Title: Tropical Expressivity of Neural Networks
Venue: NeurIPS 2024
Accepted: no
Keywords: linear regions; tropical rational maps; tropical Puiseux polynomials; monomoials; expressivity
Abstract: We propose an algebraic geometric framework to study the expressivity of linear activation neural networks.  A particular quantity that has been actively studied in the field of deep learning is the number of linear regions, which gives an estimate of the information capacity of the architecture.  To study and evaluate information capacity and expressivity, we work in the setting of tropical geometry---a combinatorial and polyhedral variant of algebraic geometry---where there are known connections between tropical rational maps and feedforward neural networks. Our work builds on and expands this relation to capitalize on the rich theory of tropical geometry to characterize and study various architectural aspects of neural networks. Our contributions are threefold: we provide a novel tropical geometric approach to selecting sampling domains among linear regions; an algebraic result allowing for a guided restriction of the sampling domain for network architectures with symmetries; and an open source library to analyze neural networks as tropical Puiseux rational maps. We provide a comprehensive set of proof-of-concept numerical experiments demonstrating the breadth of neural network architectures to which tropical geometric theory can be applied to reveal insights on expressivity characteristics of a network. Our work provides the foundations for the adaptation of both theory and existing software from computational tropical geometry and symbolic computation to deep learning.

Abstract

We propose an algebraic geometric framework to study the expressivity of linear activation neural networks. A particular quantity that has been actively studied in the field of deep learning is the number of linear regions, which gives an estimate of the information capacity of the architecture. To study and evaluate information capacity and expressivity, we work in the setting of tropical geometry-a combinatorial and polyhedral variant of algebraic geometry-where there are known connections between tropical rational maps and feedforward neural networks. Our work builds on and expands this connection to capitalize on the rich theory of tropical geometry to characterize and study various architectural aspects of neural networks. Our contributions are threefold: we provide a novel tropical geometric approach to selecting sampling domains among linear regions; an algebraic result allowing for a guided restriction of the sampling domain for network architectures with symmetries; and an open source library to analyze neural networks as tropical Puiseux rational maps. We provide a comprehensive set of proof-of-concept numerical experiments demonstrating the breadth of neural network architectures to which tropical geometric theory can be applied to reveal insights on expressivity characteristics of a network. Our work provides the foundations for the adaptation of both theory and existing software from computational tropical geometry and symbolic computation to deep learning.

1 Introduction

Deep learning has become the undisputed state-of-the-art for data analysis and has wide-reaching 21

prominence in many fields of computer science, despite still being based on a limited theoretical 22

foundation. Developing theoretical foundations to better understand the unparalleled success of deep 23

neural networks is one of the most active areas of research in modern statistical learning theory. 24

Expressivity is one of the most important approaches to quantifiably measuring the performance of a 25

deep neural network-such as how they are able to represent highly complex information implicitly 26

in their weights and to generalize from data-and therefore key to understanding the success of deep 27

learning. 28

Tropical geometry is a reinterpretation of algebraic geometry that features piecewise linear and 29

polyhedral constructions, where combinatorics naturally comes into play [e.g., 1, 2, 3]. These 30

characteristics of tropical geometry make it a natural framework for studying the linear regions in a 31

neural network-an important quantity in deep learning representing the network information capacity 32

[4, 5, 6, 7, 8, 9, 10]. The intersection of deep learning theory and tropical geometry is a relatively 33

new area of research with great potential towards the ultimate goal of understanding how and why 34

deep neural networks perform so well. In this paper, we propose a new perspective for measuring 35

and estimating the expressivity and information capacity of a neural networks by developing and 36

expanding known connections between neural networks and tropical rational functions in both theory 37

and practice. 38

Related Work. Tropical geometry has been used to characterize deep neural networks with piece39 wise linear activation functions, including two of the most popular and widely-used activation 40 functions, namely, rectified linear units (ReLUs) and maxout units. The first explicit connection 41 between tropical geometry and neural networks establishes that the decision boundary of a deep 42 neural network with ReLU activation functions is a tropical rational function [11]. Concurrently, 43 it was established that the maxout activation function fits input data by a tropical polynomial [12]. 44 These works considered neural networks whose input domain is Euclidean, which was recently 45 developed to incorporate tropically-motivated input domains, in particular, the tropical projective 46 torus [13]. Most recently, tropical geometry has been used to construct convolutional neural networks 47 that are robust to adversarial attacks via tropical decision boundaries [14]. 48

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Contributions. In this paper, we establish novel algebraic and geometric tools to quantify the expressivity of a neural network. Networks with a piecewise linear activation compute piecewise linear functions where the input space is divided into areas; the network computing a single linear function on each area. These areas are referred to as the linear regions of the network; the number of distinct linear regions is a quantifiable measure of expressivity of the network [e.g., 5]. In our work, we not only study the number of linear regions, we aim to understand their geometry . The main contributions of our work are the following.

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We provide a geometric characterization of the linear regions in a neural network via the input space: estimating the linear regions is typically carried out by random sampling from the input space, where randomness may cause some linear regions of a neural network to be missed and result in an inaccurate information capacity measure. We propose an effective sampling domain as a ball of radius R , which is a subset of the entire sampling space that hits all of the linear regions of a given neural network. We compute bounds for the radius R based on a combinatorial invariant known as the Hoffman constant , which effectively gives a geometric characterization and guarantee for the linear regions of a neural network.

We exploit geometric insight into the linear regions of a neural network to gain dramatic computational efficiency: when networks exhibit invariance under symmetry, we can restrict the sampling domain to a fundamental domain of the group action and thus reduce the number of samples required. We experimentally demonstrate that sampling from the fundamental domain provides an accurate estimate of the number of linear regions with a fraction of the compute requirements.

We provide an open source library integrated into the Open Source Computer Algebra Research (OSCAR) system [15] which converts both trained and untrained arbitrary neural networks into algebraic symbolic objects. This contribution then opens the door for the extensive theory and existing software on symbolic computation and computational tropical geometry to be used to study neural networks.

The remainder of this paper is organized as follows. We provide an overview of the technical background on tropical geometry and its connection to neural networks in Section 2. We then devote a section to each of the contributions listed above-Sections 3, 4, and 5, respectively-in which we present our theoretical contributions and numerical experiments. We close the paper with a discussion on limitations of our work and directions for future research in Section 6.

2 Technical Background

In this section, we give basic definitions from tropical geometry required to write tropical expressions for neural networks.

2.1 Tropical Polynomials

Algebraic geometry studies geometric properties of solution sets of polynomial systems that can 84 be expressed algebraically, such as their degree, dimension, and irreducible components. Tropical 85 geometry is a variant of algebraic geometry where the polynomials are defined in the tropical semiring , 86 ¯ R = ( R ∪{∞} , ⊕ , ⊙ ) where the addition and multiplication operators are given by a ⊕ b = max( a, b ) 87 and a ⊙ b = a + b , respectively. Define a ⊘ b := a -b . 88

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Using these operations, we can write polynomials as ⊕ m a m T m , where a i are coefficients, T ∈ ¯ R , and where the sum is indexed by a finite subset of N n . In our work, we consider the following generalizations of tropical polynomials.

Definition 2.1. A tropical Puiseux polynomial in the indeterminates T 1 , . . . , T n is a formal expression of the form ⊕ m a m T m where the index n runs through a finite subset of Q m ≥ 0 and T m = T m 1 1 ⊙ · · · ⊙ T m n n , and taking powers in the tropical sense.

Definition 2.2. A tropical Puiseux rational map in T 1 , . . . , T n is a tropical quotient of the form p ⊘ q where p, q are tropical Puiseux polynomials.

Tropical (Puiseux) polynomials and rational maps induce functions from R n → R , which take a point x ∈ R n to the number obtained by substituting T = x in the algebraic expression and performing the (tropical) operations. It is important to note that tropically, the formal algebraic expression contains strictly more information than the corresponding function, since different tropical expressions can induce the same function.

2.2 Tropical Expressions for Neural Networks

We now overview and recast the framework of [11], which establishes the first explicit connection between tropical geometry and neural networks, in a slightly different language for our results.

As in [11], the neural networks we will focus on are fully connected multilayer perceptrons with ReLU activation, i.e., functions R n → R m of the form σ ◦ L d ◦ σ ◦ L i -1 ◦· · · ◦ L 1 where L i : R n i -1 → R n i is an affine map and σ ( t ) = max { t, 0 } . For the remainder of this paper, we use the term 'neural network' to refer solely to these. We will always assume that the weights and biases of our neural networks are rational numbers. From a computational perspective, this is not a serious restriction since this is sufficient to describe any neural network with weights and biases given by floating point numbers. We refer to the tuple [ n, n 1 , . . . , n d -1 , m ] as the architecture of the neural network.

One of the key observations intersecting tropical geometry and deep learning is that, up to rescaling of rational weights to obtain integers, neural networks can be written as tropical rational functions [11, Theorem 5.2]. From a more computational perspective, it is usually preferable to avoid such rescaling and simply work with the original weights. The proof of Theorem 5.2 in [11] can directly be adapted to show that any neural network can be written as the function associated to a tropical Puiseux rational map. In their language, this corresponds to saying that any neural network is a tropical rational signomial with nonnegative rational exponents.

3 Sampling Domain Selection Using a Hoffman Constant

Estimating the number of linear regions of a neural network typically proceeds by sampling points from the input domain and counting the memberships of these points. To guarantee that membership is exhaustive, we seek a sampling domain as a sufficiently large ball so that all linear regions are intersected. At the same time, we would like for the ball to be as small as possible to guarantee efficient sampling. We are thus searching for the smallest ball from which we can sample in such a way that all linear regions are intersected. Given the polyhedral geometry of tropical Puiseux rational maps, it turns out that the radius of this smallest ball that we seek is closely related to the Hoffman constant , which is a combinatorial invariant.

Our contribution in this section is a definition of a Hoffman constant of a neural network; we demonstrate its relationship to the smallest sampling ball and propose algorithms to compute its true value and lower and upper bounds.

3.1 Defining a Neural Network Hoffman Constant

In simpler terms, the Hoffman constant can be expressed for a matrix as follows. Let A be an m × n matrix. For any b ∈ R m , let P ( A,b ) = { x ∈ R n : Ax ≤ b } denote the polyhedron determined by A and b . For a nonempty polyhedron P ( A,b ) , let d ( u, P ( A,b )) = min {∥ u -x ∥ : x ∈ P ( A,b ) } denote the distance from a point u ∈ R n to the polyhedron, measured under an arbitrary norm ∥ · ∥

on R n . Then there exists a constant H ( A ) only depending on A such that 136

where x + = max { x, 0 } is applied coordinate-wise [16]. The constant H ( A ) is called the Hoffman constant of A .

The Hoffman Constant for Tropical Polynomials and Rational Functions. Let f : R n → R be a tropical Puiseux polynomial and let U = { U 1 , . . . , U m } be the set of linear regions of f . Let f ( x ) = a i 1 x 1 + . . . + a in x n + b i occur on the region U i . Further, let A = [ a ij ] m × n be the matrix of coefficients in the expression of f over U . The linear region U i is defined by the following inequalities

In matrix form, (2) is equivalent to

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where 1 is a column vector of all 1's; a i is the i th row vector of A ; and b is a column vector of all b i . Denote ˜ A U i := A -1 a i and ˜ b U i := b i 1 -b . Then the linear region U i is captured by the linear system of inequalities ˜ A U i x ≤ ˜ b U i .

Definition 3.1. Let f : R n → R be a tropical Puiseux polynomial. The Hoffman constant of f is defined as

Care needs to be taken in defining a Hoffman constant for a tropical Puiseux rational map: We want to avoid having all linear regions defined by systems of linear inequalities, since there exist linear regions which are not convex. To do so, we consider convex refinements of linear regions induced by intersections of linear regions of tropical polynomials.

Definition 3.2. Let p ⊘ q be a difference of two tropical Puiseux polynomials. Let A p (respectively A q ) be the m p × n (respectively m q × n ) matrix of coefficients for p (respectively q ). The Hoffman constant of p ⊘ q is

Let f be a tropical Puiseux rational map. Then the Hoffman constant of f is defined as the minimal Hoffman constant of H ( p ⊘ q ) over all possible expressions of f = p ⊘ q .

Given the correspondence between neural networks and tropical Puiseux rational maps, the Hoffman constant is well-defined for any neural network and may be computed from the geometry and combinatorics of its linear regions.

3.2 The Minimal Effective Radius

For a neural network whose tropical Puiseux rational map is f : R n → R , let U = { U 1 , . . . , U m } be the collection of all linear regions. For any x ∈ R n , define the minimal effective radius of f at x as

̸

where B ( x, r ) is the ball of radius r centered at x . That is, R f ( x ) is the minimal radius such that the ball B ( x, r ) intersects all linear regions. It is the smallest required radius of sampling around x in order to express the full classifying capacity of the neural network f .

We start with the following lemma which relates the minimal effective radius to the Hoffman constant when f is a tropical Puiseux polynomial.

Lemma 3.3. Let f be a tropical Puiseux polynomial and x ∈ R n be any point, then

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In particular, we are interested in studying when R m and R n are equipped with the ∞ -norm. In this case, the minimal effective radius can be related to the Hoffman constant and function value of f = p ⊘ q . For a tropical Puiseux polynomial p ( x ) = max 1 ≤ i ≤ m p { a i x + b i } , let ˇ p ( x ) = min 1 ≤ j ≤ m q { a j x + b j } be its min-conjugate.

Proposition 3.4. Let f = p ⊘ q be a tropical Puiseux rational map. For any x ∈ R , we have

3.3 Computing and Estimating Hoffman Constants

The PVZ Algorithm. In [17], the authors proposed a combinatorial algorithm to compute the precise value of the Hoffman constant for a matrix A ∈ R m × n , which we refer to as the Peña-VeraZuluaga (PVZ) algorithm and sketch its main steps here.

Definition 3.5. A set-valued map Φ : R n → R m assigns a set Φ( x ) ⊆ R m . The map is surjective if Φ( R n ) = ∪ x Φ( x ) = R m . Let A ∈ R m × n . For any J ⊆ { 1 , 2 , . . . , m } , let A J be the submatrix of A consisting of rows with indices in J . The set J is called A -surjective if the set-valued map Φ( x ) = A J x + { y ∈ R J : y ≥ 0 } is surjective.

Notice that A -surjectivity is a generalization of linear independence of row vectors. We illustrate this observation in the following two examples.

Example 3.6. If J is such that A J is full-rank, then J is A -surjective, since for any y ∈ R J , there exists x ∈ R n such that y = A J x .

Example 3.7. Let A = 1 m × n be the m × n matrix whose entries are 1's. For any subset J of { 1 , . . . , m } and for any y ∈ R J , let x ∈ R n such that ∑ i x i ≤ min { y j , j ∈ J } . Then y -A J x ≥ 0 . Thus any J is A -surjective.

The PVZ algorithm is based on the following characterization of Hoffman constant.

Proposition 3.8. [17, Proposition 2] Let A ∈ R m × n . Equip R m and R n with norm ∥ · ∥ and denote its dual norm by ∥ · ∥ ∗ . Let S ( A ) be the set of all A -surjective sets. Then

where

This characterization is particularly useful when R m and R n are equipped with the ∞ -norm, since the computation of (8) reduces to a linear programming (LP) problem. The key problem is how to maximize over all A -surjective sets. To do this, the PVZ algorithm maintains three collections of sets F , I , and J where during every iteration: (i) F contains J such that J is A -surjective; (ii) I contains J such that J is not A -surjective; and (iii) J contains candidates J whose A -surjectivity will be tested.

To detect whether a candidate J ∈ J is surjective, the PVZ algorithm requires solving

If the optimal value is positive, then J is A -surjective, and J is assigned to F and all subsets of J are removed from J . Otherwise, the optimal value is 0 and there is v ∈ R J + such that A ⊤ J v = 0 . Let I ( v ) = { i ∈ J : v i > 0 } and assign I ( v ) to I . Let ˆ J ∈ J be any set containing I ( v ) . Replace all such ˆ J by sets ˆ J \{ i } , i ∈ I ( v ) which are not contained in any sets in F . The implementation used in our paper directly uses the MATLAB code provided by [17].

Lower and Upper Bounds. Alimitation of the PVZ algorithm is that during each loop, every set in J needs to be tested, and each test requires solving a LP problem. Although solving one LP problem in practice is fast, a complete while loop calls the LP solver many times.

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Here, we propose an algorithm to estimate lower and upper bounds for Hoffman constants. An intuitive way to estimate the lower bound is to sample a number of random subsets from { 1 , . . . , m } and test for A -surjectivity. This method bypasses optimizing combinatorially over S ( A ) of A -surjective sets and gives a lower bound of Hoffman constant by Proposition 3.8.

To get an upper of Hoffman constant, we use the result from [18].

Theorem 3.9. [18, Theorem 4.2] Let A ∈ R m × n . Let D ( A ) be a set of subsets of J ⊆ { 1 , . . . , m } such that A J is full rank. Let D ∗ ( A ) be the set of maximal elements in D ( A ) . Then the Hoffman constant measured under 2-norm is bounded by

where ˆ ρ ( A ) is the smallest singular value of A .

Using the fact that ∥ · ∥ 1 ≥ ∥ · ∥ 2 , and the characterization from (8), we see that the upper bound also holds when R m and R n are equipped with the ∞ -norm. However, enumerating all maximal elements in D ( A ) is not an improvement over enumerating A -surjective sets from a computational perspective. Instead, we will retain the strategy as in lower bound estimation to sample a number of sets from { 1 , 2 , . . . , m } and approximate the upper bound by (10). We verify this approach via synthetic data. The experiments are relegated to the Appendix.

4 Symmetry and the Fundamental Domain

In this section, we study a geometric characterization of the sampling domain for networks exhibiting symmetry. This corresponds to invariant neural networks .

4.1 Linear Regions of Invariant Neural Networks

The notion of invariance for a neural network describes when a manipulation of the input domain does not affect the output of the network. The manipulations we consider here are group actions.

Definition 4.1. Let σ : R n → R be a piecewise linear function, and let G be a group acting on the domain R n . σ is invariant under the group action of G if for any element g ∈ G , σ ◦ g = σ.

Given an invariant neural network, we can then define a sampling domain that takes into account the effect of the group action.

̸

Definition 4.2. Let G be a group acting on R n . A subset ∆ ⊆ R n is a fundamental domain if it satisfies two following conditions: (i) R n = ⋃ g ∈ G g · ∆ ; and (ii) g · int (∆) ∩ h · int (∆) = ∅ for all g, h ∈ G,g = h .

The fundamental domain of a group G therefore provides a periodic tiling of R n by acting on ∆ . This is very useful in the context of numerical sampling for neural networks which are invariant under some symmetry, since it means we can sample from a smaller subset of the input domain with a guarantee to find all the linear regions in the limit. This allows us, in principle, to be able to use far fewer samples while maintaining the same density of points.

Theorem 4.3. Let f : R N → R be a tropical rational map invariant under group G . Let ∆ ⊆ R N be a fundamental domain of G . Suppose L is the set of linear regions. Define the following two sets

̸

Then where | G A | is the size of the stabilizer of A .

This gives us a method for estimating the total number of linear regions from sampling in the fundamental domain using multiplicity , which we discuss next.

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4.2 Sampling from the Fundamental Domain

To demonstrate the potential performance improvements in numerical sampling exploiting symmetry in the network architecture, we consider permutation invariant neural networks inspired by deep sets [19]. Our numerical sampling approach is inspired by very recent work in this area [20].

Lemma 4.4 ([19]) . An m × m matrix W acting as a linear operator of the form W = λI m × m + γ ( 1 T 1 ) , where λ, γ ∈ R is permutation equivariant, meaning WPx = PWx for any x ∈ R m , so it commutes with any permutation matrix.

Using a weight matrix of this form, we can construct permutation invariant neural networks by setting the bias to 0, applying a ReLU activation after multiplication by W , and then summing. In this case, the network is invariant under the group action S n , so the fundamental domain is the set of points with increasing coordinates, i.e., ∆ = { ( x 1 , . . . , x n ) : x 1 ≥ x 2 ≥ . . . ≥ x n } . This splits R n into n ! tiles, so we have a clear and significant advantage in restricting sampling to the fundamental domain.

Note, however, that it is important to address the multiplicities of symmetric linear regions correctly: If a given Jacobian of shape n × 1 has no repeated elements, this means it is contained in the interior of some group action applied to the fundamental domain. This means there are n ! total linear regions with this Jacobian. If, on the other hand, there are repeated coefficients in a given Jacobian J , we consider the set C ( J ) of counts of repeated elements. For example, for J = [1 , 1 , 0] , C ( J ) = (2 , 1) .

Then the multiplicity of a given Jacobian is given by

Using this multiplicity calculation we can efficiently estimate the number of linear regions while reducing the number of point samples by a factor of n ! . This provides a dramatic gain in sampling efficiency.

In Figure 1, we present the results when Algorithm 2 is run with R = 10 , N = 10 , M = 50 . These results show that the fundamental domain estimate performs well for low dimensional inputs but appears to overcount linear regions as n increases. Despite divergence, there is still utility in this metric because we are often more concerned with obtaining an upper bound on the expressivity of a neural network than an exact figure and the fundamental domain estimate does not undercount the number of linear regions.

5 Symbolic Neural Networks

Here, we present the details on our practical contribution of a symbolic representation of neural networks as a new library integrated into OSCAR [15].

5.1 Computing Linear Regions of Tropical Puiseux Rational Maps

We present an algorithm that can compute the linear regions of any tropical Puiseux rational function. Intuitively, we do this by computing the linear regions of the numerator and denominator, and then considering intersections of such regions and how they fit together. Thus, a first step is to understand how the computation of linear regions works for tropical Puiseux polynomials. The key to our approach will be to exploit the polyhedral connection of tropical geometry and recast the problem in the language of polyhedral geometry. This, among other things, will allow us to make use the extensive polyhedral geometry library in OSCAR [15] for implementation.

One important upshot from this study is that there is a strong connection between the number of linear regions of a tropical Puiseux rational function and the number of monomials that appear in its algebraic expression. Note, however, that the two are independent, in the sense that two Puiseux rational functions could have the same number of linear regions but different numbers of (nonzero) monomials, and conversely, the same number of monomials and a different number of linear regions. For instance, computing the number of linear regions requires some combinatorial data about the intersections of the polyhedra defined by monomials.

First, we need to know how to compute the linear regions of tropical polynomials. Let P = ⊕ n a n ⊙ x n where by x n we mean x n 1 1 ⊙···⊙ x n k k and powers are taken in the tropical sense. Then

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as function R k → R , P is given by max n { a n + n 1 x 1 · · · + n k x k } . It follows that the linear regions of P are precisely the sets of the form S n = { x ∈ R n | a m + m 1 x 1 · · · + m k x k ≤ a n + n 1 x 1 · · · + n k x k for all m = n } . For any set U on which P is linear, we write L ( P, U ) for the corresponding linear map. This gives us L ( P, S n )( x ) = a n + n 1 x 1 · · · + n k x k . (11)

We now rewrite (11) using polyhedral geometry. Recall that a polyhedron in R k is a set of the form P ( A,b ) = { x ∈ R k | Ax ≤ b } . We claim that each linear region is a polyhedron: For a fixed index n , define the matrix A n to be the ( N -1) × k matrix whose rows are the vectors m -n , where m ranges over the support of the coefficients of P (ordered lexicographically) and b n to be the vector with entries a n -a m . Then S n = P ( A n , b n ) . This gives us a way to encode the computation of the linear regions of tropical Puiseux polynomials using polyhedral geometry. As a direct consequence, intersections of linear regions of tropical Puiseux polynomials are also polyhedra. In particular, there are algorithms from polyhedral geometry for determining whether such polyhedra are realizable. One of the key observations given by our algorithm is that the linear regions of tropical Puiseux rational maps are almost given by k -dimensional intersections of the linear regions of the numerator and the denominator. Indeed, note that if U is a linear region of p and V a linear region of q , then we have L ( U ∩ V, p ⊘ q ) = L ( U, p ) -L ( V, q ) . The only issue that arises is that there might be some repetition in the L ( U ∩ V, p ⊘ q ) as U ranges over the linear regions of p and V over the linear regions of q . In particular, linear regions of p ⊘ q might end up corresponding to unions of such U ∩ V .

5.2 Computing Linear Regions

Determining the linear regions of a neural network may be approached numerically or symbolically . The numerical approach exploits the fact that linear regions of a neural network correspond to regions where the gradient is constant. Thus, to estimate the number of linear regions, we can evaluate the gradient on a sample of points (e.g., a mesh) in some large box [ -R,R ] n . For sufficiently large R and a sufficiently dense sample of points, we get an accurate estimate. The symbolic approach, on the other hand, exploits the connection between neural networks and tropical Puiseux rational maps. Indeed, we can symbolically compute a Puiseux rational map that represents the neural network and then compute the number of linear regions using the approach outlined in section 5.1.

To compare each method, we ran the computations on smaller networks with varying sizes to compare run times and precision. For the symbolic approach, we generate 20 neural networks with random weights for each architecture and then compute the tropical Puiseux rational function associated to each neural network and compute the linear regions using Algorithm 3.

For the numerical approach, we also work with synthetic data and generate 1000 neural networks with random weights for each architecture. We then estimate the number of linear regions in a box of size [ -10 , 10] n and sample 1000 points from this domain.

In both cases, we use He initialization for the weights, i.e., we generate weights with distribution N (0 , 2 √ d ) where d is the input dimension. The data we obtain in this manner is summarized in Tables 10 and 11. For the symbolic approach, we also track the number of nonzero monomials to compare this quantity with the number of linear regions. For networks with 3 layers, we find the numerical estimate to be quite close, but for 4 it seems to diverge. This could be because in the numerical approach, we are only counting the number of unique Jacobians that can be found in the domain. A situation could arise where the same linear function is disconnected and hence counted twice by the symbolic approach but only once for the numerical approach.

The main observations from our experimental study are as follows. The numerical approach is faster, but offers no guarantee of precision: When running the computation for a given R and mesh grid, there seems to be no easy way of determining whether we have indeed hit all the linear regions or whether we have obtained an accurate estimate of the arrangements of these regions. It is possible to either overestimate or underestimate the number of linear regions. In particular, there is a priori no obvious way to select the parameters. We found the symbolic approach to be more precise, but slower. In general, the number of monomials seems to be far larger than the number of linear regions, which contradicts the intuition of Figure 2.

Both algorithms suffer from the curse of dimensionality: in the case of the numerical approach, the number of samples in a meshgrid grows exponenially with respect to the dimension. In the case of

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the symbolic approach, calculations with polytopes seem to scale poorly with dimension and with the complexity of the neural network.

6 Discussion: Limitations & Directions for Future Research

In this paper, we set up a framework to interpret and analyzed the expressivity of neural networks using techniques from polyhedral and tropical geometry. We demonstrated several ways in which a symbolic interpretation can often enable computational optimizations for otherwise intractable tasks and provided new insights into the inner workings of these networks. To the best of our knowledge, ours is the first work to provide practical tropical geometric theory and algorithms to numerically compute and analyze the expressivity of a neural network both in terms of inherent neural network quantities as well as tropical geometric quantities.

Despite the theoretical and practical advancement of tropical deep learning that our work offers, it is nevertheless subject to limitations, which we now discuss and which inspire directions for future research.

Experimental Limitations. The curse of dimensionality is a common theme in deep learning, and our work is unfortunately no exception. The methods introduced in this paper are quite fast for small enough networks, but scale poorly with dimension and more complex architectures.

We note that the main computational bottlenecks of the Puiseux rational function associated with a neural network are the implementation of fast multivariate Puiseux series operations. Our current computations rely on a custom implementation of this type of operation, and one potential avenue for improvement would be using such methods once they have been implemented in OSCAR [15].

For the computation of linear regions, both the numerical and symbolic approaches suffer from the curse of dimensionality. For instance, the numerical approach requires sampling on a mesh grid in a box of the form [ -R,R ] n where n is the input dimension. In particular, the number of points needed is proportional to the volume, which scales exponenially in n . Similarly, the symbolic approach relies on the computation of the Puiseux rational function associated with a neural network and polytope computations, both of which are challenging computational problems in higher dimensions.

Most of our computations rely on carrying out some elementary computations many times. Thus, another avenue of improvement for this would be to parallelize.

Structural Limitations. Much of what we are studying are basically framed as a combinatorial optimization problem, which are known to be difficult. In particular, computing the Hoffman constant is equivalent to the Stewart-Todd condition measure of a matrix and both quantities are NP-hard to compute in general cases [17, 21].

Further studying and understanding where and how symbolic computation algorithms can be made more efficient, e.g., by parallelization, would make our proposed approaches more applicable to larger neural networks. Our work effectively proposes a new intersection of symbolic computation and deep learning, so there remains infrastructure to set up to make methods from these two fields compatible.

References

Grigory Mikhalkin and Johannes Rau. Tropical geometry , volume 8. MPI for Mathematics, 2009.

David Speyer and Bernd Sturmfels. Tropical Mathematics. Mathematics Magazine , 82(3):163173, 2009.

Diane Maclagan and Bernd Sturmfels. Introduction to tropical geometry , volume 161. American Mathematical Society, 2021.

Razvan Pascanu, Guido Montufar, and Yoshua Bengio. On the number of response regions of deep feed forward networks with piece-wise linear activations. arXiv preprint arXiv:1312.6098 , 2013.

Guido F Montúfar, Razvan Pascanu, Kyunghyun Cho, and Yoshua Bengio. On the number of 393 linear regions of deep neural networks. Advances in neural information processing systems , 27, 394 2014. 395

Raman Arora, Amitabh Basu, Poorya Mianjy, and Anirbit Mukherjee. Understanding deep 396 neural networks with rectified linear units. arXiv preprint arXiv:1611.01491 , 2016. 397

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Maithra Raghu, Ben Poole, Jon Kleinberg, Surya Ganguli, and Jascha Sohl-Dickstein. On the expressive power of deep neural networks. In international conference on machine learning , pages 2847-2854. PMLR, 2017.

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Boris Hanin and David Rolnick. Deep ReLU Networks Have Surprisingly Few Activation Patterns. Advances in neural information processing systems , 32, 2019.

Huan Xiong, Lei Huang, Mengyang Yu, Li Liu, Fan Zhu, and Ling Shao. On the number of linear regions of convolutional neural networks. In International Conference on Machine Learning , pages 10514-10523. PMLR, 2020.

Alexis Goujon, Arian Etemadi, and Michael Unser. On the number of regions of piecewise 406 linear neural networks. Journal of Computational and Applied Mathematics , 441:115667, 2024. 407

Liwen Zhang, Gregory Naitzat, and Lek-Heng Lim. Tropical geometry of deep neural networks. 408 In International Conference on Machine Learning , pages 5824-5832. PMLR, 2018. 409

Vasileios Charisopoulos and Petros Maragos. A tropical approach to neural networks with 410 piecewise linear activations. arXiv preprint arXiv:1805.08749 , 2018. 411

Ruriko Yoshida, Georgios Aliatimis, and Keiji Miura. Tropical neural networks and its applica412 tions to classifying phylogenetic trees. arXiv preprint arXiv:2309.13410 , 2023. 413

Kurt Pasque, Christopher Teska, Ruriko Yoshida, Keiji Miura, and Jefferson Huang. Tropical 414 decision boundaries for neural networks are robust against adversarial attacks. arXiv preprint 415 arXiv:2402.00576 , 2024. 416

Oscar - open source computer algebra research system, version 1.0.0, 2024. 417

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Alan J Hoffman. On approximate solutions of systems of linear inequalities. In Selected Papers Of Alan J Hoffman: With Commentary , pages 174-176. World Scientific, 2003.

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Javier Pena, Juan Vera, and Luis Zuluaga. An algorithm to compute the hoffman constant of a system of linear constraints. arXiv preprint arXiv:1804.08418 , 2018.

Osman Güler, Alan J Hoffman, and Uriel G Rothblum. Approximations to solutions to systems of linear inequalities. SIAM Journal on Matrix Analysis and Applications , 16(2):688-696, 1995.

Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets. Advances in neural information processing systems , 30, 2017.

Alexis Goujon, Arian Etemadi, and Michael Unser. On the number of regions of piecewise linear neural networks. Journal of Computational and Applied Mathematics , 441:115667, 2024.

Javier F Pena, Juan C Vera, and Luis F Zuluaga. Equivalence and invariance of the chi and 429 hoffman constants of a matrix. arXiv preprint arXiv:1905.06366 , 2019. 430

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A Further Experimental Details

We ran the final computations on NVIDIA GeForce RTX 3090 GPUs. Table 7 lists the time taken by each experiment. Given that our experiments do not include training on large datasets, the experiments are not particularly expensive from the perspective memory usage, and all the code can be run on a laptop. The detail provided in the paper correspond roughly to the amount of computational resources that were used for this work, omitting trial and testing runs.

B Algorithms

Algorithm 1 Lower and approximate upper bound of Hoffman constant

Require: A : an m × n matrix; B max number of iterations; ϵ threshold of testing surjectivity.

1: Initialize H L = H U = 0 .

2: for i ∈ 1 , . . . , B do

3: Sample a random integer K .

4: Sample a random subset J from { 1 , . . . , m } of size K .

5: Solve (9). Let t be the optimal value;

6: if t > ϵ then

7: J is surjective. Update H L = max { H L , 1 t } ;

8: Compute the minimal singular value of ˆ ρ ( A J ) ;

9: if ˆ ρ ( A J ) > 0 then

return Lower bound H L and approximate upper bound H U .

10: Update H U = max { H U , 1 ˆ ρ ( A J ) } ;

Algorithm 2 Estimation of the ratio of fundamental domain sampling to regular sampling

Require: The input dimension n , R ∈ R side length for cube centered at the origin from which the samples are taken, M number of models to use, N base number of points to sample.

1: for m ∈ 1 ..M do

2: Create a permutation invariant model σ with input dimension n .

3: Sample N n points in the cube with side length R centered at the origin. Note that the number of points in the sample grows exponentially with the input dimension n .

4: Compute the Jacobian matrices of the network at each point, round to 10 decimal place to avoid numerical errors, remove duplicates, and count the number of unique Jacobians.

6: Compute the unique Jacobians similarly as for the regular sampling.

5: Sample N n n ! points from the fundamental domain of R n intersected with the sampling cube.

7: Sum the multiplicities of each Jacobian to get an estimate of the total number of linear regions.

8: Record the ratio of the fundamental domain estimate to the regular estimate. return The average ratio across M models.

C Proofs 438

C.1 Proof of Proposition 3.4 439

Proof. The polyhedra defined by 440

form a convex refinement of linear regions of f . Let 441

denote the residual of x to the polyhedron. We have 442

Algorithm 3 Linear regions of tropical Puiseux rational functions

Require: Tropical Puiseux polynomials p, q in n variables.

1: Compute the linear regions U 1 , . . . , U l of p , and set L i = L ( p, U i ) .

2: Compute the linear regions V 1 , . . . , V m of q , and set S j = L ( q, V j ) .

3: Compute the pairs ( i, j ) such that U i ∩ V j has dimension n

4: for ( i, j ) such that U i ∩ V j has dimension n do

5: Compute the linear map T ij = L i -S j

6: Set S to be the set of all T ij

7: for T ∈ S do

8: Compute the set I ( T ) indices ( i, j ) such that T = T ij .

9: Compute the set C ( T ) of connected components of

return ⋃ T ∈ S C ( T ) .

Algorithm 4 Numerical estimation of neural network linear regions

Require: The architecture of a linear activation neural network σ with scalar output, R ∈ R side length for cube centered at the origin from which the samples are taken, M number of models to use, N number of points to sample.

1: for m ∈ 1 ..M do

2: Create a model with architecture σ and initialise weights and biases using He inialisation.

3: Sample N points in the cube with side length R centered at the origin.

4: Compute the Jacobian matrices of the network at each point.

5: Round the Jacobians matrices to 5 decimal places to avoid floating point errors.

6: Remove duplicates and count the number of unique Jacobians. return The average number of linear regions.

Note that 443

Therefore, 444

which proves (6). 445

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C.2 Proof of Lemma 3.3

Proof. From the definition of minimal effective radius we have 447

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For each linear region U i characterized by ˜ A U i x ≤ ˜ b U i , by (1), d ( x, U i ) ≤ H ( ˜ A U i ) ∥ ( ˜ A U i x -˜ b U i ) + ∥ . 448 Passing to maximum we have 449

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C.3 Proof of Theorem 4.3 451

Proof. The action of G partitions U into a set of orbits [ U ] and we have |U| = ∑ [ A ] ∈ [ U ] | [ A ] | . 452

From property (i) defining a fundamental domain ∪ A ∈U A = ∪ σ ∈ G σ · ∆ , we have the trivial lower bound on the number of linear regions |U| ≥ ∑ A ∈U c | [ A ] | = | G ||U c | . Let A ∈ U . By Lagrange's theorem, the orbit of A is such that | [ A ] || G A | = | G | . Thus we have

D Numerical Calculations of the Hoffman Constant

We illustrate the computation of Hoffman constant of tropical Puiseux rational map on synthetic data. We generate tropical Puiseux rational maps by randomly generating two tropical Puiseux polynomials p and q . Specifically, suppose p has m p monomials and q has m q monomials. We construct an m p × n matrix A p and an m q × n matrix A q by uniformly sampling entries from [0 , 1] . We then form the matrix defined by (4). We then compute the exact Hoffman constant using the PVZ algorithm and estimate its lower bound and approximate its upper bound by our proposed algorithm. We record the computation time and the number of calls to solve the LP problem in the whole loop.

In the experiment we take different values of m p , m q , n and B . For each of the parameters we repeat all computation for 8 times. The true Hoffman constants, lower bounds, upper bounds, and the computation time per linear region can be found in Table 1,2,3, and the number of iterations of the PVZ algorithm and average time to solve LP during each iteration can be found in Table 4,5.6. From the tables we can see that computing the true Hoffman constants requires testing surjectivity and solving over thousands LP problems, which costs a lot of time. Although the lower bounds and approximate upper bounds can be loose, the computational time is much faster for lower bounds and upper bounds, which implies its potential to apply for real data applications.

E Tables

Tables 8 and 9 summarise the outcomes of the experiments on the computation of linear regions for 471 tropical Puiseux rational functions. For a fixed number of variables n var and number of monomials 472 n monomials, we generate n samples random Puiseux rational functions by picking random coefficients 473 and exponents using Julia's inbuilt random number generation functions, where both the numerator 474 and the denominator have n monomials monomials. We then compute the number of linear regions for 475 each of these rational functions and take the average over our all the samples that were generated. 476

F Figures 477

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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes new methods to count the number of linear regions in neural networks by viewing them as tropical Puiseux rational maps. By computing their Hoffman constant, the authors are able to identify a sampling radius which ensures that all the network’s linear regions will be intersected. They use this insight to propose algorithms for counting the number of linear regions for both invariant and traditional networks.\n\nSTRENGTHS:\nThe paper is well-written with virtually no typos and errors. The technical content is accessible and not unnecessarily convoluted and the proofs and concepts are presented clearly.\n\nWEAKNESSES:\nThe main weakness of the work, in my opinion, can be summarized in the following points:\n\n- the connections to tropical geometry and group theory are not rigorous beyond the point of simple notational fixes\n- the motivation for the work and how it fills gaps in the existing literature is unclear, and\n- the effective utility of the approach is not convincingly demonstrated by the theory or experiments.\n\n**Rigor of tropical and group theories**\n\nI believe this point is the biggest weakness of the paper. From the perspective of tropical algebra, vectors and polynomials live in $\\bar{\\mathbb{R}}$. The authors mention $\\bar{\\mathbb{R}}$ in line 87, but then this is never used in most of their work. This might seem as a notational fix, but it is not, as it introduces problems in virtually every single result in the paper. This first becomes a real issue in (4), where maximums are taken over, potentially, $\\infty$. How is it guaranteed that (4) exists in the context of tropical algebra? This is a recurring problem that appears in (5), (6), and (7). Another important issue at the intersection of group theory and tropical algebra is how groups are defined. Semirings, by construction, are objects that do not admit additive inverses. This means that, if one wants to define groups on such structures, great care needs to be taken as to how groups are defined, how they act on vector spaces, and what groups are actually permissible in this context.\n\nFrom the perspective of group theory how is the group action defined? How do the group elements act on vectors in tropical spaces? Group representations $\\rho: G \\to \\operatorname{GL}$ require the concept of an invertible matrix, however that concept is ill-defined in tropical vector spaces.\n\n(Moreover, the authors define incorrectly $\\bar{\\mathbb{R}}$. The infinity element needs to be the identity element of tropical addition: if one opts to use the max-plus semiring, $-\\infty$ should be used. If we use the min-plus semiring, $\\infty$ should be used. However, this is a notational fix.) \n\n**Motivation**\n\nIn terms of motivation it is unclear how the work is related to the existing works. There have been countless results on the number of linear regions of neural networks, and quite a few results using tropical geometry at that. What void in the literature does this paper fill? The related work paragraph lists some of the works in tropical geometry, but doesn’t highlight where these works come short and how the proposed manuscript fills that void. Moreover, there is no discussion of why the existing works on linear counting that do not utilize tropical geometry are also not able to handle the presented context.\n\n**Effective utility**\n\nAt the end of the day, I’m not sure I understand what the utility of the method is. Ignoring here the questions on motivation, the goal is to make deep learning more interpretable. However, the authors’ experiments diverge when input sizes are larger than $6$ and the networks are deeper than $4$ layers. Modern deep learning uses input sizes significantly larger than $100$ and decade old networks are deeper than $10$ layers. So what effectively are we learning about deep networks?\n\nUnfortunately, there are some larger issues with the method and experiments. Beyond the concerns from the perspective of tropical geometry, we have zero guarantees about the upper and lower bounds of the Hoffman constant. There is no asymptotic analysis on the sample complexities of the bounds, no analyses about tightness or optimality, or even, at the very least, an analysis that the bounds are not vacuous or trivial. On line 272 the authors claim that even though their estimate diverges, that’s acceptable because frequently we’re interested is an upper bound on expressivity. However, how can we guarantee that there the number of regions is not undercounted (I couldn’t find a proof)? For that statement to be true the bound needs to be tight, but from their own experiments (Tables 1 and 2), the true $H$ ends up being larger than the upper bound, which is used to calculate the radius and eventually the radius. Obviously, then, the computed bounds are not representative: then, since the estimate Hoffman constant is not accurate and the algorithm diverges, what essentially do we gain?\n\nQUESTIONS:\nI have some more fine-grained comments and questions.\n\n**Major**\n\nThere is a recurring discussion about massively improving efficiency. However, that efficiency is when compared against what? There is not a single comparison against existing works, so the improvement is over what? Regardless, as the authors discuss in Algorithm 2 and in the section about limitations, there is an *exponential* scaling with the number of inputs. Even if the authors introduce a factorial reduction for invariant networks, they require exponential sampling, which leads to no real improvement, as is evident by their really small parameter values and large running times. How where these sample numbers chosen in the first place? I couldn’t find any asymptotic analysis that guarantees with high probability that the whole space will be sampled with such a number of samples. Overall, the computational complexity of the approach is prohibitive, as it required $N^n$ Jacobian computations, which is nigh impossible for parameter values found in real networks.\n\nThere is a significant lack of attribution to prior work. In the introduction, textbooks on tropical geometry are cited as well as papers on the counting of linear regions. However, when the intersection of tropical geometry and machine/deep learning is discussed, zero references are provided, whereas the field has been fairly active in the last 5 years with dozens of highly cited papers. Similarly, in the related works section the final citation has no connection to the presented paper beyond being at the intersection of tropical geometry and machine learning. However, there are much older and much more well-cited papers to include as general references on the field, both as general reviews and as specific applications (tropical compression of neural networks comes to mind). As a final comment on this train of thought, would the authors like to elaborate on line 352? Specifically, both [11] and [12] analyze the expressivity of neural networks using tropical geometric quantities (through zonotopes and Newton polytopes). How do the authors view their work as the first in doing that?\n\nAs a last major comment, there is another lack of preciseness, however to a lesser extend as to what I mentioned above. There is a recurring discussion about one method being more precise than the other: however, what does that mean? Preciseness is never defined, and it’s not straightforward how the authors compute the ground truth of the number of linear regions. For example, how can we say that Table 10 is more precise than Table 11? In a similar vain, there are underlying assumptions that are not discussed or communicated. For example, what are the assumptions of 4.3 from the perspective of group theory? Does the theorem work for any, possibly infinite group? Similarly, there are constraints in Definition 3.5 that are not communicated. The way A-surjectivity is introduced, it is required that $m < n$. However, that makes explicit assumptions about the networks that can be analyzed, which is never communicated or considered.\n\n**Minor**\n\nThere are some minor things that are unclear in the text. For example, it is unclear what Section 5.2 refers to. Is it a table or a plot, and which one? In Theorem 4.3 $\\mathcal{L}$ is mentioned which is never used or defined, and it is unclear what the term bias is referring to in 256 (I’m assuming $\\lambda$?). Table 1 (and others) are lacking headers and it’s impossible to understand what the different columns represent. Finally, the term maximal elements is used with respect to sets of sets. However, sets lack canonical order, so it’s unclear what “maximal element” means in this context.\n\nOverall the notation is good, but at a few points it is confusing. Matrix indices usually refer to columns, not rows, which can lead to confusion and the choice of $\\rho$ for the singular values (over the universally accepted $\\sigma$) can hurt readability. In 329 the input dimension is denoted by $d$, in contrast to the rest of the manuscript and the notation $\\min\\{y_j, j\\in J\\}$ is not consistent with set theory nor optimization (and also in conflict with the authors own notation, for example, in (10) or 295. Finally, $S_n$ is used as the permutation group but also as a set in 295.\n\nLIMITATIONS:\nI think the authors accurately identify the main limitations of their approach, which relates to the large computational complexity.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 1, "DATE": "2024-07-10"}

comments: SUMMARY:
This study investigates the expressive power of deep fully-connected ReLU networks (or a piecewise linear function) from the perspective of tropical geometry. The number of linear regions gives an estimate of the information capacity of the network, and the authors provide a novel tropical geometric approach to selecting sampling domains among linear regions.

STRENGTHS:
- An effective sampling domain is proposed as a ball of radius bounded by Hoffman constant, a combinatorial invariant

- The proposed sampling algorithm is doable and implemented.

WEAKNESSES:
- The proposed algorithms suffer from the curse of dimensionality

QUESTIONS:
- The authors mention in the abstract that the number of linear regions is an estimate of information capacity of the network. I need more clarifications, because this fact bridges tropical geometry and machine learning study.

- (minor) l.133: I was a bit confused here. Is the matrix-vector product $Ax$ in the sense of tropical algebra or in the ordinal sense? What does a “vector-inequality” $Ax \le b$ mean?

LIMITATIONS:
it is discussed in Section 6
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the expressivity of neural networks as captured by the number of linear regions using tools from tropical geometry. There are three main contributions, two of which are theoretical and the other is about open source library that allows the analysis of neural networks as Puiseux rational maps. \n\nThe first theoretical contribution is that they propose a new approach for selecting sampling domains among the linear regions and the second is a way to prune the search space for network architectures with symmetries.\n\nPrior work on tropical geometry and deep neural nets have analyzed ReLU and maxout units. Contrary to prior works, this work makes an effort to understand the geometry of the linear regions not just their number. To do so the authors propose a way of sampling the domain that leads to more accurate estimates compared to random sampling from the input space, which is a previously used alternative that can result in some missed linear regions and hence in inaccurate estimates about the information capacity of the neural network. This insight about sampling, allows then the authors to reduce the time to estimate the linear regions of special types of neural networks that exhibit some symmetry. This essentially reduces the number of samples needed and they experimentally verify their results.\n\nFinally, the authors release OSCAR an open source library.\n\nSTRENGTHS:\n-connections of neural networks expressibity with tropical geometry, though they have been exploited in the past, are strengthened in this paper\n\n-the paper presents a nice story that leads to faster sampling methods, both in theory and in practice.\n\nWEAKNESSES:\n-the main weakness I see in the paper is that, though well-motivated and interesting, it lacks technical depth. For example, there is essentially one main result stated as Theorem 4.3, and some intermediate results stated as Lemma 3.3 and Proposition 3.4. On the one hand, the latter two are simple observations about Puiseux polynomials, and on the other hand the proof of the Theorem 4.3 is not more than 3 lines (as shown in Appendix C). As such, I believe it is a nice transfer of ideas from tropical geometry to neural networks, but given that the connection was already there and used in prior works more than 10 years back, I don't think the better sampling algorithm is solid enough.\n\nQUESTIONS:\n-Would it be possible perhaps to strengthen the paper by proving depth-width separation results analogous to Telgarsky \"Benefits of Depth in Neural Networks\" using your techniques?\n\n-Is there some notion of optimality associated with your sampling algorithm? Could it be further improved?\n\nLIMITATIONS:\nSee weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-15"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work provides a geometric characterization of the linear regions in a neural network via the input space. Although linear regions are usually estimated by randomly sampling from the input space, stochasticity may cause some linear regions of a neural network to be missed. This paper proposes an effective sampling domain as a ball of radius R and computes bounds for the radius R based on a combinatorial invariant known as the Hoffman constant, which gives a geometric characterization for the linear regions of a neural network. Further, the paper exploits geometric insight into the linear regions of a neural network to gain dramatic computational efficiency when networks exhibit invariance under symmetry. Lastly, the paper provides code for converting trained and untrained neural networks into algebraic symbolic objects, useful for precisely the kinds of analysis this paper performs.\n\nSTRENGTHS:\n1. The authors present an interesting and novel way to analyze the capacity of a neural network using fundamental notions from tropical geometry.\n\n2. The paper and theory were very clearly presented. In terms of writing, the presentation of the relevant tropical geometry for purposes of Hoffman constant estimation Section 3 was excellent.\n\nWEAKNESSES:\n1. My greatest concern in this paper stems from the experiments for upper and lower bound estimation for Hoffman constants given in Tables 1, 2, and 3 in the appendix. It seems the experimental upper and lower bounds computed there do not actually bound the true Hoffman constant. I understand that the upper bound may be loose due to the way it is estimated, but the lower bound should always be below the true Hoffman constant, as per my understanding. Yet for, say, the first of eight computations in table 1, the lower bound $H_L$ is $0.5460$, which is clearly above the true value of $H$, given to be $0.3298$. For that example, the upper bound $H^U$ is given to be $0.2081$, which is clearly not above the true value. This pattern continues, and the lower and upper bounds for the Hoffman constants seem to fluctuate somewhat arbitrarily around the true Hoffman constants, which is concerning. I am currently assuming that there is some kind of mistake with these experimental values and would like a clarification from the authors regarding this.\n\n2. Due to the curse of dimensionality, this method for estimating the expressiveness of neural networks can only be applied to simple neural networks in practice. This is very apparent due to the way the numerical approach requires sampling on a mesh grid in an $n$-dimensional box (but is also true for the symbolic approach, that relies on the computation of the Puiseux rational function associated with a neural network, which becomes increasingly quite hard in higher dimensions). To the credit of the authors, they are up-front about this limitation, but this does significantly hinder the applicability of the presented results.\n\nQUESTIONS:\nMy questions to the authors are listed below:\n\n1. Why do the computed upper and lower bounds for the Hoffman constants in Tables 1, 2, and 3, seem wrong? Is there a mistake with this computation, and if so, can you provide the correct tables so that I may judge how tight/loose the bounds are? I elaborated on this in the first weakness of the \"Weaknesses\" section above. \n\n### Additional Comments and Minor Corrections\n\nThe writing in this paper is quite good and the material is very clearly introduced. Nonetheless, I would like to give a non-exhaustive list of minor corrections below:\n\nL103, 105: Please use \\citet for in-text citations. This happens several times, but I will only mention it once.\n\nL349: \"to interpret and analyzed\" -> \"to interpret and analyze\"\n\nLIMITATIONS:\nThe authors adequately discuss the limitations of their work in Section 6.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-16"}