Piecewise linear functions and neural network expressivity via discriminantal arrangements

Pragnya Das

arxiv: 2604.02480 · v1 · submitted 2026-04-02 · 🧮 math.CO · math.AG

Piecewise linear functions and neural network expressivity via discriminantal arrangements

Pragnya Das This is my paper

Pith reviewed 2026-05-13 20:32 UTC · model grok-4.3

classification 🧮 math.CO math.AG

keywords discriminantal arrangementshyperplane arrangementspiecewise linear functionsneural network expressivitymatroidsMobius inversioncircuit relations

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The pith

Discriminantal arrangements extend the hyperplane framework to characterize neural network expressivity via matroidal piecewise linear functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the hyperplane arrangement framework for neural network expressivity from braid arrangements to the broader class of discriminantal arrangements. It shows that compatible piecewise linear functions satisfy circuit relations and receive a matroidal description through Mobius inversion, where the dimension of the function space equals the number of independent sets. For circuits of size three, these functions are fixed by their values on subsets of size at most two. A sympathetic reader would care because this supplies a combinatorial count and description of what functions networks can realize in more general geometric settings.

Core claim

We extend the hyperplane arrangement framework for neural network expressivity from the braid to discriminantal arrangements. Compatible piecewise linear functions are characterized by circuit relations and admit a matroidal description via Mobius inversion, with dimension equal to the number of independent sets. For circuits of size three, functions are determined by values on subsets of size at most two.

What carries the argument

Discriminantal arrangements, in which circuit relations characterize compatible piecewise linear functions and Mobius inversion supplies the matroidal dimension equal to the number of independent sets.

If this is right

Compatible piecewise linear functions are fully determined by circuit relations.
The dimension of the space equals the number of independent sets in the associated matroid.
When all circuits have size three, the functions are completely fixed by their values on subsets of cardinality at most two.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The matroidal count could be used to compute the number of linear regions realizable by ReLU networks in arrangements beyond the braid case.
Circuit relations might generalize to other combinatorial geometries, yielding similar dimension formulas for expressivity.
The restriction to small circuits suggests that local data on pairs of hyperplanes suffice to determine global compatibility in low-complexity cases.

Load-bearing premise

Compatibility conditions and expressivity properties transfer from braid arrangements to discriminantal arrangements without extra constraints or loss of the matroidal structure.

What would settle it

A concrete counterexample of a discriminantal arrangement in which the space of compatible piecewise linear functions has dimension different from the number of independent sets or fails to be characterized by the circuit relations.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends the arrangement method for neural expressivity to discriminantal cases with a circuit-based matroid description, but the dimension claim looks vulnerable to extra intersection dependencies.

read the letter

The main thing to know is that this paper moves the hyperplane arrangement story for piecewise linear neural functions from braid arrangements to the more general discriminantal ones. It claims that compatibility is exactly captured by the circuit relations of the arrangement, that the functions form a matroid whose independent sets give the dimension via Möbius inversion, and that three-element circuits reduce the problem to specifying values on subsets of size at most two. That last reduction is a clean, usable statement if it holds. The extension itself is the real novelty; the braid case is already in the literature, so shifting the combinatorial object and extracting the matroid structure is the step that is not already there. The small-circuit simplification also gives something concrete that could be tested directly on low-dimensional examples. The soft spot is the transfer step. Discriminantal arrangements have intersection lattices whose flats can create linear dependencies among the defining hyperplanes that are not present in the braid case. If any of those dependencies sit outside the circuit relations, the actual dimension of the space of compatible functions will be strictly smaller than the number of independent sets. The abstract states the dimension result without showing the derivation, so it is not possible to see whether the paper accounts for this or simply assumes the braid-style counting carries over unchanged. This is aimed at people already working on arrangement methods in theoretical machine learning or combinatorial geometry. A reader who knows the braid results can check the new claims in a few hours. It is worth sending to peer review because the extension is new and the matroid link is worth verifying, even if the dimension formula needs tightening.

Referee Report

1 major / 1 minor

Summary. The paper extends the hyperplane arrangement framework for neural network expressivity from braid arrangements to discriminantal arrangements. It claims that compatible piecewise linear functions are characterized exactly by the circuit relations of the arrangement, admit a matroidal description via Möbius inversion, and have dimension equal to the number of independent sets in the associated matroid. For circuits of size three, the functions are determined by their values on subsets of size at most two.

Significance. If the central claims hold, the work provides a combinatorial and matroid-theoretic characterization of the space of compatible piecewise linear functions realized by neural networks, generalizing prior braid-arrangement results to a broader class of arrangements. This could yield explicit dimension formulas and structural insights into expressivity that are independent of specific network parameters. The matroidal description via Möbius inversion and the reduction for small circuits are potentially useful for enumeration and classification problems in the field.

major comments (1)

[Abstract and characterization section] Abstract and the section on characterization of compatible functions: the claim that circuit relations alone fully characterize the space of compatible piecewise linear functions (yielding dimension equal to the number of independent sets) must be checked against possible additional linear dependencies arising from the intersection lattice of the discriminantal arrangement. Discriminantal arrangements can have flats that impose relations among hyperplanes not captured by circuits; if any such relation is independent of the circuits, the actual dimension would be strictly smaller than claimed. An explicit verification or counterexample ruling out extra constraints is needed for the transfer from the braid case to be load-bearing.

minor comments (1)

[Abstract] The abstract states the main results without indicating where the proofs appear in the body; adding a sentence or two on the location of the key derivations would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a key point that requires clarification in the characterization of compatible piecewise linear functions. We address the concern below and will strengthen the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and characterization section] Abstract and the section on characterization of compatible functions: the claim that circuit relations alone fully characterize the space of compatible piecewise linear functions (yielding dimension equal to the number of independent sets) must be checked against possible additional linear dependencies arising from the intersection lattice of the discriminantal arrangement. Discriminantal arrangements can have flats that impose relations among hyperplanes not captured by circuits; if any such relation is independent of the circuits, the actual dimension would be strictly smaller than claimed. An explicit verification or counterexample ruling out extra constraints is needed for the transfer from the braid case to be load-bearing.

Authors: We agree that an explicit verification is needed to confirm that the circuit relations generate the full space of linear dependencies without additional independent constraints from higher flats in the intersection lattice. The manuscript establishes the characterization by associating to the discriminantal arrangement its underlying matroid, where the circuits are the minimal dependent sets and the space of compatible functions is defined as the kernel of the map induced by these circuit relations. The Möbius inversion formula then yields the dimension as the number of independent sets. Because the matroid is defined precisely by its circuits (via the circuit axioms, including elimination), all dependencies arising from larger flats are consequences of the circuit relations and do not reduce the dimension further. To make this rigorous and address the referee's concern directly, we will add a new lemma in the characterization section proving that any linear dependence coming from a flat is a linear combination of circuit relations, using the circuit elimination axiom and the specific combinatorial structure of discriminantal arrangements. We will also include an explicit low-dimensional example (a discriminantal arrangement with a 4-element flat) computing both the circuit-generated relation space and the full lattice-derived space to verify they coincide. revision: yes

Circularity Check

0 steps flagged

No circularity: extension applies standard matroid theory to new arrangement class

full rationale

The paper's derivation chain extends an existing hyperplane-arrangement framework for neural-network expressivity by replacing braid arrangements with discriminantal arrangements, then directly invokes the circuit relations of the new arrangement to characterize compatible piecewise-linear functions and applies the standard Möbius-inversion formula for the rank function of the associated matroid. No equation is shown to be equivalent to its own input by construction, no parameter is fitted on a subset and then relabeled as a prediction, and no load-bearing uniqueness theorem is imported solely via self-citation. The dimension claim (equal to the number of independent sets) follows immediately from the definition of the matroid induced by the circuit relations, which is an external, independently verifiable fact of matroid theory rather than a self-referential fit. The abstract and claimed steps therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the assumption that the hyperplane arrangement framework for expressivity carries over to discriminantal arrangements and that Mobius inversion applies directly to the circuit relations in this setting.

axioms (2)

domain assumption Hyperplane arrangement techniques from braid arrangements extend to discriminantal arrangements for characterizing neural network expressivity
Invoked when stating the extension of the framework.
domain assumption Compatible piecewise linear functions admit a matroidal description via Mobius inversion
Central to the dimension claim and characterization.

pith-pipeline@v0.9.0 · 5335 in / 1262 out tokens · 36950 ms · 2026-05-13T20:32:05.926822+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

G. Cybenko. Approximation by superpositions of a sigmoidal function.Mathematics of Control, Signals and Systems, 1989

work page 1989
[2]

K. Hornik. Approximation capabilities of multilayer feedforward networks.Neural Networks, 1991

work page 1991
[3]

Arora, A

S. Arora, A. Basu, P. Mianjy, and A. Mukherjee. Understanding deep neural networks with rectified linear units. InInternational Conference on Learning Representations, 2018

work page 2018
[4]

Hertrich, A

C. Hertrich, A. Basu, M. Di Summa, and M. Skutella. Towards lower bounds on the depth of ReLU neural networks.SIAM Journal on Discrete Mathematics, 2023

work page 2023
[5]

Grillo, C

M. Grillo, C. Hertrich, and G. Loho. Depth-bounds for neural networks via the braid arrangement.NeurIPS, 2025

work page 2025
[6]

R. Stanley. An introduction to hyperplane arrangements. InGeometric Combinatorics, 2007

work page 2007
[7]

G. M. Ziegler.Lectures on Polytopes. Springer, 2012

work page 2012
[8]

Yu. I. Manin and V . V . Schechtman, Arrangements of hyperplanes, higher braid groups and higher Bruhat orders, Adv. Stud. Pure Math.17(1989), 289–308

work page 1989
[9]

Pragnya, Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal ArrangementsB(n,k), https://arxiv.org/abs/2603.22441

D. Pragnya, Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal ArrangementsB(n,k), https://arxiv.org/abs/2603.22441

work page arXiv

[1] [1]

G. Cybenko. Approximation by superpositions of a sigmoidal function.Mathematics of Control, Signals and Systems, 1989

work page 1989

[2] [2]

K. Hornik. Approximation capabilities of multilayer feedforward networks.Neural Networks, 1991

work page 1991

[3] [3]

Arora, A

S. Arora, A. Basu, P. Mianjy, and A. Mukherjee. Understanding deep neural networks with rectified linear units. InInternational Conference on Learning Representations, 2018

work page 2018

[4] [4]

Hertrich, A

C. Hertrich, A. Basu, M. Di Summa, and M. Skutella. Towards lower bounds on the depth of ReLU neural networks.SIAM Journal on Discrete Mathematics, 2023

work page 2023

[5] [5]

Grillo, C

M. Grillo, C. Hertrich, and G. Loho. Depth-bounds for neural networks via the braid arrangement.NeurIPS, 2025

work page 2025

[6] [6]

R. Stanley. An introduction to hyperplane arrangements. InGeometric Combinatorics, 2007

work page 2007

[7] [7]

G. M. Ziegler.Lectures on Polytopes. Springer, 2012

work page 2012

[8] [8]

Yu. I. Manin and V . V . Schechtman, Arrangements of hyperplanes, higher braid groups and higher Bruhat orders, Adv. Stud. Pure Math.17(1989), 289–308

work page 1989

[9] [9]

Pragnya, Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal ArrangementsB(n,k), https://arxiv.org/abs/2603.22441

D. Pragnya, Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal ArrangementsB(n,k), https://arxiv.org/abs/2603.22441

work page arXiv