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Piecewise linear functions and neural network expressivity via discriminantal arrangements

Pragnya Das

Abstract

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.

Piecewise linear functions and neural network expressivity via discriminantal arrangements

Abstract

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.

Paper Structure

This paper contains 28 sections, 22 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperplane Arrangements
  4. Continuous Piecewise Linear Functions
  5. Discriminantal Arrangements and Circuits
  6. Circuit-Induced Linear Functionals
  7. Circuit-Constrained Function Spaces
  8. Definition of the Function Space
  9. Compatibility and Induced Set Functions
  10. Dimension
  11. Matroidal and Algebraic Structure of Circuit-Constrained Function Spaces
  12. Matroidal Interpretation
  13. Möbius Transform and Support Constraints
  14. The Case k=2: Explicit Structure and Interpretation
  15. Example: A Discriminantal Arrangement for n=3, k=2
  16. ...and 13 more sections

Key Result

Lemma 3.2

Let $\sigma \subset \mathbb{R}^n$ be a simplicial cone generated by linearly independent rays $v_1,\dots,v_n$. Let $G \subset \sigma$ be a set of points such that $\operatorname{aff}(G) = \mathbb{R}^n$. Then any function $f : G \to \mathbb{R}$ extends to at most one affine function on $\sigma$. In p

Theorems & Definitions (58)

  • Definition 2.1: Discriminantal-type dependence structure
  • Definition 2.2: Discriminantal arrangement $A(n,k)$
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • ...and 48 more