Hyperplane Neural Codes and the Polar Complex

TL;DR

This paper explores the properties of hyperplane codes generated by neural networks, demonstrating that their associated polar complexes are shellable, which explains many known properties of these codes.

Contribution

It establishes the shellability of the polar complex for stable hyperplane codes, linking combinatorial properties to neural code structures.

Findings

Polar complex of stable hyperplane codes is shellable.

Shellability explains many properties of hyperplane codes.

Most known properties follow from polar complex shellability.

Abstract

Hyperplane codes are a class of convex codes that arise as the output of a one layer feed-forward neural network. Here we establish several natural properties of stable hyperplane codes in terms of the {\it polar complex} of the code, a simplicial complex associated to any combinatorial code. We prove that the polar complex of a stable hyperplane code is shellable and show that most currently known properties of the hyperplane codes follow from the shellability of the appropriate polar complex.