 , S 2 are piece-wise linear surfaces on Z of order k 1 and k 2 , then S 1 ∪ S 2 is a piece-wise linear surface on Z of order at most max { k 1 , k 2 } .
Revised: If S 1 , S 2 are piece-wise linear surfaces on Z of order k 1 and k 2 , then S 1 ∪ S 2 is a piece-wise linear surface on Z of order at most max { k 1 , k 2 } .

Original: Proof.
Revised: Proof.

Original: Let S 1 =
Revised: Let S 1 =

Original: (cid:83) l ∈ [ k 1 ] ,i ∈ [ n l ] H li and S 2 = (cid:83) l ∈ [ k 2 ] ,
Revised: (cid:83) l ∈ [ k 1 ] ,i ∈ [ n l ] H li and S 2 = (cid:83) l ∈ [ k 2 ] ,

Original: i ∈
Revised: i ∈

Original: [ m l ] G li be the pwl.
Revised: [ m l ] G li be the pwl.

Original: surface representations of
Revised: surface representations of

Original: S 1 , S 2 .
Revised: S 1 , S 2 .

Original: Given H li , consider the partition
Revised: Given H li , consider the partition

Original: P 
Revised: P :

Original: [Equation]
Revised: [Equation]

Original: We can write H li
Revised: We can write H li

Original: = (cid:83) P ∈ P H li ∩ P and denote the nonempty intersections H li
Revised: = (cid:83) P ∈ P H li ∩ P and denote the nonempty intersections H li

Original: ∩ P as (cid:8) ¯ H li,j (cid:9) j .
Revised: ∩ P as (cid:8) ¯ H li,j (cid:9) j .

Original: Similarly, we decompose G li =
Revised: Similarly, we decompose G li =

Original: (cid:83) j ¯ G li,j .
Revised: (cid:83) j ¯ G li,j .

Original: Then S 1 ∪ S 2 = (cid:83) l ∈
Revised: Then S 1 ∪ S 2 = (cid:83) l ∈

Original: [max { k 1 ,k 2 } ] ( (cid:83) i,j ¯ H li,j ∪ (cid:83) i,j ¯ G li,j ) , where each ¯ H li,j and ¯ G li,j is a piece-wise hyperplane wrt. 
Revised: [max { k 1 ,k 2 } ] ( (cid:83) i,j ¯ H li,j ∪ (cid:83) i,j ¯ G li,j ) , where each ¯ H li,j and ¯ G li,j is a piece-wise hyperplane wrt. P [Equation]

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: Given sets Z and S ⊆
Revised: Given sets Z and S ⊆

Original: Z , we introduce the notation
Revised: Z , we introduce the notation

Original: [Equation]
Revised: [Equation]

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

Original: (The dependence on Z is suppressed.)
Revised: (The dependence on Z is suppressed.)

Original: By Lemma A.1, (cid:3) i S is itself a pwl.
Revised: By Lemma A.1, (cid:3) i S is itself a pwl.

Original: surface on Z of order at most i .
Revised: surface on Z of order at most i .

Original: Lemma A.2.
Revised: Lemma A.2.

Original: For i ≤ j and any set S , we have (cid:3) i (cid:3) j S = (cid:3) j (cid:3) i S
Revised: For i ≤ j and any set S , we have (cid:3) i (cid:3) j S = (cid:3) j (cid:3) i S

Original: = (cid:3) i S .
Revised: = (cid:3) i S .

Original: Proof.
Revised: Proof.

Original: We will need these deﬁnitions:
Revised: We will need these deﬁnitions:

Original: [Equation]
Revised: [Equation]

Original: Consider ﬁrst the equality (cid:3) j (cid:3) i S = (cid:3) i S .
Revised: Consider ﬁrst the equality (cid:3) j (cid:3) i S = (cid:3) i S .

Original: We know that (cid:3) j
Revised: We know that (cid:3) j

Original: (cid:3) i S ⊆
Revised: (cid:3) i S ⊆

Original: (cid:3) i S because the square operator always yields a subset.
Revised: (cid:3) i S because the square operator always yields a subset.

Original: At the same time, (cid:3) i S ⊆ (cid:3) j (cid:3) i S , because (cid:3) i S satisﬁes the condition for membership in (6).
Revised: At the same time, (cid:3) i S ⊆ (cid:3) j (cid:3) i S , because (cid:3) i S satisﬁes the condition for membership in (6).

Original: To prove the equality (cid:3) i (cid:3) j S = (cid:3) i S , we use the inclusion (cid:3) j S ⊆ S to deduce (cid:3) i (cid:3) j S ⊆ (cid:3) i S .
Revised: To prove the equality (cid:3) i (cid:3) j S = (cid:3) i S , we use the inclusion (cid:3) j S ⊆ S to deduce (cid:3) i (cid:3) j S ⊆ (cid:3) i S .

Original: Now let S (cid:48)(cid:48) ⊆
Revised: Now let S (cid:48)(cid:48) ⊆

Original: S be one of the sets under the union in (3), i.e. it is a pwl.
Revised: S be one of the sets under the union in (3), i.e. it is a pwl.

Original: surface of order at most i .
Revised: surface of order at most i .

Original: Then it is also a pwl.
Revised: Then it is also a pwl.

Original: surface of order at most j , implying S (cid:48)(cid:48) ⊆
Revised: surface of order at most j , implying S (cid:48)(cid:48) ⊆

Original: (cid:3) j S .
Revised: (cid:3) j S .

Original: This means S (cid:48)(cid:48) is also one of the sets under the union in (5), proving that (cid:3) i S ⊆ (cid:3) i (cid:3) j S .
Revised: This means S (cid:48)(cid:48) is also one of the sets under the union in (5), proving that (cid:3) i S ⊆ (cid:3) i (cid:3) j S .

Original: Lemma A.3.
Revised: Lemma A.3.

Original: Let Z and S ⊆
Revised: Let Z and S ⊆

Original: Z be sets.
Revised: Z be sets.

Original: Then one can write (cid:3) k +1 S = (cid:3) k S ∪ (cid:83)
Revised: Then one can write (cid:3) k +1 S = (cid:3) k S ∪ (cid:83)

Original: i H i where H i are piece-wise hyperplanes wrt.
Revised: i H i where H i are piece-wise hyperplanes wrt.

Original: P Z ( (cid:3) k S ) .
Revised: P Z ( (cid:3) k S ) .

Original: Proof.
Revised: Proof.

Original: Let (cid:3) k +1 S =
Revised: Let (cid:3) k +1 S =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li be the pwl.
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li be the pwl.

Original: surface representation of (cid:3) k +1 S .
Revised: surface representation of (cid:3) k +1 S .

Original: If κ ≤ k , then (cid:3) k +1 S = (cid:3) k S and we are done.
Revised: If κ ≤ k , then (cid:3) k +1 S = (cid:3) k S and we are done.

Original: Otherwise, (cid:83) l ∈ [ k ] ,i ∈ [ n k ] H li ⊆ (cid:3) k S , implying (cid:3) k +1 S ⊆
Revised: Otherwise, (cid:83) l ∈ [ k ] ,i ∈ [ n k ] H li ⊆ (cid:3) k S , implying (cid:3) k +1 S ⊆

Original: (cid:3)
Revised: (cid:3)

Original: k S ∪ (cid:91) i ∈
Revised: k S ∪ (cid:91) i ∈

Original: [ n k +1 ]
Revised: [ n k +1 ]

Original: H k
Revised: H k

Original: +1 i .
Revised: +1 i .

Original: At the same time, (cid:3) k S ∪ (cid:83) i ∈ [ n k +

Revised: At the same time, (cid:3) k S ∪ (cid:83) i ∈ [ n k +

Original: ] H k
Revised: ] H k

Original: +

Revised: +

Original: i is a pwl.
Revised: i is a pwl.

Original: surface of order at most k + 1 because (cid:3) k S is a pwl.
Revised: surface of order at most k + 1 because (cid:3) k S is a pwl.

Original: surface of order at most k and H k
Revised: surface of order at most k and H k

Original: +1 i can be decomposed into piece-wise hyperplanes wrt.
Revised: +1 i can be decomposed into piece-wise hyperplanes wrt.

Original: P Z ( (cid:3) k S ) .
Revised: P Z ( (cid:3) k S ) .

Original: Therefore, (cid:3) k S ∪ (cid:83) i ∈ [ n k +

Revised: Therefore, (cid:3) k S ∪ (cid:83) i ∈ [ n k +

Original: H k +1 i ⊆
Revised: H k +1 i ⊆

Original: (cid:3) k +1 S , implying in fact equality.
Revised: (cid:3) k +1 S , implying in fact equality.

Original: Deﬁnition A.4 (Canonical representation of a pwl. surface) .
Revised: Deﬁnition A.4 (Canonical representation of a pwl. surface) .

Original: Let S be a pwl.
Revised: Let S be a pwl.

Original: surface on Z .
Revised: surface on Z .

Original: The pwl. surface representation S =
Revised: The pwl. surface representation S =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li is called canonical if (cid:83) l ∈ [ k ] ,i ∈ [ n l ] H li = (cid:3) k S for all k ∈ [ κ ] , and each H li is distinct.
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li is called canonical if (cid:83) l ∈ [ k ] ,i ∈ [ n l ] H li = (cid:3) k S for all k ∈ [ κ ] , and each H li is distinct.

Original: Lemma A.4.
Revised: Lemma A.4.

Original: If S = (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li is a pwl.
Revised: If S = (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li is a pwl.

Original: surface in canonical form, then κ is the order of S .
Revised: surface in canonical form, then κ is the order of S .

Original: Proof.
Revised: Proof.

Original: Denote the order of S by λ .
Revised: Denote the order of S by λ .

Original: By the deﬁnition of order, λ ≤ κ , and S = (cid:3) λ S .
Revised: By the deﬁnition of order, λ ≤ κ , and S = (cid:3) λ S .

Original: Then, since S =
Revised: Then, since S =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li is a canonical representation, we have
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li is a canonical representation, we have

Original: [Equation]
Revised: [Equation]

Original: It follows that κ = λ .
Revised: It follows that κ = λ .

Original: Lemma A.5.
Revised: Lemma A.5.

Original: Every pwl. surface has a canonical representation.
Revised: Every pwl. surface has a canonical representation.

Original: Proof.
Revised: Proof.

Original: The inclusion (cid:83) l ∈ [ k ] ,i ∈ [ n l ] H li ⊆ (cid:3) k S holds for any representation.
Revised: The inclusion (cid:83) l ∈ [ k ] ,i ∈ [ n l ] H li ⊆ (cid:3) k S holds for any representation.

Original: We will show the other inclusion by induction in the order of S .
Revised: We will show the other inclusion by induction in the order of S .

Original: If S is order one, (cid:3) 1 S ⊆ S
Revised: If S is order one, (cid:3) 1 S ⊆ S

Original: = (cid:83) i ∈ [ n 

Revised: = (cid:83) i ∈ [ n

Original: H 1 i holds for any representation and we are done.
Revised: H 1 i holds for any representation and we are done.

Original: Now assume the lemma holds up to order κ − 1 , and let S be order κ .
Revised: Now assume the lemma holds up to order κ − 1 , and let S be order κ .

Original: Then by Lemma A.3, S = (cid:3) κ S = (cid:3) κ − 1 S ∪ (cid:83)
Revised: Then by Lemma A.3, S = (cid:3) κ S = (cid:3) κ − 1 S ∪ (cid:83)

Original: i H κi , where H κi are piece-wise hyperplanes wrt.
Revised: i H κi , where H κi are piece-wise hyperplanes wrt.

Original: P Z ( (cid:3) κ − 1 S ) .
Revised: P Z ( (cid:3) κ − 1 S ) .

Original: By the inductive assumption, (cid:3) κ − 1 S has a canonical representation, say (cid:3) κ − 1 S
Revised: By the inductive assumption, (cid:3) κ − 1 S has a canonical representation, say (cid:3) κ − 1 S

Original: =
Revised: =

Original: (cid:83) l ∈ [ κ − 1] ,i ∈ [ n l ] H li .
Revised: (cid:83) l ∈ [ κ − 1] ,i ∈ [ n l ] H li .

Original: We claim that S
Revised: We claim that S

Original: =
Revised: =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li is a canonical representation of S .
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li is a canonical representation of S .

Original: If k = κ , then clearly (cid:3) k S ⊆ S =
Revised: If k = κ , then clearly (cid:3) k S ⊆ S =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li .
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li .

Original: If k ∈ [ κ − 1] , then by Lemma A.2, (cid:3) k S = (cid:3) k (cid:3) κ − 1 S
Revised: If k ∈ [ κ − 1] , then by Lemma A.2, (cid:3) k S = (cid:3) k (cid:3) κ − 1 S

Original: =
Revised: =

Original: (cid:83) l ∈ [ k ] ,i ∈ [ n l ] H li , where we have used the canonical representation of (cid:3) κ − 1 S .
Revised: (cid:83) l ∈ [ k ] ,i ∈ [ n l ] H li , where we have used the canonical representation of (cid:3) κ − 1 S .

Original: Finally, distinctness of H li can be ensured by throwing away duplicates.
Revised: Finally, distinctness of H li can be ensured by throwing away duplicates.

Original: Lemma A.6.
Revised: Lemma A.6.

Original: Let Z be an open set.
Revised: Let Z be an open set.

Original: If S is a piece-wise linear surface on Z , and if S = (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li and S =
Revised: If S is a piece-wise linear surface on Z , and if S = (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li and S =

Original: (cid:83) k ∈ [ κ ] ,j ∈ [ m k ] G kj are two canonical representations of S , then for all l ∈ [ κ ] , n l = m l and there exists a permutation π : [ n l ] → [ n l ] such that H li
Revised: (cid:83) k ∈ [ κ ] ,j ∈ [ m k ] G kj are two canonical representations of S , then for all l ∈ [ κ ] , n l = m l and there exists a permutation π : [ n l ] → [ n l ] such that H li

Original: =
Revised: =

Original: G lπ ( i ) .
Revised: G lπ ( i ) .

Original: In other words, the canonical representation is unique up to within-order indexing.
Revised: In other words, the canonical representation is unique up to within-order indexing.

Original: Proof.
Revised: Proof.

Original: Let k ∈
Revised: Let k ∈

Original: [ κ ] .
Revised: [ κ ] .

Original: Because both representations are canonical, we have
Revised: Because both representations are canonical, we have

Original: [Equation]
Revised: [Equation]

Original: where H ki and G kj are piece-wise hyperplanes wrt.
Revised: where H ki and G kj are piece-wise hyperplanes wrt.

Original: P Z ( (cid:3) k − 1 S ) .
Revised: P Z ( (cid:3) k − 1 S ) .

Original: Then for each P ∈ P
Revised: Then for each P ∈ P

Original: Z
Revised: Z

Original: ( (cid:3) k − 1 S ) ,
Revised: ( (cid:3) k − 1 S ) ,

Original: [Equation]
Revised: [Equation]

Original: where on both sides above we have a union of hyperplanes on an open set.
Revised: where on both sides above we have a union of hyperplanes on an open set.

Original: The claim follows.
Revised: The claim follows.

Original: Deﬁnition A.5 (Dependency graph of a pwl. surface) .
Revised: Deﬁnition A.5 (Dependency graph of a pwl. surface) .

Original: Let S be a piece-wise linear surface on Z , and let S
Revised: Let S be a piece-wise linear surface on Z , and let S

Original: =
Revised: =

Original: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li be its canonical representation.
Revised: (cid:83) l ∈ [ κ ] ,i ∈ [ n l ] H li be its canonical representation.

Original: We deﬁne the dependency graph of
Revised: We deﬁne the dependency graph of

Original: S as the directed graph that has the piece-wise hyperplanes (cid:8) H li (cid:9) l,i as vertices, and has an edge
Revised: S as the directed graph that has the piece-wise hyperplanes (cid:8) H li (cid:9) l,i as vertices, and has an edge

Original: H li → H kj iff l < k and H li ∩ cl H kj (cid:54) = ∅ .
Revised: H li → H kj iff l < k and H li ∩ cl H kj (cid:54) = ∅ .

Original: A. R E LU NETWORKS AND FOLDS
Revised: A. R E LU NETWORKS AND FOLDS

Original: We denote by σ the ReLU function: σ ( u ) i = max { 0 , u i } for i ∈ [dim( u )] .
Revised: We denote by σ the ReLU function: σ ( u ) i = max { 0 , u i } for i ∈ [dim( u )] .

Original: Deﬁnition A.6 (ReLU network) .
Revised: Deﬁnition A.6 (ReLU network) .

Original: Let Z ⊆
Revised: Let Z ⊆

Original: R d 0 with d 0 ≥ 2 be a nonempty open set, and let θ (cid:44) ( W 1 , b 1 , . .
Revised: R d 0 with d 0 ≥ 2 be a nonempty open set, and let θ (cid:44) ( W 1 , b 1 , . .

Original: , W L , b L ) be the network’s parameters, with W l ∈ R d l × d l − 1 , b l ∈ R d l , and
Revised: , W L , b L ) be the network’s parameters, with W l ∈ R d l × d l − 1 , b l ∈ R d l , and

Original: d L = 1 .
Revised: d L = 1 .

Original: A ReLU network parameterised by θ is the function h θ : Z → R , deﬁned by
Revised: A ReLU network parameterised by θ is the function h θ : Z → R , deﬁned by

Original: [Equation]
Revised: [Equation]

Original: (11) where h l θ ( z ) =
Revised: (11) where h l θ ( z ) =

Original: W l · z + b l .
Revised: W l · z + b l .

Original: For 1 ≤ l ≤ k ≤ L , we also denote
Revised: For 1 ≤ l ≤ k ≤ L , we also denote

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: For a ReLU network h θ : Z → R and l ∈ [ L − 1] , denote Z l θ (cid:44) (cid:8) ˇ h 1: l θ ( z ) | z ∈ Z (cid:9) .
Revised: For a ReLU network h θ : Z → R and l ∈ [ L − 1] , denote Z l θ (cid:44) (cid:8) ˇ h 1: l θ ( z ) | z ∈ Z (cid:9) .

Original: Also, for convenience, deﬁne Z 0 θ (cid:44) Z .
Revised: Also, for convenience, deﬁne Z 0 θ (cid:44) Z .

Original: (We will omit the subscript θ when it is clear from the context.)
Revised: (We will omit the subscript θ when it is clear from the context.)

Original: We write f | A to denote the restriction of the function f to the domain A .
Revised: We write f | A to denote the restriction of the function f to the domain A .

Original: Deﬁnition A.7 (Activation indicator) .
Revised: Deﬁnition A.7 (Activation indicator) .

Original: A tuple I (cid:44) ( I 1 , . . .
Revised: A tuple I (cid:44) ( I 1 , . . .

Original: , I L − 1 ) is called an activation indicator if I l = diag( i l ) ∈ R d l × d l and i l ∈ { 0 , 1 } d l for l ∈
Revised: , I L − 1 ) is called an activation indicator if I l = diag( i l ) ∈ R d l × d l and i l ∈ { 0 , 1 } d l for l ∈

Original: [ L − 1] .
Revised: [ L − 1] .

Original: It is called non-trivial if i l (cid:54) = 0 for all l ∈
Revised: It is called non-trivial if i l (cid:54) = 0 for all l ∈

Original: [ L − 1] and non-trivial up to k if i l (cid:54) = 0 for all l ∈ [ k ] .
Revised: [ L − 1] and non-trivial up to k if i l (cid:54) = 0 for all l ∈ [ k ] .

Original: Given a parameter vector θ (cid:44) ( W 1 , b 1 , . . .
Revised: Given a parameter vector θ (cid:44) ( W 1 , b 1 , . . .

Original: , W L , b L ) and an activation indicator I , we introduce the notation
Revised: , W L , b L ) and an activation indicator I , we introduce the notation

Original: [Equation]
Revised: [Equation]

Original: (We will omit the argument θ when it is clear from the context.)
Revised: (We will omit the argument θ when it is clear from the context.)

Original: These quantities characterise the different linear pieces implemented by the network’s units.
Revised: These quantities characterise the different linear pieces implemented by the network’s units.

Original: Also deﬁne I θ ( z ) (cid:44) ( I θ 1 ( z ) , . . .
Revised: Also deﬁne I θ ( z ) (cid:44) ( I θ 1 ( z ) , . . .

Original: , I θ L − 1 ( z )) as the activation indicator for a speciﬁc input:
Revised: , I θ L − 1 ( z )) as the activation indicator for a speciﬁc input:

Original: I θ l ( z )[ i, i ] (cid:44) 1 (cid:8) h 1: l i ( z ) ≥ 0 (cid:9) for all ( l, i ) .
Revised: I θ l ( z )[ i, i ] (cid:44) 1 (cid:8) h 1: l i ( z ) ≥ 0 (cid:9) for all ( l, i ) .

Original: Lemma A.7.
Revised: Lemma A.7.

Original: In a ReLU network with parameters θ = ( W 1 , b 1 , . . .
Revised: In a ReLU network with parameters θ = ( W 1 , b 1 , . . .

Original: , W L , b L ) , the pre-activations satisfy h 1: l i ( z ) ∈ (cid:8) w l i ( θ , I ) ·
Revised: , W L , b L ) , the pre-activations satisfy h 1: l i ( z ) ∈ (cid:8) w l i ( θ , I ) ·

Original: z + b l i ( θ , I ) (cid:9) I , where the indexing runs over all possible activation indicators I .
Revised: z + b l i ( θ , I ) (cid:9) I , where the indexing runs over all possible activation indicators I .

Original: More precisely, h 1: l i ( z ) =
Revised: More precisely, h 1: l i ( z ) =

Original: w li ( θ , I θ ( z )) · z + b li ( θ , I θ ( z )) .
Revised: w li ( θ , I θ ( z )) · z + b li ( θ , I θ ( z )) .

Original: Proof.
Revised: Proof.

Original: Left as exercise.
Revised: Left as exercise.

Original: Deﬁnition A.8 (Fold-set) .
Revised: Deﬁnition A.8 (Fold-set) .

Original: Let Z be an open set, and f : Z → R a continuous, piece-wise linear function.
Revised: Let Z be an open set, and f : Z → R a continuous, piece-wise linear function.

Original: We deﬁne the fold-set of f , denoted by F ( f ) , as the set of all points at which f is nondifferentiable.
Revised: We deﬁne the fold-set of f , denoted by F ( f ) , as the set of all points at which f is nondifferentiable.

Original: Deﬁnition A.9 (Positive / negative in a neighbourhood) .
Revised: Deﬁnition A.9 (Positive / negative in a neighbourhood) .

Original: Let Z be an open set.
Revised: Let Z be an open set.

Original: The function f : Z → R is positive (negative) in the neighbourhood of z ∈ Z if for any (cid:15) > 

Revised: The function f : Z → R is positive (negative) in the neighbourhood of z ∈ Z if for any (cid:15) >

Original: there exists z (cid:48) ∈ B (cid:15) ( z ) such that f ( z (cid:48) ) > 

Revised: there exists z (cid:48) ∈ B (cid:15) ( z ) such that f ( z (cid:48) ) >

Original: ( f ( z (cid:48) ) < 0 ).
Revised: ( f ( z (cid:48) ) < 0 ).

Original: Deﬁnition A.10 (Unit fold-set) .
Revised: Deﬁnition A.10 (Unit fold-set) .

Original: Let h θ : Z → R be a ReLU network.
Revised: Let h θ : Z → R be a ReLU network.

Original: We deﬁne the unit ( l, i ) fold-set of h θ , denoted F l i ( h θ ) , as the set of all z ∈ Z where
Revised: We deﬁne the unit ( l, i ) fold-set of h θ , denoted F l i ( h θ ) , as the set of all z ∈ Z where

Original: h 1: l θ [ i ]( z ) = 0 and h 1: l θ [ i ] is positive in the neighbourhood of z .
Revised: h 1: l θ [ i ]( z ) = 0 and h 1: l θ [ i ] is positive in the neighbourhood of z .

Original: Lemma A.8.
Revised: Lemma A.8.

Original: Let Z be an open set, and f : Z → R a continuous piece-wise linear function.
Revised: Let Z be an open set, and f : Z → R a continuous piece-wise linear function.

Original: Then F ( σ ◦ f ) consists of those z ∈ Z that satisfy
Revised: Then F ( σ ◦ f ) consists of those z ∈ Z that satisfy

Original: • f ( z ) > 0 and z ∈ F ( f ) , or
Revised: • f ( z ) > 0 and z ∈ F ( f ) , or

Original: • f ( z ) = 

Revised: • f ( z ) =

Original: and f is positive in the neighbourhood of z .
Revised: and f is positive in the neighbourhood of z .

Original: Proof.
Revised: Proof.

Original: We will prove that if z satisﬁes any of the two conditions, then z ∈ F ( σ ◦ f ) , and if it violates both, then z ∈ F ( σ ◦ f ) c .
Revised: We will prove that if z satisﬁes any of the two conditions, then z ∈ F ( σ ◦ f ) , and if it violates both, then z ∈ F ( σ ◦ f ) c .

Original: We begin with the latter implication.
Revised: We begin with the latter implication.

Original: Let z be such that f ( z )
Revised: Let z be such that f ( z )

Original: > 0 and z / ∈ F ( f ) , i.e. f is differentiable at z .
Revised: > 0 and z / ∈ F ( f ) , i.e. f is differentiable at z .

Original: Since f is piece-wise linear, there exists (cid:15) > 0 such that all of B (cid:15) ( z ) lies inside a single linear region of f and f ( B (cid:15) ( z )) ⊆
Revised: Since f is piece-wise linear, there exists (cid:15) > 0 such that all of B (cid:15) ( z ) lies inside a single linear region of f and f ( B (cid:15) ( z )) ⊆

Original: (0 , ∞ ] .
Revised: (0 , ∞ ] .

Original: Then, on B (cid:15) ( z ) , the ReLU behaves like an identity, implying σ ◦ f is differentiable at z , proving that z ∈ F ( σ ◦ f ) c .
Revised: Then, on B (cid:15) ( z ) , the ReLU behaves like an identity, implying σ ◦ f is differentiable at z , proving that z ∈ F ( σ ◦ f ) c .

Original: Next, consider z such that f ( z ) = 0 .
Revised: Next, consider z such that f ( z ) = 0 .

Original: For it to violate the second condition, there must exist a ball B (cid:15) ( z ) around z such that f ( B (cid:15) ( z )) ⊆
Revised: For it to violate the second condition, there must exist a ball B (cid:15) ( z ) around z such that f ( B (cid:15) ( z )) ⊆

Original: ( −∞ , 0] .
Revised: ( −∞ , 0] .

Original: (This is also true if f ( z ) < 0 .)
Revised: (This is also true if f ( z ) < 0 .)

Original: Then, on B (cid:15) ( z ) , the ReLU behaves like a constant zero, implying that σ ◦ f is differentiable at z .
Revised: Then, on B (cid:15) ( z ) , the ReLU behaves like a constant zero, implying that σ ◦ f is differentiable at z .

Original: We now prove the other implication.
Revised: We now prove the other implication.

Original: If f ( z ) > 0 and z ∈ F ( f ) , then there exists (cid:15) > 0 such that f ( B (cid:15) ( z )) ⊆ (0 , ∞ ] , which guarantees that the ReLU behaves like an identity on B (cid:15) ( z ) .
Revised: If f ( z ) > 0 and z ∈ F ( f ) , then there exists (cid:15) > 0 such that f ( B (cid:15) ( z )) ⊆ (0 , ∞ ] , which guarantees that the ReLU behaves like an identity on B (cid:15) ( z ) .

Original: In this ball, we have σ ◦ f = f , so z ∈ F ( σ ◦ f ) .
Revised: In this ball, we have σ ◦ f = f , so z ∈ F ( σ ◦ f ) .

Original: If f ( z ) = 0 and f is positive in the neighbourhood of z , we distinguish several cases.
Revised: If f ( z ) = 0 and f is positive in the neighbourhood of z , we distinguish several cases.

Original: If z / ∈ F ( f ) , then there exists a ball B δ ( z ) on which f behaves linearly, i.e. σ ( f ( z ))
Revised: If z / ∈ F ( f ) , then there exists a ball B δ ( z ) on which f behaves linearly, i.e. σ ( f ( z ))

Original: =
Revised: =

Original: σ ( w (cid:124) z
Revised: σ ( w (cid:124) z

Original: + b ) , implying z ∈ F ( σ ◦ f ) .
Revised: + b ) , implying z ∈ F ( σ ◦ f ) .

Original: If z ∈ F ( f ) and, in addition, there exists a ball B δ ( z ) such that f ( B δ ( z )) ⊆
Revised: If z ∈ F ( f ) and, in addition, there exists a ball B δ ( z ) such that f ( B δ ( z )) ⊆

Original: [0 , ∞ ) , then the ReLU behaves like an identity on B δ ( z ) and z ∈ F ( σ ◦ f ) .
Revised: [0 , ∞ ) , then the ReLU behaves like an identity on B δ ( z ) and z ∈ F ( σ ◦ f ) .

Original: The ﬁnal case is z ∈ F ( f ) such that f attains both positive and negative values in its neighbourhood.
Revised: The ﬁnal case is z ∈ F ( f ) such that f attains both positive and negative values in its neighbourhood.

Original: Since f is piece-wise linear, there exist p , n such that f ( z + (cid:15) n ) < 0 < f
Revised: Since f is piece-wise linear, there exist p , n such that f ( z + (cid:15) n ) < 0 < f

Original: ( z + (cid:15) p ) , and z + (cid:15) p , z + (cid:15) n / ∈ F ( f ) for all (cid:15) ∈ (0 , 1] .
Revised: ( z + (cid:15) p ) , and z + (cid:15) p , z + (cid:15) n / ∈ F ( f ) for all (cid:15) ∈ (0 , 1] .

Original: Then ∇ ( σ ◦ f )( z + (cid:15) p ) (cid:54) = 0 and ∇ ( σ ◦ f )( z + (cid:15) n ) = 0 , yielding z ∈ F ( σ ◦ f ) .
Revised: Then ∇ ( σ ◦ f )( z + (cid:15) p ) (cid:54) = 0 and ∇ ( σ ◦ f )( z + (cid:15) n ) = 0 , yielding z ∈ F ( σ ◦ f ) .

Original: Lemma A.9.
Revised: Lemma A.9.

Original: Let Z be an open set, and let f 1 , . . .
Revised: Let Z be an open set, and let f 1 , . . .

Original: , f n : Z → R be continuous, piece-wise linear functions.
Revised: , f n : Z → R be continuous, piece-wise linear functions.

Original: For any w 1 , . . . , w n ∈ R , deﬁne f = (cid:80) ni
Revised: For any w 1 , . . . , w n ∈ R , deﬁne f = (cid:80) ni

Original: =1 w i f i .
Revised: =1 w i f i .

Original: Then F ( f ) ⊆
Revised: Then F ( f ) ⊆

Original: (cid:83) ni =1 F ( f i ) .
Revised: (cid:83) ni =1 F ( f i ) .

Original: Proof.
Revised: Proof.

Original: Left as exercise.
Revised: Left as exercise.

Original: A. G ENERAL AND TRANSPARENT R E LU NETWORKS
Revised: A. G ENERAL AND TRANSPARENT R E LU NETWORKS

Original: Lemma A.10.
Revised: Lemma A.10.

Original: For all θ except a closed zero-measure set,
Revised: For all θ except a closed zero-measure set,

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: for all activation indicators I and all k ≤ l .
Revised: for all activation indicators I and all k ≤ l .

Original: Proof.
Revised: Proof.

Original: First, notice that (16) is just a special case of (17) with I l equal to the identity matrix.
Revised: First, notice that (16) is just a special case of (17) with I l equal to the identity matrix.

Original: It therefore sufﬁces to prove (17).
Revised: It therefore sufﬁces to prove (17).

Original: To further simplify, we will prove the statement for a single ﬁxed activation indicator I .
Revised: To further simplify, we will prove the statement for a single ﬁxed activation indicator I .

Original: Then if Θ ( I ) is the set of networks for which (17) holds given I , and Θ ( I ) contains all networks except a closed zero-measure set, then also (cid:84) I Θ
Revised: Then if Θ ( I ) is the set of networks for which (17) holds given I , and Θ ( I ) contains all networks except a closed zero-measure set, then also (cid:84) I Θ

Original: ( I ) contains all networks except a closed zero-measure set, proving the lemma.
Revised: ( I ) contains all networks except a closed zero-measure set, proving the lemma.

Original: Let us hence ﬁx I , and let k ∈
Revised: Let us hence ﬁx I , and let k ∈

Original: [ L ] .
Revised: [ L ] .

Original: We proceed by induction.
Revised: We proceed by induction.

Original: For the initial step, notice that the matrix I k W k is just W k with some rows replaced by zeroes.
Revised: For the initial step, notice that the matrix I k W k is just W k with some rows replaced by zeroes.

Original: The rank of such a matrix is the same as the matrix obtained by removing the zero rows, which has size (rank( I k ) , d k − 1 ) .
Revised: The rank of such a matrix is the same as the matrix obtained by removing the zero rows, which has size (rank( I k ) , d k − 1 ) .

Original: For all W k except a closed zero-measure set, this matrix has rank min { d k − 1 , rank( I k ) } .
Revised: For all W k except a closed zero-measure set, this matrix has rank min { d k − 1 , rank( I k ) } .

Original: For the inductive step, denote ¯ W i :
Revised: For the inductive step, denote ¯ W i :

Original: =
Revised: =

Original: I
Revised: I

Original: i W i · · · I k W k and
Revised: i W i · · · I k W k and

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: We assume that rank( ¯ W i − 1 ) =
Revised: We assume that rank( ¯ W i − 1 ) =

Original: r i − 1 and want to prove the same for i .
Revised: r i − 1 and want to prove the same for i .

Original: Notice that for all W i except a closed zero-measure set, any r i rows of W i are linearly independent and their span intersects with ker( ¯ W (cid:124)
Revised: Notice that for all W i except a closed zero-measure set, any r i rows of W i are linearly independent and their span intersects with ker( ¯ W (cid:124)

Original: i − 1 ) only at 0 .
Revised: i − 1 ) only at 0 .

Original: To see this, recall that by the inductive assumption, rank( ¯ W (cid:124)
Revised: To see this, recall that by the inductive assumption, rank( ¯ W (cid:124)

Original: i − 1 ) =
Revised: i − 1 ) =

Original: r i − 1 , so ker( ¯ W (cid:124)
Revised: r i − 1 , so ker( ¯ W (cid:124)

Original: i − 1 )
Revised: i − 1 )

Original: has dimension d i − 1 − r i − 1 .
Revised: has dimension d i − 1 − r i − 1 .

Original: We can concatenate any r i -subset of rows of W i to the basis of ker( ¯ W (cid:124)
Revised: We can concatenate any r i -subset of rows of W i to the basis of ker( ¯ W (cid:124)

Original: i − 1 ) to obtain a matrix of size ( r i + d
Revised: i − 1 ) to obtain a matrix of size ( r i + d

Original: i − 1 − r i − 1 , d i − 1 ) , which is a wide matrix, because r i ≤ r i − 1 .
Revised: i − 1 − r i − 1 , d i − 1 ) , which is a wide matrix, because r i ≤ r i − 1 .

Original: Hence, its rows are linearly independent for all W i except a closed zero-measure set.
Revised: Hence, its rows are linearly independent for all W i except a closed zero-measure set.

Original: We now prove that rank( I i W i
Revised: We now prove that rank( I i W i

Original: ¯ W i − 1 )
Revised: ¯ W i − 1 )

Original: = min (cid:8) rank( ¯ W i − 1 ) , rank( I i ) (cid:9) (cid:44) r
Revised: = min (cid:8) rank( ¯ W i − 1 ) , rank( I i ) (cid:9) (cid:44) r

Original: i .
Revised: i .

Original: The “ ≤ ” direction is immediate.
Revised: The “ ≤ ” direction is immediate.

Original: For the “ ≥ ” direction, we distinguish between two cases.
Revised: For the “ ≥ ” direction, we distinguish between two cases.

Original: If rank( I i ) ≤ rank( ¯ W i − 1 ) , let v 1 , . . .
Revised: If rank( I i ) ≤ rank( ¯ W i − 1 ) , let v 1 , . . .

Original: v r
Revised: v r

Original: i be the (linearly independent) nonzero rows of I
Revised: i be the (linearly independent) nonzero rows of I

Original: i W i .
Revised: i W i .

Original: We want to show that (cid:8) v (cid:124)
Revised: We want to show that (cid:8) v (cid:124)

Original: j ¯ W i − 1 (cid:9) j are linearly independent, i.e. that I
Revised: j ¯ W i − 1 (cid:9) j are linearly independent, i.e. that I

Original: i W i
Revised: i W i

Original: ¯ W i − 1 has at least r i linearly independent rows.
Revised: ¯ W i − 1 has at least r i linearly independent rows.

Original: If (cid:80) r i j =1 λ j v (cid:124)
Revised: If (cid:80) r i j =1 λ j v (cid:124)

Original: j ¯ W i − 1 = 0 , then (cid:80) r i j =

Revised: j ¯ W i − 1 = 0 , then (cid:80) r i j =

Original: λ j v j ∈ ker( ¯ W (cid:124)
Revised: λ j v j ∈ ker( ¯ W (cid:124)

Original: i − 1 ) , which by assumption implies (cid:80) λ
Revised: i − 1 ) , which by assumption implies (cid:80) λ

Original: j v j = 0 .
Revised: j v j = 0 .

Original: By the independence of { v j } , we obtain λ j = 0 , i.e. (cid:8) v (cid:124)
Revised: By the independence of { v j } , we obtain λ j = 0 , i.e. (cid:8) v (cid:124)

Original: j ¯ W i − 1 (cid:9) j are linearly independent, and rank( I i W
Revised: j ¯ W i − 1 (cid:9) j are linearly independent, and rank( I i W

Original: i
Revised: i

Original: ¯ W i − 1 )
Revised: ¯ W i − 1 )

Original: =
Revised: =

Original: r i .
Revised: r i .

Original: If rank( I i ) > rank( ¯ W i − 1 ) , we can reduce the problem to the case rank( I i ) ≤ rank( ¯ W i − 1 ) by observing that rank( I i W i
Revised: If rank( I i ) > rank( ¯ W i − 1 ) , we can reduce the problem to the case rank( I i ) ≤ rank( ¯ W i − 1 ) by observing that rank( I i W i

Original: ¯ W i − 1 ) ≥ rank( J i W i
Revised: ¯ W i − 1 ) ≥ rank( J i W i

Original: ¯ W i − 1 ) if J i equals I
Revised: ¯ W i − 1 ) if J i equals I

Original: i only with some 1’s replaced by 0’s.
Revised: i only with some 1’s replaced by 0’s.

Original: We can thus take any such J i and apply the argument from the previous paragraph to obtainrank( [Equation] i [Equation]
Revised: We can thus take any such J i and apply the argument from the previous paragraph to obtain

Original: ≥

Original: [Equation]
Revised: [Equation]

Original: i

Original: [Equation]

Original: ≥ r i .

Original: Lemma A.11.
Revised: Lemma A.11.

Original: For all θ except a closed zero-measure set, the following holds.
Revised: For all θ except a closed zero-measure set, the following holds.

Original: Let ( l, i ) , ( k, j ) be any units, let I be an activation indicator non-trivial up to l − 1 , and let J be an activation indicator non-trivial up to k − 1 , such that ( l, i, I 1: l − 1 ) (cid:54) =
Revised: Let ( l, i ) , ( k, j ) be any units, let I be an activation indicator non-trivial up to l − 1 , and let J be an activation indicator non-trivial up to k − 1 , such that ( l, i, I 1: l − 1 ) (cid:54) =

Original: ( k, j, J 1: k − 1 ) .
Revised: ( k, j, J 1: k − 1 ) .

Original: Then, for all scalars c ∈ R , it holds that [ w li ( θ , I ) , b li ( θ , I )]
Revised: Then, for all scalars c ∈ R , it holds that [ w li ( θ , I ) , b li ( θ , I )]

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

Original: c [ w kj ( θ , J ) , b kj ( θ , J )] .
Revised: c [ w kj ( θ , J ) , b kj ( θ , J )] .

Original: Proof.
Revised: Proof.

Original: First, we exclude from consideration all θ =
Revised: First, we exclude from consideration all θ =

Original: ( W 1 , b 1 , . . .
Revised: ( W 1 , b 1 , . . .

Original: , W L , b L ) such that e (cid:124) i W l I l − 1 W l − 1 · · · I k W k e j = 0 for some l, k, i, j , and some I non-trivial up to l − 1 .
Revised: , W L , b L ) such that e (cid:124) i W l I l − 1 W l − 1 · · · I k W k e j = 0 for some l, k, i, j , and some I non-trivial up to l − 1 .

Original: Since for any ﬁxed ( l, k, i, j, I ) , the set of θ satisfying the above is the set of roots of a non-trivial polynomial in θ , it is zero-measure and closed.
Revised: Since for any ﬁxed ( l, k, i, j, I ) , the set of θ satisfying the above is the set of roots of a non-trivial polynomial in θ , it is zero-measure and closed.

Original: Because there are only ﬁnitely many conﬁgurations of ( l, k, i, j, I ) , we have thus excluded a closed zero-measure set of parameters.
Revised: Because there are only ﬁnitely many conﬁgurations of ( l, k, i, j, I ) , we have thus excluded a closed zero-measure set of parameters.

Original: We will denote its complement Θ ∗ .
Revised: We will denote its complement Θ ∗ .

Original: From now on, we assume θ ∈ Θ ∗ .
Revised: From now on, we assume θ ∈ Θ ∗ .

Original: Notice that the case c = 0 of the lemma is thus automatically satisﬁed, since w li ( θ , I ) (cid:44) e (cid:124)
Revised: Notice that the case c = 0 of the lemma is thus automatically satisﬁed, since w li ( θ , I ) (cid:44) e (cid:124)

Original: i W l I l − 1 W l − 1 · · · I 1 W 1 (cid:54) = 

Revised: i W l I l − 1 W l − 1 · · · I 1 W 1 (cid:54) =

Original: by the deﬁnition of Θ ∗ .
Revised: by the deﬁnition of Θ ∗ .

Original: In the following, we can therefore assume c (cid:54) = 0 and treat ( l, i, I ) and ( k, j, J ) symmetrically.
Revised: In the following, we can therefore assume c (cid:54) = 0 and treat ( l, i, I ) and ( k, j, J ) symmetrically.

Original: Denote by Θ ¬ ⊆
Revised: Denote by Θ ¬ ⊆

Original: Θ ∗ the set of parameters θ for which the lemma does not hold; we need to show that Θ ¬ is closed and zero-measure.
Revised: Θ ∗ the set of parameters θ for which the lemma does not hold; we need to show that Θ ¬ is closed and zero-measure.

Original: We start by showing the latter property by contradiction.
Revised: We start by showing the latter property by contradiction.

Original: Suppose Θ ¬ is positive-measure.
Revised: Suppose Θ ¬ is positive-measure.

Original: We know that for all θ ∈ Θ ¬ , there exist triples ( l, i, I ) , ( k, j, J ) as stated in the lemma, and a scalar c ∈ R such that [ w li ( θ , I ) , b li ( θ , I )]
Revised: We know that for all θ ∈ Θ ¬ , there exist triples ( l, i, I ) , ( k, j, J ) as stated in the lemma, and a scalar c ∈ R such that [ w li ( θ , I ) , b li ( θ , I )]

Original: =
Revised: =

Original: c [ w kj ( θ , J ) , b kj ( θ , J )] .
Revised: c [ w kj ( θ , J ) , b kj ( θ , J )] .

Original: Let
Revised: Let

Original: C denote the set of all triplet-pairs (( l, i, I ) , ( k, j, J )) satisfying the conditions of the lemma; then the previous statement can be written as
Revised: C denote the set of all triplet-pairs (( l, i, I ) , ( k, j, J )) satisfying the conditions of the lemma; then the previous statement can be written as

Original: [Equation]
Revised: [Equation]

Original: Since C is ﬁnite, there exist (( l, i, I ) , ( k, j, J ))
Revised: Since C is ﬁnite, there exist (( l, i, I ) , ( k, j, J ))

Original: ∈ C for which the set under the union (call it Θ (cid:48) ) is positive-measure.
Revised: ∈ C for which the set under the union (call it Θ (cid:48) ) is positive-measure.

Original: We now consider two cases.
Revised: We now consider two cases.

Original: If ( l, i ) (cid:54) =
Revised: If ( l, i ) (cid:54) =

Original: ( k, j ) , then observe that Θ (cid:48) must contain some θ , θ (cid:48) such that θ =
Revised: ( k, j ) , then observe that Θ (cid:48) must contain some θ , θ (cid:48) such that θ =

Original: ( W 1 , b 1 , . . .
Revised: ( W 1 , b 1 , . . .

Original: , W L , b L ) and θ (cid:48)
Revised: , W L , b L ) and θ (cid:48)

Original: =
Revised: =

Original: ( W 1 , b 1 , . . .
Revised: ( W 1 , b 1 , . . .

Original: , W l , b l + δ e i , . . .
Revised: , W l , b l + δ e i , . . .

Original: , W L , b L ) , where δ (cid:54) =and l ≥ k .
Revised: , W L , b L ) , where δ (cid:54) =and l ≥ k .

Original: By membership in Θ (cid:48) , there exist c, c (cid:48) ∈
Revised: By membership in Θ (cid:48) , there exist c, c (cid:48) ∈

Original: R such that
Revised: R such that

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

Original: [Equation]
Revised: [Equation]

Original: Notice that w li , w kj do not depend on the b l [ i ] -component of θ , and neither does b kj because k ≤ l and ( k, j ) (cid:54) =
Revised: Notice that w li , w kj do not depend on the b l [ i ] -component of θ , and neither does b kj because k ≤ l and ( k, j ) (cid:54) =

Original: ( l, i ) .
Revised: ( l, i ) .

Original: It follows that [ w kj ( θ , J ) , b kj ( θ , J )] =
Revised: It follows that [ w kj ( θ , J ) , b kj ( θ , J )] =

Original: [ w kj ( θ (cid:48) , J ) , b kj ( θ (cid:48) , J )]
Revised: [ w kj ( θ (cid:48) , J ) , b kj ( θ (cid:48) , J )]

Original: =: v .
Revised: =: v .

Original: Notice also that [ w l i ( θ (cid:48) , I ) , b l i ( θ (cid:48) , I )]
Revised: Notice also that [ w l i ( θ (cid:48) , I ) , b l i ( θ (cid:48) , I )]

Original: = [ w l i ( θ , I ) , b l i ( θ , I )
Revised: = [ w l i ( θ , I ) , b l i ( θ , I )

Original: + δ ] .
Revised: + δ ] .

Original: Putting everything together, we have that
Revised: Putting everything together, we have that

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

Original: 23) which implies ( c (cid:48) − c ) v = [ 0 , δ ] , and in particular w kj ( θ , J ) = 0 .
Revised: 23) which implies ( c (cid:48) − c ) v = [ 0 , δ ] , and in particular w kj ( θ , J ) = 0 .

Original: This contradicts the assumption that θ ∈ Θ ∗ and completes the proof for the case ( l, i ) (cid:54) =
Revised: This contradicts the assumption that θ ∈ Θ ∗ and completes the proof for the case ( l, i ) (cid:54) =

Original: ( k, j ) .
Revised: ( k, j ) .

Original: If ( l, i ) = ( k, j ) , then it must be that I 1: l − 1 (cid:54) =
Revised: If ( l, i ) = ( k, j ) , then it must be that I 1: l − 1 (cid:54) =

Original: J 1: l − 1 .
Revised: J 1: l − 1 .

Original: Wlog, let ( λ, ι ) ∈
Revised: Wlog, let ( λ, ι ) ∈

Original: [ l − 1] × [ d λ ] be such that
Revised: [ l − 1] × [ d λ ] be such that

Original: I λ [ ι, ι ] = 1 and J λ [ ι, ι ] = 0 .
Revised: I λ [ ι, ι ] = 1 and J λ [ ι, ι ] = 0 .

Original: Then there exist θ , θ (cid:48)
Revised: Then there exist θ , θ (cid:48)

Original: ∈ Θ (cid:48) such that θ =
Revised: ∈ Θ (cid:48) such that θ =

Original: ( W 1 , b 1 , . . .
Revised: ( W 1 , b 1 , . . .

Original: , W L , b L ) and θ (cid:48)
Revised: , W L , b L ) and θ (cid:48)

Original: =
Revised: =

Original: ( W 1 , b 1 , . . .
Revised: ( W 1 , b 1 , . . .

Original: , W λ ,
Revised: , W λ ,

Original: b λ + δ e ι , . . .
Revised: b λ + δ e ι , . . .

Original: , W L , b L ) , where δ (cid:54) = 0 .
Revised: , W L , b L ) , where δ (cid:54) = 0 .

Original: Then there exist c, c (cid:48)
Revised: Then there exist c, c (cid:48)

Original: ∈ R such that
Revised: ∈ R such that

Original: [Equation]
Revised: [Equation]

Original: where as before, w li does not depend on the b λ [ ι ] -component.
Revised: where as before, w li does not depend on the b λ [ ι ] -component.

Original: For b li we now have b li ( θ (cid:48) , J ) =
Revised: For b li we now have b li ( θ (cid:48) , J ) =

Original: b li ( θ , J ) and b li ( θ (cid:48) , I ) =
Revised: b li ( θ , J ) and b li ( θ (cid:48) , I ) =

Original: b li ( θ , I ) + d , where d = δ e (cid:124)
Revised: b li ( θ , I ) + d , where d = δ e (cid:124)

Original: i W l I l − 1 · · · W λ +1 e ι and by membership in Θ ∗ , d (cid:54) = 0 .
Revised: i W l I l − 1 · · · W λ +1 e ι and by membership in Θ ∗ , d (cid:54) = 0 .

Original: From here, we can proceed as in the case ( l, i ) (cid:54) =
Revised: From here, we can proceed as in the case ( l, i ) (cid:54) =

Original: ( k, j ) , completing the proof for Θ ¬ being zero-measure.
Revised: ( k, j ) , completing the proof for Θ ¬ being zero-measure.

Original: Finally, we show that Θ ¬ is closed.
Revised: Finally, we show that Θ ¬ is closed.

Original: Let θ ∈ Θ ∗ \ Θ ¬ , i.e. for all (( l, i, I ) , ( k, j, J ))
Revised: Let θ ∈ Θ ∗ \ Θ ¬ , i.e. for all (( l, i, I ) , ( k, j, J ))

Original: ∈ C , the vectors [ w li ( θ , I ) , b li ( θ , I )] and [ w kj ( θ , J ) , b kj ( θ , J )] are non-colinear.
Revised: ∈ C , the vectors [ w li ( θ , I ) , b li ( θ , I )] and [ w kj ( θ , J ) , b kj ( θ , J )] are non-colinear.

Original: Since w li , b li , w kj , b kj are continuous functions in θ , there exists a small enough (cid:15) > 0 such that [ w li ( θ (cid:48) , I ) , b li ( θ (cid:48) , I )] and [ w kj ( θ (cid:48) , J ) , b kj ( θ (cid:48) , J )] are non-colinear for all θ (cid:48)
Revised: Since w li , b li , w kj , b kj are continuous functions in θ , there exists a small enough (cid:15) > 0 such that [ w li ( θ (cid:48) , I ) , b li ( θ (cid:48) , I )] and [ w kj ( θ (cid:48) , J ) , b kj ( θ (cid:48) , J )] are non-colinear for all θ (cid:48)

Original: ∈ B (cid:15) ( θ ) and all (( l, i, I ) , ( k, j, J )) ∈ C .
Revised: ∈ B (cid:15) ( θ ) and all (( l, i, I ) , ( k, j, J )) ∈ C .

Original: Hence, Θ ∗ \ Θ ¬ is open, and Θ ¬ is closed.
Revised: Hence, Θ ∗ \ Θ ¬ is open, and Θ ¬ is closed.

Original: Lemma A.12.
Revised: Lemma A.12.

Original: For all ReLU nets h : Z → R except a closed zero-measure set,
Revised: For all ReLU nets h : Z → R except a closed zero-measure set,

Original: [Equation]
Revised: [Equation]

Original: for all units ( l, i ) .
Revised: for all units ( l, i ) .

Original: Proof.
Revised: Proof.

Original: We provide a proof for a single unit ( l, i ) ; the extension to all units follows from the ﬁnite number of units.
Revised: We provide a proof for a single unit ( l, i ) ; the extension to all units follows from the ﬁnite number of units.

Original: Let G , H , denote the sets deﬁned on the right-hand sides of (26) and (27) respectively.
Revised: Let G , H , denote the sets deﬁned on the right-hand sides of (26) and (27) respectively.

Original: Clearly, H ⊆
Revised: Clearly, H ⊆

Original: F li ( h ) ⊆
Revised: F li ( h ) ⊆

Original: G .
Revised: G .

Original: We will show that G ⊆ H .
Revised: We will show that G ⊆ H .

Original: Let Y ⊆
Revised: Let Y ⊆

Original: R denote the set of all local optima of the function z
Revised: R denote the set of all local optima of the function z

Original: (cid:55)→ W l [ i, :] · ˇ h 1: l − 1 ( z ) .
Revised: (cid:55)→ W l [ i, :] · ˇ h 1: l − 1 ( z ) .

Original: Due to piece-wise linearity of the function, and the ﬁnite number of pieces, Y is ﬁnite.
Revised: Due to piece-wise linearity of the function, and the ﬁnite number of pieces, Y is ﬁnite.

Original: It follows that for all ReLU networks except a closed zero-measure set, − b l [ i ] / ∈ Y .
Revised: It follows that for all ReLU networks except a closed zero-measure set, − b l [ i ] / ∈ Y .

Original: It is thus guaranteed that h 1: l i never attains a local maximum or minimum at zero.
Revised: It is thus guaranteed that h 1: l i never attains a local maximum or minimum at zero.

Original: No z ∈ G can therefore be a local maximum or minimum, implying that h 1: l i is both positive and negative in the neighbourhood of z .
Revised: No z ∈ G can therefore be a local maximum or minimum, implying that h 1: l i is both positive and negative in the neighbourhood of z .

Original: Hence, z ∈ H .
Revised: Hence, z ∈ H .

Original: Deﬁnition A.11 (General ReLU network) .
Revised: Deﬁnition A.11 (General ReLU network) .

Original: A ReLU network is general if it satisﬁes Lemmas A.10,
Revised: A ReLU network is general if it satisﬁes Lemmas A.10,

Original: A.11 and A.12.
Revised: A.11 and A.12.

Original: All ReLU networks except a closed zero-measure set are general.
Revised: All ReLU networks except a closed zero-measure set are general.

Original: Lemma A.13.
Revised: Lemma A.13.

Original: If h is a general ReLU network, then F ( h 1: l i ) =
Revised: If h is a general ReLU network, then F ( h 1: l i ) =

Original: (cid:83) d l − 1 j =1 F (ˇ h 1: l − 1 j ) for all ( l, i ) .
Revised: (cid:83) d l − 1 j =1 F (ˇ h 1: l − 1 j ) for all ( l, i ) .

Original: Proof.
Revised: Proof.

Original: The inclusion F ( h 1: l i ) ⊆
Revised: The inclusion F ( h 1: l i ) ⊆

Original: (cid:83) d l − 1 j =1 F ( ˇ h 1: l − 1 j ) follows from Lemma A.9.
Revised: (cid:83) d l − 1 j =1 F ( ˇ h 1: l − 1 j ) follows from Lemma A.9.

Original: For the other inclusion, let z ∈ F (ˇ h 1: l − 1 k ) for some k ∈ [ d l − 1 ] .
Revised: For the other inclusion, let z ∈ F (ˇ h 1: l − 1 k ) for some k ∈ [ d l − 1 ] .

Original: Then there exist sequences of points z 1 ( (cid:15) ) , z 2 ( (cid:15) ) ∈B (cid:15) [Equation] z
Revised: Then there exist sequences of points z 1 ( (cid:15) ) , z 2 ( (cid:15) ) ∈

Original: [Equation]

Original: [Equation] =1 F (ˇ h [Equation] j ) such that I ( z 1 ( (cid:15) ))
Revised: B (cid:15) ( z ) \ (cid:83) d l − 1 j =1 F (ˇ h 1: l − 1 j ) such that I ( z 1 ( (cid:15) ))

Original: =:
Revised: =:

Original: I and I ( z 2 ( (cid:15) ))
Revised: I and I ( z 2 ( (cid:15) ))

Original: =:
Revised: =:

Original: J are independent of (cid:15) , and ∇ ˇ h 1: l − 1 k ( z 1 ( (cid:15) )
Revised: J are independent of (cid:15) , and ∇ ˇ h 1: l − 1 k ( z 1 ( (cid:15) ))

Original: [Equation]
Revised: (cid:54) =

Original: = [Equation] ( z 2 ( (cid:15) )) .
Revised: ∇ ˇ h 1: l − 1 k ( z 2 ( (cid:15) )) .

Original: We consider three cases based on the (non-)triviality of I and J .
Revised: We consider three cases based on the (non-)triviality of I and J .

Original: First, suppose both I and J are trivial up to l − 1 .
Revised: First, suppose both I and J are trivial up to l − 1 .

Original: Then by Lemma A.7, ∇ ˇ h 1: l − 1 k ( z 1 ( (cid:15) ))
Revised: Then by Lemma A.7, ∇ ˇ h 1: l − 1 k ( z 1 ( (cid:15) ))

Original: =
Revised: =

Original: I l − 1 [ k, k ] w l − 1 k ( I ) = 0 , (28) and similarly ∇ ˇ h 1: l − 1 k ( z 2 ( (cid:15) ))
Revised: I l − 1 [ k, k ] w l − 1 k ( I ) = 0 , (28) and similarly ∇ ˇ h 1: l − 1 k ( z 2 ( (cid:15) ))

Original: = 0 , which contradicts ∇ ˇ h 1: l − 1 k ( z 1 ( (cid:15) )
Revised: = 0 , which contradicts ∇ ˇ h 1: l − 1 k ( z 1 ( (cid:15) )

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

Original: ∇ ˇ h 1: l − 1 k ( z 2 ( (cid:15) ) .
Revised: ∇ ˇ h 1: l − 1 k ( z 2 ( (cid:15) ) .

Original: Hence, at least one of I , J , must be non-trivial up to l − 1 .
Revised: Hence, at least one of I , J , must be non-trivial up to l − 1 .

Original: Second, say both I and J are non-trivial up to l − 1 .
Revised: Second, say both I and J are non-trivial up to l − 1 .

Original: From ∇ ˇ h 1: l − 1 k ( z 1 ( (cid:15) )) (cid:54) = ∇ ˇ h 1: l − 1 k ( z 2 ( (cid:15) ))
Revised: From ∇ ˇ h 1: l − 1 k ( z 1 ( (cid:15) )) (cid:54) = ∇ ˇ h 1: l − 1 k ( z 2 ( (cid:15) ))

Original: it follows that
Revised: it follows that

Original: I 1: l − 1 (cid:54) =
Revised: I 1: l − 1 (cid:54) =

Original: J 1: l − 1 , we can therefore apply Lemma A.11 to ( l, i, I ) and ( l, i, J ) .
Revised: J 1: l − 1 , we can therefore apply Lemma A.11 to ( l, i, I ) and ( l, i, J ) .

Original: We obtain w li ( I ) (cid:54) =
Revised: We obtain w li ( I ) (cid:54) =

Original: w li ( J ) , implying ∇ h 1: l i ( z 1 ( (cid:15) ))
Revised: w li ( J ) , implying ∇ h 1: l i ( z 1 ( (cid:15) ))

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

Original: ∇ h 1: l i ( z 2 ( (cid:15) )) .
Revised: ∇ h 1: l i ( z 2 ( (cid:15) )) .

Original: Thus, z must be a fold-point of h 1: l i .
Revised: Thus, z must be a fold-point of h 1: l i .

Original: Finally, say I is trivial up to l − 1 and J is non-trivial up to l − 1 .
Revised: Finally, say I is trivial up to l − 1 and J is non-trivial up to l − 1 .

Original: Then ∇ h 1: l i ( z 1 ( (cid:15) ))
Revised: Then ∇ h 1: l i ( z 1 ( (cid:15) ))

Original: =
Revised: =

Original: w li ( I ) = 0 , whereas Lemma A.11 applied to ( l, i, J ) with c = 0 yields ∇ h 1: l i ( z 2 ( (cid:15) ))
Revised: w li ( I ) = 0 , whereas Lemma A.11 applied to ( l, i, J ) with c = 0 yields ∇ h 1: l i ( z 2 ( (cid:15) ))

Original: =
Revised: =

Original: w li ( J ) (cid:54) = 0 .
Revised: w li ( J ) (cid:54) = 0 .

Original: Hence, ∇ h 1: l i ( z 1 ( (cid:15) ))
Revised: Hence, ∇ h 1: l i ( z 1 ( (cid:15) ))

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

Original: ∇ h 1: l i ( z 2 ( (cid:15) )) and z must be a fold-point of h 1: l i .
Revised: ∇ h 1: l i ( z 2 ( (cid:15) )) and z must be a fold-point of h 1: l i .

Original: Deﬁnition A.12 (Transparent ReLU network) .
Revised: Deﬁnition A.12 (Transparent ReLU network) .

Original: A ReLU network h :
Revised: A ReLU network h :

Original: Z → R is called transparent up to layer m , if for all z ∈ Z and l ∈ [ m ] , there exists
Revised: Z → R is called transparent up to layer m , if for all z ∈ Z and l ∈ [ m ] , there exists

Original: i ∈
Revised: i ∈

Original: [ d l ] such that h 1: l i ( z ) ≥ 0 , or in other words, rank( I l ( z )) ≥ 1 .
Revised: [ d l ] such that h 1: l i ( z ) ≥ 0 , or in other words, rank( I l ( z )) ≥ 1 .

Original: If h is transparent up to layer L − 1 , we say it is transparent .
Revised: If h is transparent up to layer L − 1 , we say it is transparent .

Original: Lemma A.14.
Revised: Lemma A.14.

Original: Let h : Z → R be a ReLU network, and let λ ∈
Revised: Let h : Z → R be a ReLU network, and let λ ∈

Original: [ L ] .
Revised: [ L ] .

Original: If h is general, then h λ : L | int Z λ −is general.
Revised: If h is general, then h λ : L | int Z λ −is general.

Original: If h is transparent, then h λ : L | int Z λ − 1 is transparent.
Revised: If h is transparent, then h λ : L | int Z λ − 1 is transparent.

Original: Proof.
Revised: Proof.

Original: We will abbreviate h λ : L | int Z λ − 1 as h λ : L .
Revised: We will abbreviate h λ : L | int Z λ − 1 as h λ : L .

Original: Assume h is general.
Revised: Assume h is general.

Original: Then h λ : L clearly satisﬁes
Revised: Then h λ : L clearly satisﬁes

Original: Lemma A.10, and for all ( l, i ) , W l [ i, :] (cid:54) = 0 (cid:124) .
Revised: Lemma A.10, and for all ( l, i ) , W l [ i, :] (cid:54) = 0 (cid:124) .

Original: Next, we prove that h λ :
Revised: Next, we prove that h λ :

Original: L satisﬁes Lemma A.11.
Revised: L satisﬁes Lemma A.11.

Original: Suppose this was not the case; then there exist units ( λ − 1 + l, i ) , ( λ − 1 + k, j ) , and non-trivial activation indicators
Revised: Suppose this was not the case; then there exist units ( λ − 1 + l, i ) , ( λ − 1 + k, j ) , and non-trivial activation indicators

Original: I =
Revised: I =

Original: ( I λ , . . .
Revised: ( I λ , . . .

Original: , I λ − 1+ l ) , J =
Revised: , I λ − 1+ l ) , J =

Original: ( J λ , . .
Revised: ( J λ , . .

Original: , I λ − 1+ k ) , with ( l, i, I ) (cid:54) =
Revised: , I λ − 1+ k ) , with ( l, i, I ) (cid:54) =

Original: ( k, j, J ) , and a scalar C ∈ R such that
Revised: ( k, j, J ) , and a scalar C ∈ R such that

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: Then for any non-trivial indicator ( I 1 , . .
Revised: Then for any non-trivial indicator ( I 1 , . .

Original: , I λ − 1 ) (cid:44) ( J 1 , . . .
Revised: , I λ − 1 ) (cid:44) ( J 1 , . . .

Original: , J λ − 1 ) , we obtain by post-multiplying (29),
Revised: , J λ − 1 ) , we obtain by post-multiplying (29),

Original: [Equation]
Revised: [Equation]

Original: and for all ι ∈ [ λ − 1] ,
Revised: and for all ι ∈ [ λ − 1] ,

Original: [Equation]
Revised: [Equation]

Original: The ﬁrst equality means that w li ( I ) =
Revised: The ﬁrst equality means that w li ( I ) =

Original: C · w kj ( J ) , and the second equality implies b li ( I ) =
Revised: C · w kj ( J ) , and the second equality implies b li ( I ) =

Original: C · b kj ( J ) .
Revised: C · b kj ( J ) .

Original: However, that contradicts the fact that h satisﬁes Lemma A.11.
Revised: However, that contradicts the fact that h satisﬁes Lemma A.11.

Original: The last condition of generality is Lemma A.12.
Revised: The last condition of generality is Lemma A.12.

Original: Suppose h λ : L does not satisfy the lemma.
Revised: Suppose h λ : L does not satisfy the lemma.

Original: Then there exists a unit ( l, i ) such that
Revised: Then there exists a unit ( l, i ) such that

Original: [Equation]
Revised: [Equation]

Original: (cid:8)
Revised: (cid:8)

Original: [Equation]
Revised: [Equation]

Original: Z l − 1 | h λ : l i is positive and negative in the neighbourhood of z (cid:9) , i.e. there exists z ∈ int Z l − 1 such that h λ : l i ( z ) = 0 , and for some (cid:15) > 0 either h λ : l i ( B (cid:15) ( z )) ⊆
Revised: l − 1 | h λ : l i is positive and negative in the neighbourhood of z (cid:9) , i.e. there exists z ∈ int Z l − 1 such that h λ : l i ( z ) = 0 , and for some (cid:15) > 0 either h λ : l i ( B (cid:15) ( z )) ⊆

Original: ( −∞ , 0] or h λ :
Revised: ( −∞ , 0] or h λ :

Original: l i ( B (cid:15) ( z )) ⊆
Revised: l i ( B (cid:15) ( z )) ⊆

Original: [0 , ∞ ) .
Revised: [0 , ∞ ) .

Original: However, then there exists z (cid:

Revised: However, then there exists z (cid:

Original: ) ∈
Revised: ) ∈

Original: Z such that ˇ h 1: l − 1 ( z (cid:48) )
Revised: Z such that ˇ h 1: l − 1 ( z (cid:48) )

Original: =
Revised: =

Original: z , and for z (cid:48) we obtain h 1: l i ( z (cid:48) )
Revised: z , and for z (cid:48) we obtain h 1: l i ( z (cid:48) )

Original: = 0 , and by continuity, there is δ > 

Revised: = 0 , and by continuity, there is δ >

Original: such that either h 1: l i ( B δ ( z (cid:48) )) ⊆ ( −∞ , 0] or h 1: l i ( B δ ( z (cid:48) )) ⊆
Revised: such that either h 1: l i ( B δ ( z (cid:48) )) ⊆ ( −∞ , 0] or h 1: l i ( B δ ( z (cid:48) )) ⊆

Original: [0 , ∞ ) .
Revised: [0 , ∞ ) .

Original: This contradicts the fact that h satisﬁes Lemma A.12.
Revised: This contradicts the fact that h satisﬁes Lemma A.12.

Original: We have thus shown that if h is general, then h λ : L | int Z λ − 1 is general.
Revised: We have thus shown that if h is general, then h λ : L | int Z λ − 1 is general.

Original: Finally, assume h is transparent, i.e. for all z ∈ Z and l ∈ [ L − 1] , there exists
Revised: Finally, assume h is transparent, i.e. for all z ∈ Z and l ∈ [ L − 1] , there exists

Original: i ∈
Revised: i ∈

Original: [ d l ] such that h 1: l i ( z ) ≥ 0 .
Revised: [ d l ] such that h 1: l i ( z ) ≥ 0 .

Original: Then also for all z ∈ int Z λ − 1 and l ∈ { λ, . . .
Revised: Then also for all z ∈ int Z λ − 1 and l ∈ { λ, . . .

Original: , L − 1 } , there exists i ∈
Revised: , L − 1 } , there exists i ∈

Original: [ d l ] such that h λ : l i ( z ) ≥ 0 .
Revised: [ d l ] such that h λ : l i ( z ) ≥ 0 .

Original: Hence, h λ : L is transparent.
Revised: Hence, h λ : L is transparent.

Original: Lemma A.15.
Revised: Lemma A.15.

Original: a) For all ReLU networks h : Z → R and all l ∈ [ L ] , i ∈
Revised: a) For all ReLU networks h : Z → R and all l ∈ [ L ] , i ∈

Original: [ d l ] , we have F ( h 1: l i ) ⊆
Revised: [ d l ] , we have F ( h 1: l i ) ⊆

Original: (cid:83)
Revised: (cid:83)

Original: k ∈
Revised: k ∈

Original: [ l − 1] ,j ∈ [ d k ] F kj ( h ) .
Revised: [ l − 1] ,j ∈ [ d k ] F kj ( h ) .

Original: In particular, F ( h ) ⊆
Revised: In particular, F ( h ) ⊆

Original: (cid:83)
Revised: (cid:83)

Original: k ∈
Revised: k ∈

Original: [ L − 1] ,j ∈ [ d l ] F kj ( h ) .
Revised: [ L − 1] ,j ∈ [ d l ] F kj ( h ) .

Original: b) For all general ReLU networks h : Z → R transparent up to layer l − 1 , we have F ( h 1: l i ) =
Revised: b) For all general ReLU networks h : Z → R transparent up to layer l − 1 , we have F ( h 1: l i ) =

Original: (cid:83) k ∈
Revised: (cid:83) k ∈

Original: [ l − 1] ,j ∈ [ d k ] F k j ( h ) .
Revised: [ l − 1] ,j ∈ [ d k ] F k j ( h ) .

Original: In particular, for all general transparent ReLU networks,
Revised: In particular, for all general transparent ReLU networks,

Original: F ( h ) =
Revised: F ( h ) =

Original: (cid:83)
Revised: (cid:83)

Original: k ∈
Revised: k ∈

Original: [ L − 1] ,j ∈ [ d k ] F kj ( h ) .
Revised: [ L − 1] ,j ∈ [ d k ] F kj ( h ) .

Original: Proof.
Revised: Proof.

Original: We give a proof of b) only.
Revised: We give a proof of b) only.

Original: A proof of a) can be obtained by replacing some equalities by inclusions.
Revised: A proof of a) can be obtained by replacing some equalities by inclusions.

Original: We will prove by induction that F ( h 1: l i ) =
Revised: We will prove by induction that F ( h 1: l i ) =

Original: (cid:83) k ∈
Revised: (cid:83) k ∈

Original: [ l − 1] ,j ∈ [ d k ] F kj ( h ) if h is general and transparent up to layer l − 1 .
Revised: [ l − 1] ,j ∈ [ d k ] F kj ( h ) if h is general and transparent up to layer l − 1 .

Original: For l = 1 , the function h 1: l i is linear, so F ( h ) = ∅
Revised: For l = 1 , the function h 1: l i is linear, so F ( h ) = ∅

Original: and the claim holds trivially.
Revised: and the claim holds trivially.

Original: Now assume that F ( h 1: l i ) =
Revised: Now assume that F ( h 1: l i ) =

Original: (cid:83) k ∈
Revised: (cid:83) k ∈

Original: [ l − 1] ,j ∈ [ d k ] F kj ( h ) holds; we will prove the same statement for l + 1 .
Revised: [ l − 1] ,j ∈ [ d k ] F kj ( h ) holds; we will prove the same statement for l + 1 .

Original: By Lemma A.8 and Lemma A.13, we have
Revised: By Lemma A.8 and Lemma A.13, we have

Original: [Equation]
Revised: [Equation]

Original: Since (cid:83) d l i =
Revised: Since (cid:83) d [Equation] =

Original: [Equation]
Revised: [Equation]

Original: z ∈ Z | h
Revised: ∈ [Equation] | h

Original: [Equation]
Revised: [Equation]

Original: l i ( z
Revised: l i [Equation] z

Original: [Equation]
Revised: [Equation]

Original: > 0 (cid:9) ⊆
Revised: > [Equation] ⊆

Original: Z , we obtain
Revised: Z , we obtain

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

Original: It remains to show the reverse inclusion; we do so by contradiction.
Revised: It remains to show the reverse inclusion; we do so by contradiction.

Original: Suppose ¯ z ∈ (cid:83) k ∈
Revised: Suppose ¯ z ∈ (cid:83) k ∈

Original: [ l ] ,j ∈ [ d k ] F kj ( h ) \ F ( h 1: l +1 ι ) , or equivalently
Revised: [ l ] ,j ∈ [ d k ] F kj ( h ) \ F ( h 1: l +1 ι ) , or equivalently

Original: [Equation]
Revised: [Equation]

Original: Because h is transparent, there exists
Revised: Because h is transparent, there exists

Original: i ∈
Revised: i ∈

Original: [ d l ] : h 1: l i (¯ z ) ≥ 0 , so for this i we have h 1: l i (¯ z ) = 0 .
Revised: [ d l ] : h 1: l i (¯ z ) ≥ 0 , so for this i we have h 1: l i (¯ z ) = 0 .

Original: However, by Lemma A.12, this implies ¯ z ∈ F li ( h ) ⊆
Revised: However, by Lemma A.12, this implies ¯ z ∈ F li ( h ) ⊆

Original: F ( h 1: l +1 ι ) .
Revised: F ( h 1: l +1 ι ) .

Original: Lemma A.16.
Revised: Lemma A.16.

Original: Let h : Z → R be a ReLU network.
Revised: Let h : Z → R be a ReLU network.

Original: Then F l +1 i ( h ) is a union of piece-wise hyperplanes wrt.
Revised: Then F l +1 i ( h ) is a union of piece-wise hyperplanes wrt.

Original: P Z ( (cid:83) k ∈
Revised: P Z ( (cid:83) k ∈

Original: [ l ] ,j ∈ [ d k ] F kj ( h )) .
Revised: [ l ] ,j ∈ [ d k ] F kj ( h )) .

Original: Proof.
Revised: Proof.

Original: Since P Z ( F ( h 1: l +1 i )) is the partition of the input space into the linear regions of h 1: l +1 i , and
Revised: Since P Z ( F ( h 1: l +1 i )) is the partition of the input space into the linear regions of h 1: l +1 i , and

Original: F ( h 1: l +1 i ) ⊆
Revised: F ( h 1: l +1 i ) ⊆

Original: (cid:83)
Revised: (cid:83)

Original: k ∈
Revised: k ∈

Original: [ l ] ,j ∈ [ d k ] F kj ( h ) by Lemma A.15, the function h 1: l +1 i is also linear on the regions of
Revised: [ l ] ,j ∈ [ d k ] F kj ( h ) by Lemma A.15, the function h 1: l +1 i is also linear on the regions of

Original: P Z ( (cid:83) k ∈
Revised: P Z ( (cid:83) k ∈

Original: [ l ] ,j ∈ [ d k ] F kj ( h )) .
Revised: [ l ] ,j ∈ [ d k ] F kj ( h )) .

Original: For any P ∈ P Z ( (cid:83) k ∈ [ l ] ,j ∈ [ d k ] F kj ( h )) , denote the slope and bias of h 1: l +1 i on P by w ( P ) , b ( P ) .
Revised: For any P ∈ P Z ( (cid:83) k ∈ [ l ] ,j ∈ [ d k ] F kj ( h )) , denote the slope and bias of h 1: l +1 i on P by w ( P ) , b ( P ) .

Original: Then
Revised: Then

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: and h
Revised: and h

Revised: [Equation]

Revised: +

Original: [Equation]
Revised: [Equation]

Original: is positive in the neighbourhood of z }
Revised: is positive in the neighbourhood of z }

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: The positivity condition guarantees that ( w ( P ) , b ( P )) (cid:54) =
Revised: The positivity condition guarantees that ( w ( P ) , b ( P )) (cid:54) =

Original: ( 0 , 0) , so P ∩ F l +1 i is either an empty set or a piece-wise hyperplane.
Revised: ( 0 , 0) , so P ∩ F l +1 i is either an empty set or a piece-wise hyperplane.

Original: Corollary A.1.
Revised: Corollary A.1.

Original: Let h : Z → R be a ReLU network.
Revised: Let h : Z → R be a ReLU network.

Original: Then the set (cid:83) l ∈ [ κ ] ,i ∈
Revised: Then the set (cid:83) l ∈ [ κ ] ,i ∈

Original: [ d l ] F li ( h ) is a pwl.
Revised: [ d l ] F li ( h ) is a pwl.

Original: surface of order at most κ .
Revised: surface of order at most κ .

Original: In particular, if h is general and transparent, then F ( h ) =
Revised: In particular, if h is general and transparent, then F ( h ) =

Original: (cid:83)
Revised: (cid:83)

Original: l ∈
Revised: l ∈

Original: [ L − 1] ,i ∈
Revised: [ L − 1] ,i ∈

Original: [ d l ] F li ( h ) is a pwl.
Revised: [ d l ] F li ( h ) is a pwl.

Original: surface of order at most L − 1 .
Revised: surface of order at most L − 1 .

Original: A. M AIN RESULT
Revised: A. M AIN RESULT

Original: Lemma A.17.
Revised: Lemma A.17.

Original: For any bounded domain X and any architecture ( d 1 , . . .
Revised: For any bounded domain X and any architecture ( d 1 , . . .

Original: , d L − 1 ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 , there exists a nonempty open set of transparent ReLU networks h : X → R such that for l ∈
Revised: , d L − 1 ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 , there exists a nonempty open set of transparent ReLU networks h : X → R such that for l ∈

Original: [ L − 1] ,
Revised: [ L − 1] ,

Original: • dim Z l = d l , and
Revised: • dim Z l = d l , and

Original: • the set F ( h l : L | int Z l − 1 ) is a pwl. surface whose dependency graph contains d l directed paths of length ( L − 1 − l ) with distinct starting vertices.
Revised: • the set F ( h l : L | int Z l − 1 ) is a pwl. surface whose dependency graph contains d l directed paths of length ( L − 1 − l ) with distinct starting vertices.

Original: Proof.
Revised: Proof.

Original: We give an explicit construction; we ﬁrst state it
Revised: We give an explicit construction; we ﬁrst state it

Original: and then we prove its properties.
Revised: and then we prove its properties.

Original: For l = 1 , we choose the parameters ( W l , b l ) as follows.
Revised: For l = 1 , we choose the parameters ( W l , b l ) as follows.

Original: Let P 1 be some nonempty open convex subset of X , and deﬁne
Revised: Let P 1 be some nonempty open convex subset of X , and deﬁne

Original: [Equation]
Revised: [Equation]

Original: Since P l has at least two elements, the strict separation theorem implies that W l 1 is nonempty.
Revised: Since P l has at least two elements, the strict separation theorem implies that W l 1 is nonempty.

Original: It is also open.
Revised: It is also open.

Original: We can therefore choose ( W l [1 , :] , b l [1]) from W l 1 , and deﬁne
Revised: We can therefore choose ( W l [1 , :] , b l [1]) from W l 1 , and deﬁne

Original: [Equation]
Revised: [Equation]

Original: for all
Revised: for all

Original: [Equation]
Revised: [Equation]

Original: such that W l [1 , :] z
Revised: such that W l [1 , :] z

Original: + b l [1] ≤ 0 } .
Revised: + b l [1] ≤ 0 } .

Original: [Equation]

Original: The set A := (cid:8) z ∈ conv cl Z l | W l [1 , :] z
Revised: The set A := (cid:8) z ∈ conv cl Z l | W l [1 , :] z

Original: + b l [1] ≤ 0 (cid:9) is nonempty, convex and compact, and by construction there exists z ∈ P l \ A .
Revised: + b l [1] ≤ 0 (cid:9) is nonempty, convex and compact, and by construction there exists z ∈ P l \ A .

Original: Again, the strict separation theorem implies W l 2 is nonempty.
Revised: Again, the strict separation theorem implies W l 2 is nonempty.

Original: It is also open by the boundedness of A .
Revised: It is also open by the boundedness of A .

Original: We then choose ( W l [2 , :] , b l [2]) from W l 2 , and deﬁne
Revised: We then choose ( W l [2 , :] , b l [2]) from W l 2 , and deﬁne

Original: [Equation]
Revised: [Equation]

Original: for all i ∈ [
Revised: for all i ∈ [

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Note that int ¯ Q 2 is nonempty, open and convex.
Revised: Note that int ¯ Q 2 is nonempty, open and convex.

Original: Also, there exist z 1 , z 2 ∈ ¯ Q 2 such that
Revised: Also, there exist z 1 , z 2 ∈ ¯ Q 2 such that

Original: [Equation]
Revised: [Equation]

Original: For i ∈ { 3 , . . .
Revised: For i ∈ { 3 , . . .

Original: , d l } , we then choose ( W l
Revised: , d l } , we then choose ( W l

Original: [ i, :] , b l [ i ]) as follows.
Revised: [ i, :] , b l [ i ]) as follows.

Original: Let A :=
Revised: Let A :=

Original: conv
Revised: conv

Original: z 1 , . .
Revised: z 1 , . .

Original: , z i − 1 } and let z ∈ int ¯ Q i − 1 \ A .
Revised: , z i − 1 } and let z ∈ int ¯ Q i − 1 \ A .

Original: Such z exists because dim ¯ Q i − 1 =
Revised: Such z exists because dim ¯ Q i − 1 =

Original: dim P l = d l − 1 , and dim A ≤ i − 2 ≤ d l − 2 ≤ d l − 1 − 2 .
Revised: dim P l = d l − 1 , and dim A ≤ i − 2 ≤ d l − 2 ≤ d l − 1 − 2 .

Original: By strict separation and boundedness, there exists a nonempty open set of hyperplanes that strictly separate A and z .
Revised: By strict separation and boundedness, there exists a nonempty open set of hyperplanes that strictly separate A and z .

Original: Let ( W l
Revised: Let ( W l

Original: [ i, :] , b l [ i ]) be any of them, oriented such that W l
Revised: [ i, :] , b l [ i ]) be any of them, oriented such that W l

Original: [ i, :]
Revised: [ i, :]

Original: z j + b l [ i ] > 0 for j ∈ [ i − 1] .
Revised: z j + b l [ i ] > 0 for j ∈ [ i − 1] .

Original: Then denote by z i ∈ int ¯ Q i − 1 any point that satisﬁes W l [ i, :]
Revised: Then denote by z i ∈ int ¯ Q i − 1 any point that satisﬁes W l [ i, :]

Original: z
Revised: z

Original: i + b l [ i ] = 0 ; it exists by convexity.
Revised: i + b l [ i ] = 0 ; it exists by convexity.

Original: Then int ¯ Q i is nonempty and convex, and the construction for { i + 1 , . . .
Revised: Then int ¯ Q i is nonempty and convex, and the construction for { i + 1 , . . .

Original: , d l } goes through.
Revised: , d l } goes through.

Original: For layers l ∈ { 2 , . . .
Revised: For layers l ∈ { 2 , . . .

Original: , L − 1 } , we use a similar construction as for l = 1 .
Revised: , L − 1 } , we use a similar construction as for l = 1 .

Original: Denote
Revised: Denote

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: P l is nonempty (because int ¯ Q d l − 1 is nonempty), open (because W l − 1 is wide) and convex.
Revised: P l is nonempty (because int ¯ Q d l − 1 is nonempty), open (because W l − 1 is wide) and convex.

Original: We assume W l − 1 is full-rank; this holds for all choices of W l − 1 except a closed zero-measure set.
Revised: We assume W l − 1 is full-rank; this holds for all choices of W l − 1 except a closed zero-measure set.

Original: By the construction of ¯ Q d l − 1 , there exist z 1 , . . .
Revised: By the construction of ¯ Q d l − 1 , there exist z 1 , . . .

Original: , zd
Revised: , z

Original: [Equation]

Original: ∈

Original: [Equation]
Revised: [Equation]

Original: 1 such that or i ∈
Revised: such that or i ∈

Original: [ d l − 1 ] ,
Revised: [ d l − 1 ] ,

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: Denote their images z (cid:48) i :=
Revised: Denote their images z (cid:48) i :=

Original: W l − 1 z
Revised: W l − 1 z

Original: i + b l − 1 .
Revised: i + b l − 1 .

Original: Then by openness, there exists a ball around each z (cid:48)
Revised: Then by openness, there exists a ball around each z (cid:48)

Original: i such that
Revised: i such that

Original: [Equation]
Revised: [Equation]

Original: implying that each z (cid:48)
Revised: implying that each z (cid:48)

Original: i has a d l − 1 -dimensional, relatively open neighbourhood B [Equation] ( z
Revised: i has a d l − 1 -dimensional, relatively open neighbourhood B (cid:15) ( z

Original: [Equation]
Revised: [Equation]

Original: i [Equation] ∩{ z [Equation]
Revised: i ) ∩{ z (cid:48)

Original: [ i ] = 0 } ⊆
Revised: [ i ] = 0 } ⊆

Original: ¯
Revised: ¯

Original: P l whose elements satisfy z (cid:48)
Revised: P l whose elements satisfy z (cid:48)

Original: [ i ] = 

Revised: [ i ] =

Original: and z (cid:48)
Revised: and z (cid:48)

Original: [ j ] > 0 for j (cid:54) =
Revised: [ j ] > 0 for j (cid:54) =

Original: i .
Revised: i .

Original: It follows that the set
Revised: It follows that the set

Original: W

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

Original: , b

Original: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: for j (cid:54)
Revised: j (cid:54)

Original: [Equation]
Revised: [Equation]

Original: i, and w
Revised: i, and w

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

Original: contains a nonempty open subset.
Revised: contains a nonempty open subset.

Original: We can therefore choose ( W l [1 , :] , b l [1]) from W l 1 .
Revised: We can therefore choose ( W l [1 , :] , b l [1]) from W l 1 .

Original: For the choice of ( W l [ i, :] , b l [ i ]) for i ∈ { 2 , . . .
Revised: For the choice of ( W l [ i, :] , b l [ i ]) for i ∈ { 2 , . . .

Original: , d l } , we use the same procedure as in the ﬁrst layer.
Revised: , d l } , we use the same procedure as in the ﬁrst layer.

Original: Finally, choose ( W L , b L ) arbitrarily.
Revised: Finally, choose ( W L , b L ) arbitrarily.

Original: We will show that this construction satisﬁes the lemma.
Revised: We will show that this construction satisﬁes the lemma.

Original: The networks are transparent because of how we deﬁne W l 2 : for all x ∈ X and l ∈ [ L − 1] , either h 1: l 1 ( x ) > 0 or h 1: l 2 ( x ) > 0 .
Revised: The networks are transparent because of how we deﬁne W l 2 : for all x ∈ X and l ∈ [ L − 1] , either h 1: l 1 ( x ) > 0 or h 1: l 2 ( x ) > 0 .

Original: Also, dim Z l = d l because Z l contains int ¯ Q d l , which is nonempty and open.
Revised: Also, dim Z l = d l because Z l contains int ¯ Q d l , which is nonempty and open.

Original: Now let l ∈
Revised: Now let l ∈

Original: [ L ] .
Revised: [ L ] .

Original: We can think of the function h l : L | int Z l − 1 as an ( L − l + 1) -layer
Revised: We can think of the function h l : L | int Z l − 1 as an ( L − l + 1) -layer

Original: ReLU network parameterised by ( W l , b l , . .
Revised: ReLU network parameterised by ( W l , b l , . .

Original: , W L , b L ) .
Revised: , W L , b L ) .

Original: Because h is transparent, also h l : L | int Z l − 1 is transparent.
Revised: Because h is transparent, also h l : L | int Z l − 1 is transparent.

Original: Corollary A.1 then implies that S :
Revised: Corollary A.1 then implies that S :

Original: = F ( h l : L | int Zl − 1 )
Revised: = F ( h l : L | int Z

Original: = (cid:83)

Original: [Equation]
Revised: [Equation]

Original: F kj (
Revised: − 1 )

Original: [Equation] |
Revised: =

Original: [Equation]
Revised: [Equation]

Original: 1 ) is a pwl. surface.
Revised: ) is a pwl. surface.

Original: Let S =
Revised: Let S =

Original: (cid:83)
Revised: (cid:83)

Original: k ∈
Revised: k ∈

Original: [ L − l ] ,j ∈ [ n λ ]
Revised: [ L − l ] ,j ∈ [ n λ ]

Original: H kj be its canonical representation, and let G denote its dependency graph.
Revised: H kj be its canonical representation, and let G denote its dependency graph.

Original: To ﬁnd the required paths in G , we ﬁrst identify some important vertices.
Revised: To ﬁnd the required paths in G , we ﬁrst identify some important vertices.

Original: For λ ∈ [ L − l ] , denote
Revised: For λ ∈ [ L − l ] , denote

Original: [Equation]
Revised: [Equation]

Original: This set is nonempty and open because P l + λ is nonempty and open.
Revised: This set is nonempty and open because P l + λ is nonempty and open.

Original: Next, for any unit ( λ, ι ) ,
Revised: Next, for any unit ( λ, ι ) ,

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: By the deﬁnition of W l + λ − 1and the fact that h l : l + λ − 1 ( Z + λ )
Revised: By the deﬁnition of W l + λ − 1and the fact that h l : l + λ − 1 ( Z + λ )

Original: [Equation]
Revised: [Equation]

Original: P l + λ , the set A :=
Revised: P l + λ , the set A :=

Original: F λι ( h l : L | int Z l − 1 )
Revised: F λι ( h l : L | int Z l − 1 )

Original: ∩ Z + λ is nonempty.
Revised: ∩ Z + λ is nonempty.

Original: Also, by the deﬁnition of { P l (cid:48) } l (cid:48) , h l : l + λ − 1 ι is in fact linear on Z + λ , so A is a hyperplane on Z + λ .
Revised: Also, by the deﬁnition of { P l (cid:48) } l (cid:48) , h l : l + λ − 1 ι is in fact linear on Z + λ , so A is a hyperplane on Z + λ .

Original: Therefore there exists a piece-wise hyperplane H k ( λ,ι ) j ( λ,ι ) from the canonical representation of S that contains A ; by Lemma A.11, all { H k ( λ,ι ) j ( λ,ι ) } λ,ι are distinct from each other.
Revised: Therefore there exists a piece-wise hyperplane H k ( λ,ι ) j ( λ,ι ) from the canonical representation of S that contains A ; by Lemma A.11, all { H k ( λ,ι ) j ( λ,ι ) } λ,ι are distinct from each other.

Original: We now show that G contains the edge
Revised: We now show that G contains the edge

Original: H k ( λ,i ) j ( λ,i )
Revised: H k ( λ,i ) j ( λ,i )

Original: → H k ( λ +1 ,ι ) j
Revised: → H k ( λ +1 ,ι ) j

Original: ( λ +1 ,ι ) for λ ∈
Revised: ( λ +1 ,ι ) for λ ∈

Original: [ L − l − 1] and all ( i, ι ) .
Revised: [ L − l − 1] and all ( i, ι ) .

Original: By the deﬁnition of W l + λ , there exists ¯ z ∈ ¯ P l + λ such that ¯ z
Revised: By the deﬁnition of W l + λ , there exists ¯ z ∈ ¯ P l + λ such that ¯ z

Original: [ i ] = 0 , ¯ z [ j ] > 0 for j (cid:54)
Revised: [ i ] = 0 , ¯ z [ j ] > 0 for j (cid:54)

Original: =
Revised: =

Original: i , and W l + λ [ ι, :]¯ z + b l + λ [ ι ] = 0 .
Revised: i , and W l + λ [ ι, :]¯ z + b l + λ [ ι ] = 0 .

Original: There also exists a ball B (cid:15) (¯ z ) ⊆
Revised: There also exists a ball B (cid:15) (¯ z ) ⊆

Original: ( W l + λ − 1 P l + λ − 1 + b l + λ − 1 ) ∩ { z
Revised: ( W l + λ − 1 P l + λ − 1 + b l + λ − 1 ) ∩ { z

Original: [ j ] > 0 for j (cid:54) = i } .
Revised: [ j ] > 0 for j (cid:54) = i } .

Original: Then because of how { P l (cid:48) } l (cid:48) are deﬁned, there exists ¯ z (cid:

Revised: Then because of how { P l (cid:48) } l (cid:48) are deﬁned, there exists ¯ z (cid:

Original: ) ∈
Revised: ) ∈

Original: P l such that h l : l + λ − 1 (¯ z (cid:48) )
Revised: P l such that h l : l + λ − 1 (¯ z (cid:48) )

Original: =
Revised: =

Original: ¯ z , so it satisﬁes hl :
Revised: ¯ z , so it satisﬁes h

Original: [Equation]
Revised: [Equation]

Original: + [Equation] [Equation] (cid:48)
Revised: (56) h l

Original: [Equation]
Revised: [Equation]

Original: 0 , (56) h l : l + k − 1 j (¯ z (cid:48) ) > 0 , for k ∈ [ λ ] , ( k, j ) (cid:54) =
Revised: (¯ z (cid:48) ) > 0 , for k ∈ [ λ ] , ( k, j ) (cid:54) =

Original: ( λ, i ) , (57) h l : l + λ ι (¯ z (cid:48) )
Revised: ( λ, i ) , (57) h l : l + λ ι ( [Equation]

Original: = 0 .
Revised: [Equation]

Original: It follows
Revised: It follows

Original: that ¯ z (cid:48) ∈ F λi ( h l : L | int Z l − 1 )
Revised: that ¯ z (cid:48) ∈ F λi ( h l : L | int Z l − 1 )

Original: ∩ Z + λ
Revised: ∩ Z + λ

Original: ⊆
Revised: ⊆

Original: H k
Revised: H k

Original: ( λ,i )
Revised: ( λ,i )

Original: j ( λ,i ) .
Revised: j ( λ,i ) .

Original: At the same time, the preimage ( h l : l + λ − 1 ) − 1 ( B (cid:15) (¯ z )) is open by continuity, and contains ¯ z (cid:48) .
Revised: At the same time, the preimage ( h l : l + λ − 1 ) − 1 ( B (cid:15) (¯ z )) is open by continuity, and contains ¯ z (cid:48) .

Original: So there exists a ball B (cid:15) (¯ z (cid:48) ) ⊆
Revised: So there exists a ball B (cid:15) (¯ z (cid:48) ) ⊆

Original: P l such that all z (cid:48) ∈ B (cid:15) (¯ z (cid:48) ) satisfy h l : l + k − 1 j ( z (cid:48) ) > 0 , for k ∈ [ λ ] , ( k, j ) (cid:54) =
Revised: P l such that all z (cid:48) ∈ B (cid:15) (¯ z (cid:48) ) satisfy h l : l + k − 1 j ( z (cid:48) ) > 0 , for k ∈ [ λ ] , ( k, j ) (cid:54) =

Original: ( λ, i ) .
Revised: ( λ, i ) .

Original: On this ball, h l : l + λ − 1 i is linear, so the set A := B (cid:15) (¯ z (cid:48) )
Revised: On this ball, h l : l + λ − 1 i is linear, so the set A := B (cid:15) (¯ z (cid:48) )

Original: ∩ (cid:8) h l : l + λ − 1 i ( z (cid:48) )
Revised: ∩ (cid:8) h l : l + λ − 1 i ( z (cid:48) )

Original: > 0 (cid:9) is an open half-ball.
Revised: > 0 (cid:9) is an open half-ball.

Original: On A , h l : l + λ ι is linear as well, and the set (cid:8) z (cid:48) | h l : l + λ ι ( z (cid:48) )
Revised: On A , h l : l + λ ι is linear as well, and the set (cid:8) z (cid:48) | h l : l + λ ι ( z (cid:48) )

Original: = 

Revised: =

Original: (cid:9) intersects the center of the half-ball, ¯ z (cid:48) .
Revised: (cid:9) intersects the center of the half-ball, ¯ z (cid:48) .

Original: Therefore there exists a sequence of points { z (cid:48) n } ⊆
Revised: Therefore there exists a sequence of points { z (cid:48) n } ⊆

Original: P l such that z (cid:48) n → ¯ z (cid:48) and h l : l + k − 1 ( z (cid:48) ) > 0 , for k ∈ [ λ ] , (60) h l : l + λ ι ( z (cid:48) )
Revised: P l such that z (cid:48) n → ¯ z (cid:48) and h l : l + k − 1 ( z (cid:48) ) > 0 , for k ∈ [ λ ] , (60) h l : l + λ ι ( z (cid:48) )

Original: = 0 .
Revised: = 0 .

Original: We obtain that ¯ z (cid:48) ∈ cl( F λ +1 ι
Revised: We obtain that ¯ z (cid:48) ∈ cl( F λ +1 ι

Original: ( h l : L | int Z l − 1 )
Revised: ( h l : L | int Z l − 1 )

Original: ∩ Z + λ +

Revised: ∩ Z + λ +

Original: ) ⊆
Revised: ) ⊆

Original: cl H k
Revised: cl H k

Original: ( λ +1 ,ι ) j
Revised: ( λ +1 ,ι ) j

Original: ( λ +1 ,ι ) , which implies cl H k 
Revised: ( λ +1 ,ι ) , which implies cl H k (

Original: [Equation]
Revised: [Equation]

Original: It remains to show that k ( λ, i ) <
Revised: It remains to show that k ( λ, i ) <

Original: k ( λ +1 , ι ) .
Revised: k ( λ +1 , ι ) .

Original: Consider again the ball B (cid:15) (¯ z (cid:48) ) .
Revised: Consider again the ball B (cid:15) (¯ z (cid:48) ) .

Original: By Lemma A.11, h l : l + λ ι is a different linear function on B (cid:15) (¯ z (cid:48) )∩ (cid:8) h [Equation]
Revised: By Lemma A.11, h l : l + λ ι is a different linear function on B (cid:15) (¯ z (cid:48) )

Original: [Equation] + λ [Equation] 1 i ( z
Revised: ∩ (cid:8) h l : l + λ − 1 i ( z

Original: [Equation]
Revised: [Equation]

Original: ) > 0 (cid:9) and on B 
Revised: ) > 0 (cid:9) and on B (cid:15)

Original: [Equation]
Revised: [Equation]

Original: Hence, H k ( λ +1 ,ι ) j
Revised: Hence, H k ( λ +1 ,ι ) j

Original: ( λ +1 ,ι ) is not a piece-wise hyperplane wrt.
Revised: ( λ +1 ,ι ) is not a piece-wise hyperplane wrt.

Original: any partition that does not include H k ( λ,i )
Revised: any partition that does not include H k ( λ,i )

Original: j ( λ,i ) .
Revised: j ( λ,i ) .

Original: We obtain that k ( λ, i ) <
Revised: We obtain that k ( λ, i ) <

Original: k ( λ + 1 , ι ) , proving that G contains the edge
Revised: k ( λ + 1 , ι ) , proving that G contains the edge

Original: H k
Revised: H k

Original: ( λ,i )
Revised: ( λ,i )

Original: j ( λ,i )
Revised: j ( λ,i )

Original: → H k ( λ +1 ,ι ) j
Revised: → H k ( λ +1 ,ι ) j

Original: ( λ +1 ,ι ) .
Revised: ( λ +1 ,ι ) .

Original: Finally, observe that the d l paths H k (1 ,i ) j (1 ,i )
Revised: Finally, observe that the d l paths H k (1 ,i ) j (1 ,i )

Original: → H k (2 , 1) j (2 , 1) → · · · → H k ( L − l, 1) j
Revised: → H k (2 , 1) j (2 , 1) → · · · → H k ( L − l, 1) j

Original: ( L − l, 1) have length ( L − l − 1) , and distinct starting vertices.
Revised: ( L − l, 1) have length ( L − l − 1) , and distinct starting vertices.

Original: This proves the theorem.
Revised: This proves the theorem.

Original: Lemma A.18.
Revised: Lemma A.18.

Original: For all general ReLU networks h : Z → R , the following holds.
Revised: For all general ReLU networks h : Z → R , the following holds.

Original: Denote S =
Revised: Denote S =

Original: (cid:83) l ∈ [ λ ] ,i ∈
Revised: (cid:83) l ∈ [ λ ] ,i ∈

Original: [ d l ] F li ( h ) and let S = (cid:83) k ∈ [ κ ] ,j ∈ [ n k ] H kj be the canonical representation of S .
Revised: [ d l ] F li ( h ) and let S = (cid:83) k ∈ [ κ ] ,j ∈ [ n k ] H kj be the canonical representation of S .

Original: Then H kj ⊆
Revised: Then H kj ⊆

Original: F li ( h ) for some ( l, i ) with l ≥ k .
Revised: F li ( h ) for some ( l, i ) with l ≥ k .

Original: Moreover, if the dependency graph of S contains a directed path of length m starting at H kj , then l ≤ λ − m .
Revised: Moreover, if the dependency graph of S contains a directed path of length m starting at H kj , then l ≤ λ − m .

Original: Proof.
Revised: Proof.

Original: Because the representation is canonical, we have
Revised: Because the representation is canonical, we have

Original: [Equation]
Revised: [Equation]

Original: which implies H kj ⊆ (cid:83) l ≥ k,i F li ( h ) .
Revised: which implies H kj ⊆ (cid:83) l ≥ k,i F li ( h ) .

Original: By piece-wise linearity, we can write
Revised: By piece-wise linearity, we can write

Original: [Equation]
Revised: [Equation]

Original: where P runs over the linear regions of h 1: l i .
Revised: where P runs over the linear regions of h 1: l i .

Original: Moreover, by the deﬁnition of F li ( h ) , all w li ( P ) (cid:54) = 0 .
Revised: Moreover, by the deﬁnition of F li ( h ) , all w li ( P ) (cid:54) = 0 .

Original: Combined with Lemma A.11, we obtain that each nonempty set on the right-hand side of (64) is a different hyperplane on an open set.
Revised: Combined with Lemma A.11, we obtain that each nonempty set on the right-hand side of (64) is a different hyperplane on an open set.

Original: Therefore there exists one for which H kj ⊆
Revised: Therefore there exists one for which H kj ⊆

Original: F li ( h ) ∩ P
Revised: F li ( h ) ∩ P

Original: ⊆
Revised: ⊆

Original: F li ( h ) .
Revised: F li ( h ) .

Original: Now assume that the dependency graph of S contains a directed path of length m starting at H kj =:
Revised: Now assume that the dependency graph of S contains a directed path of length m starting at H kj =:

Original: H k 0 j 0 ; denote the path H k 0 j 

Revised: H k 0 j 0 ; denote the path H k 0 j

Original: → H k 1 j 1 → · · · → H k m j m .
Revised: → H k 1 j 1 → · · · → H k m j m .

Original: By the ﬁrst part of the lemma, we know that
Revised: By the ﬁrst part of the lemma, we know that

Original: H k
Revised: H k

Revised: ι

Revised: [Equation]

Original: ι j ι ⊆ F l ι i ι ( h ) for some l ι .
Revised: ⊆ [Equation] l ι i ι ( h ) for some l ι .

Original: Let ι ∈ [ m ] ; we will show that l ι − 1 < l ι .
Revised: Let ι ∈ [ m ] ; we will show that l ι − 1 < l ι .

Original: Because of the edge H k ι − 1 j ι − 1 → H k ι j ι , we know that H k ι − 1 j ι − 1 and cl
Revised: Because of the edge H k ι − 1 j ι − 1 → H k ι j ι , we know that H k ι − 1 j ι − 1 and cl

Original: H
Revised: H

Original: k ι j ι intersect, and that H
Revised: k ι j ι intersect, and that H

Original: k ι j ι is a piece-wise hyperplane wrt.
Revised: k ι j ι is a piece-wise hyperplane wrt.

Original: some partition P for which H k ι − 1 j ι − 1 is a boundary.
Revised: some partition P for which H k ι − 1 j ι − 1 is a boundary.

Original: Let ˆ z be any point of intersection.
Revised: Let ˆ z be any point of intersection.

Original: By openness, there exists a ball B (cid:15) (ˆ z ) such that H k ι − 1 j ι − 1 is a hyperplane on B (cid:15) (ˆ z ) , and H k
Revised: By openness, there exists a ball B (cid:15) (ˆ z ) such that H k ι − 1 j ι − 1 is a hyperplane on B (cid:15) (ˆ z ) , and H k

Original: ι j ι is a hyperplane on one half-ball deﬁned by B (cid:15) (ˆ z ) and H k ι − 1 j ι − 1 .
Revised: ι j ι is a hyperplane on one half-ball deﬁned by B (cid:15) (ˆ z ) and H k ι − 1 j ι − 1 .

Original: If it was the case that l ι − 1 ≥ l ι , then F l ι i ι ( h ) would be a hyperplane on B (cid:15) (ˆ z ) , i.e. there would have to exist some piecewise hyperplane on the opposite half-ball as H k ι − 1 j ι − 1 , but included in the same hyperplane.
Revised: If it was the case that l ι − 1 ≥ l ι , then F l ι i ι ( h ) would be a hyperplane on B (cid:15) (ˆ z ) , i.e. there would have to exist some piecewise hyperplane on the opposite half-ball as H k ι − 1 j ι − 1 , but included in the same hyperplane.

Original: However, by Lemma A.11, no two piece-wise hyperplanes in S are included in a single hyperplane, so we get a contradiction.
Revised: However, by Lemma A.11, no two piece-wise hyperplanes in S are included in a single hyperplane, so we get a contradiction.

Original: Hence, we obtain l 0 < l 1 < · · · < l m ≤ λ , which yields l 0 ≤ λ − m .
Revised: Hence, we obtain l 0 < l 1 < · · · < l m ≤ λ , which yields l 0 ≤ λ − m .

Original: Lemma A.19.
Revised: Lemma A.19.

Original: Let ( w , b ) , ( c , a ) ∈ R d × R and let F ⊆ R d with dim F = d − 1 .
Revised: Let ( w , b ) , ( c , a ) ∈ R d × R and let F ⊆ R d with dim F = d − 1 .

Original: If w (cid:124) z
Revised: If w (cid:124) z

Original: + b =and c (cid:124) z + a = 0 for all z ∈ F , then either ( w , b ) =
Revised: + b =and c (cid:124) z + a = 0 for all z ∈ F , then either ( w , b ) =

Original: ( 0 , 0) , ( c , a ) =
Revised: ( 0 , 0) , ( c , a ) =

Original: ( 0 , 0) , or there exists β ∈ R : ( c , a )
Revised: ( 0 , 0) , or there exists β ∈ R : ( c , a )

Original: =
Revised: =

Original: β ( w , b ) .
Revised: β ( w , b ) .

Original: Proof.
Revised: Proof.

Original: Since dim F = d − 1 , there exist d afﬁnely independent vectors f 0 , . .
Revised: Since dim F = d − 1 , there exist d afﬁnely independent vectors f 0 , . .

Original: , f d − 1 in F .
Revised: , f d − 1 in F .

Original: Hence there are d − 1 linearly independent vectors v 1 := f 1 − f 0 , . .
Revised: Hence there are d − 1 linearly independent vectors v 1 := f 1 − f 0 , . .

Original: . , v d − 1 := f d − 1 − f 0 , such that w (cid:124) v i =
Revised: . , v d − 1 := f d − 1 − f 0 , such that w (cid:124) v i =

Original: c (cid:124) v i = 0 .
Revised: c (cid:124) v i = 0 .

Original: In other words, both w and c lie in the orthogonal complement of the span of v 1 , . . . , v d − 1 .
Revised: In other words, both w and c lie in the orthogonal complement of the span of v 1 , . . . , v d − 1 .

Original: If w = 0 , then necessarily b = 0 , and similarly for ( c , a ) .
Revised: If w = 0 , then necessarily b = 0 , and similarly for ( c , a ) .

Original: If w (cid:54) = 0 (cid:54) =
Revised: If w (cid:54) = 0 (cid:54) =

Original: c , then because the orthogonal complement is one-dimensional, there exists β ∈ R such that c = β w .
Revised: c , then because the orthogonal complement is one-dimensional, there exists β ∈ R such that c = β w .

Original: Then c (cid:124)
Revised: Then c (cid:124)

Original: z
Revised: z

Original: + a − β ( w (cid:124)
Revised: + a − β ( w (cid:124)

Original: z + b ) =
Revised: z + b ) =

Original: a − βb = 

Revised: a − βb =

Original: and the lemma follows.
Revised: and the lemma follows.

Original: Theorem A.1.
Revised: Theorem A.1.

Original: Consider a bounded domain X and any architecture ( d 1 , . . .
Revised: Consider a bounded domain X and any architecture ( d 1 , . . .

Original: , d L − 1 ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 .
Revised: , d L − 1 ) with d 0 ≥ d 1 ≥ · · · ≥ d L − 1 ≥ 2 .

Original: Let h θ : X → R be a general ReLU network satisfying Lemma A.17, and let h η : X → R be any general ReLU network such that h θ ( x ) =
Revised: Let h θ : X → R be a general ReLU network satisfying Lemma A.17, and let h η : X → R be any general ReLU network such that h θ ( x ) =

Original: h η ( x ) for all x ∈ X .
Revised: h η ( x ) for all x ∈ X .

Original: Denote η (cid:44) ( W (cid:48) 1 , b (cid:48) 1 , . . .
Revised: Denote η (cid:44) ( W (cid:48) 1 , b (cid:48) 1 , . . .

Original: , W (cid:48)
Revised: , W (cid:48)

Original: L , b (cid:48) L ) .
Revised: L , b (cid:48) L ) .

Original: Then there exist permutation matrices P 1 , . . .
Revised: Then there exist permutation matrices P 1 , . . .

Original: P L − 1 , and positiveentry diagonal matrices M 1 , . .
Revised: P L − 1 , and positiveentry diagonal matrices M 1 , . .

Original: , M L − 1 , such that
Revised: , M L − 1 , such that

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: First, notice that h η is transparent.
Revised: First, notice that h η is transparent.

Original: To see this, observe that h θ is transparent, i.e. rank( I θ l ( x )) ≥ 1 for all l ∈
Revised: To see this, observe that h θ is transparent, i.e. rank( I θ l ( x )) ≥ 1 for all l ∈

Original: [ L − 1] and x ∈ X .
Revised: [ L − 1] and x ∈ X .

Original: By Lemma A.10, ∇ x h η ( x ) = ∇ x h θ ( x ) (cid:54) = 

Revised: By Lemma A.10, ∇ x h η ( x ) = ∇ x h θ ( x ) (cid:54) =

Original: (cid:124) , implying that h η is transparent.
Revised: (cid:124) , implying that h η is transparent.

Original: We proceed by induction.
Revised: We proceed by induction.

Original: Let l = 1 .
Revised: Let l = 1 .

Original: Then we have
Revised: Then we have

Original: [Equation]
Revised: [Equation]

Original: which implies F ( h l : L θ | int Z l − 1 θ )
Revised: which implies F ( h l : L θ | int Z l − 1 θ )

Original: =
Revised: =

Original: F ( h l : L η | int Z l − 1 θ ) .
Revised: F ( h l : L η | int Z l − 1 θ ) .

Original: (For notational convenience, we will omit the domain restriction for now.)
Revised: (For notational convenience, we will omit the domain restriction for now.)

Original: Because both networks are general and transparent, Corollary A. implies that the set
Revised: Because both networks are general and transparent, Corollary A. implies that the set

Original: [Equation]
Revised: [Equation]

Original: is a pwl.
Revised: is a pwl.

Original: surface of order at most L − l .
Revised: surface of order at most L − l .

Original: By Lemma A.17, its graph contains d l directed paths of length ( L − 1 − l ) with distinct starting vertices.
Revised: By Lemma A.17, its graph contains d l directed paths of length ( L − 1 − l ) with distinct starting vertices.

Original: Denote these vertices H 1 , . . .
Revised: Denote these vertices H 1 , . . .

Original: , H d l .
Revised: , H d l .

Original: By Lemma A.18, H i ⊆
Revised: By Lemma A.18, H i ⊆

Original: F λι ( h l : L θ ) for some ( λ, ι ) with λ ≤ ( L − l ) − ( L − 1 − l ) = 1 .
Revised: F λι ( h l : L θ ) for some ( λ, ι ) with λ ≤ ( L − l ) − ( L − 1 − l ) = 1 .

Original: We thus obtain (cid:83) i ∈
Revised: We thus obtain (cid:83) i ∈

Original: [ d l ] H i ⊆
Revised: [ d l ] H i ⊆

Original: (cid:83)
Revised: (cid:83)

Original: i ∈
Revised: i ∈

Original: [ d l ] F 1 i ( h l : L θ ) , where on the left-hand side we have a union of d l hyperplanes, and on the righthand side we have a union of at most d l hyperplanes.
Revised: [ d l ] F 1 i ( h l : L θ ) , where on the left-hand side we have a union of d l hyperplanes, and on the righthand side we have a union of at most d l hyperplanes.

Original: It follows that (cid:83) i ∈
Revised: It follows that (cid:83) i ∈

Original: [ d l ] H i = (cid:83)
Revised: [ d l ] H i = (cid:83)

Original: i ∈
Revised: i ∈

Original: [ d l ] F 1 i ( h l : L θ ) , and by applying the same argument to h η , we get (cid:83)
Revised: [ d l ] F 1 i ( h l : L θ ) , and by applying the same argument to h η , we get (cid:83)

Original: i ∈
Revised: i ∈

Original: [ d l ] F 1 i ( h l : L θ ) =
Revised: [ d l ] F 1 i ( h l : L θ ) =

Original: (cid:83)
Revised: (cid:83)

Original: i ∈
Revised: i ∈

Original: [ d l ] F 1 i ( h l : L η ) .
Revised: [ d l ] F 1 i ( h l : L η ) .

Original: Therefore there must exist a permutation π : [ d l ] → [ d l ] such that F 1 i ( h l : L θ )
Revised: Therefore there must exist a permutation π : [ d l ] → [ d l ] such that F 1 i ( h l : L θ )

Original: =
Revised: =

Original: F 1 π ( i ) ( h l : L η ) for all i .
Revised: F 1 π ( i ) ( h l : L η ) for all i .

Original: Then by
Revised: Then by

Original: Lemma A.19, there exist scalars m 1 , . . .
Revised: Lemma A.19, there exist scalars m 1 , . . .

Original: m d l , such that
Revised: m d l , such that

Original: [Equation]
Revised: [Equation]

Original: We know that m i (cid:54)
Revised: We know that m i (cid:54)

Original: = 

Revised: =

Original: because the folds F 1 i ( h l : L θ ) , F 1 i ( h l : L η ) , are nonempty; otherwise (cid:83) i ∈
Revised: because the folds F 1 i ( h l : L θ ) , F 1 i ( h l : L η ) , are nonempty; otherwise (cid:83) i ∈

Original: [ d l ] H i could not be a union of d l hyperplanes.
Revised: [ d l ] H i could not be a union of d l hyperplanes.

Original: We have thus shown that there exists a permutation matrix
Revised: We have thus shown that there exists a permutation matrix

Original: P l ∈ R d l × d l and a nonzero-entry diagonal matrix M l ∈ R d l × d l such that W l = M l P l W (cid:48) l and b l = M l P l b (cid:48) l .
Revised: P l ∈ R d l × d l and a nonzero-entry diagonal matrix M l ∈ R d l × d l such that W l = M l P l W (cid:48) l and b l = M l P l b (cid:48) l .

Original: Next, we show that the diagonal entries of M l are positive.
Revised: Next, we show that the diagonal entries of M l are positive.

Original: Let z − , z + ∈ int Z l − 1 θ be such that
Revised: Let z − , z + ∈ int Z l − 1 θ be such that

Original: I θ ( z − ) and I θ ( z + ) differ only in I θ l
Revised: I θ ( z − ) and I θ ( z + ) differ only in I θ l

Original: [ i, i ] .
Revised: [ i, i ] .

Original: Wlog, let I θ l
Revised: Wlog, let I θ l

Original: [ i, i ]( z − ) =
Revised: [ i, i ]( z − ) =

Original: and I θ l
Revised: and I θ l

Original: [ i, i ]( z + ) = 1 , and denote the row span of I θ l ( z − ) W l by W .
Revised: [ i, i ]( z + ) = 1 , and denote the row span of I θ l ( z − ) W l by W .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: Since h θ is general and d l ≤ d l − 1 , the matrix W l has full row rank.
Revised: Since h θ is general and d l ≤ d l − 1 , the matrix W l has full row rank.

Original: This means that the rows of W l form a basis, in which the representation of ∇ z h l : L θ ( z − ) has one more zero coefﬁcient compared to ∇ z h l : L θ ( z + ) .
Revised: This means that the rows of W l form a basis, in which the representation of ∇ z h l : L θ ( z − ) has one more zero coefﬁcient compared to ∇ z h l : L θ ( z + ) .

Original: In other words, for two points z − , z + ∈ int Z l − 1 θ whose indicators differ only in
Revised: In other words, for two points z − , z + ∈ int Z l − 1 θ whose indicators differ only in

Original: I θ l
Revised: I θ l

Original: [ i, i ] , the point for which I θ l [ i, i ]( z ) = 0 is also the one for which ∇ z h l : L θ ( z ) has more zero coefﬁcients when expressed in the row basis of W l .
Revised: [ i, i ] , the point for which I θ l [ i, i ]( z ) = 0 is also the one for which ∇ z h l : L θ ( z ) has more zero coefﬁcients when expressed in the row basis of W l .

Original: Now, observe that if I θ ( z − ) and I θ ( z + ) differ only in
Revised: Now, observe that if I θ ( z − ) and I θ ( z + ) differ only in

Original: I θ l
Revised: I θ l

Original: [ i, i ] , then I η ( z − ) and I η ( z + ) differ only in I η l [ π ( i ) , π ( i )] .
Revised: [ i, i ] , then I η ( z − ) and I η ( z + ) differ only in I η l [ π ( i ) , π ( i )] .

Original: Because W l = M l P l W (cid:48) l , the number of zero coefﬁcients of ∇ z h l : L η ( z − ) = ∇ z h l : L θ ( z − ) in the row basis of W (cid:48) l is the same as the number of zero coefﬁcients of ∇ z h l : L θ ( z − ) in the row basis of W l .
Revised: Because W l = M l P l W (cid:48) l , the number of zero coefﬁcients of ∇ z h l : L η ( z − ) = ∇ z h l : L θ ( z − ) in the row basis of W (cid:48) l is the same as the number of zero coefﬁcients of ∇ z h l : L θ ( z − ) in the row basis of W l .

Original: It follows that I η l [ π ( i ) , π ( i )]
Revised: It follows that I η l [ π ( i ) , π ( i )]

Original: ( z − ) =
Revised: ( z − ) =

Original: and I η l [ π ( i ) , π ( i )]
Revised: and I η l [ π ( i ) , π ( i )]

Original: ( z + ) = 1 .
Revised: ( z + ) = 1 .

Original: Hence, m i is positive.
Revised: Hence, m i is positive.

Original: For the inductive step, let l ∈ { 2 , . . .
Revised: For the inductive step, let l ∈ { 2 , . . .

Original: , L − 1 } , and assume that there exist permutation matrices
Revised: , L − 1 } , and assume that there exist permutation matrices

Original: P 1 , . .
Revised: P 1 , . .

Original: , P l − 1 , and positive-entry diagonal matrices M 1 , . . .
Revised: , P l − 1 , and positive-entry diagonal matrices M 1 , . . .

Original: , M l − 1 , such that (65) holds up to layer l − 1 .
Revised: , M l − 1 , such that (65) holds up to layer l − 1 .

Original: Then h 1: l − 1 θ ≡ M l − 1 P l − 1 h 1: l − 1 η .
Revised: Then h 1: l − 1 θ ≡ M l − 1 P l − 1 h 1: l − 1 η .

Original: Since h 1: L θ ≡ h 1: L η , it follows that
Revised: Since h 1: L θ ≡ h 1: L η , it follows that

Original: [Equation]
Revised: [Equation]

Original: 69) where ˜ η := ( W (cid:48)
Revised: 69) where ˜ η := ( W (cid:48)

Original: l P − 1 l − 1 M − 1 l − 1 , b (cid:48) l , W (cid:48)
Revised: l P − 1 l − 1 M − 1 l − 1 , b (cid:48) l , W (cid:48)

Original: l +1 , b (cid:48)
Revised: l +1 , b (cid:48)

Original: l +1 , . . .
Revised: l +1 , . . .

Original: , W (cid:48)
Revised: , W (cid:48)

Original: L , b (cid:48) L ) .
Revised: L , b (cid:48) L ) .

Original: We can therefore apply the same argument to h l : L θ | int Z l − 1 θ and h l : L ˜ η
Revised: We can therefore apply the same argument to h l : L θ | int Z l − 1 θ and h l : L ˜ η

Original: | int Z l − 1 θ as we presented above for the case l = 1 .
Revised: | int Z l − 1 θ as we presented above for the case l = 1 .

Original: We obtain that there exists a permutation matrix P l ∈ R d l × d l and a positive-entry diagonal matrix M l ∈ R d
Revised: We obtain that there exists a permutation matrix P l ∈ R d l × d l and a positive-entry diagonal matrix M l ∈ R d

Original: l × d l such that
Revised: l × d l such that

Original: [Equation]
Revised: [Equation]

Original: Finally, consider the last layer.
Revised: Finally, consider the last layer.

Original: We know that h 1: L − 1 θ ≡ M L − 1 P L − 1 h 1: L − 1 η , which implies h L θ ≡ h L η ◦ P − 1 L − 1 M − 1 L − 1 , i.e. h L θ and h L η ◦ P − 1 L − 1 M − 1 L − 1 are identical linear functions supported on the full-dimensional domain Z L − 1 θ .
Revised: We know that h 1: L − 1 θ ≡ M L − 1 P L − 1 h 1: L − 1 η , which implies h L θ ≡ h L η ◦ P − 1 L − 1 M − 1 L − 1 , i.e. h L θ and h L η ◦ P − 1 L − 1 M − 1 L − 1 are identical linear functions supported on the full-dimensional domain Z L − 1 θ .

Original: It follows that W L = W (cid:48)
Revised: It follows that W L = W (cid:48)

Original: L P − 1 L − 1 M − 1 L − 1 and b L = b (cid:48)
Revised: L P − 1 L − 1 M − 1 L − 1 and b L = b (cid:48)

Original: L .
Revised: L .