The dual of the space of interactions in neural network models

TL;DR

This paper explores the dual space of neural network interactions, proposing methods to identify and eliminate unstable patterns, and analyzing phase transitions in pattern stability using real data.

Contribution

It introduces a dual perspective on the Gardner problem, defining an integer linear system to characterize unstable patterns and proposing Monte Carlo methods for their removal.

Findings

Unstable patterns can be systematically identified and removed.

Phase transitions in pattern stability can occur non-linearly.

The approach is demonstrated on real neural data.

Abstract

In this work the Gardner problem of inferring interactions and fields for an Ising neural network from given patterns under a local stability hypothesis is addressed under a dual perspective. By means of duality arguments an integer linear system is defined whose solution space is the dual of the Gardner space and whose solutions represent mutually unstable patterns. We propose and discuss Monte Carlo methods in order to find and remove unstable patterns and uniformly sample the space of interactions thereafter. We illustrate the problem on a set of real data and perform ensemble calculation that shows how the emergence of phase dominated by unstable patterns can be triggered in a non-linear discontinuous way.