# Storage capacity in symmetric binary perceptrons

## TL;DR

This paper investigates the storage capacity of symmetric binary perceptrons, introducing two variants and analyzing their capacity thresholds using probabilistic methods and the replica technique, revealing different organizational structures of solutions.

## Contribution

The paper provides the first rigorous analysis of the capacity of symmetric binary perceptrons, demonstrating when annealed approximations are valid and proposing a solution organization conjecture.

## Key findings

- Capacity equals annealed computation for certain symmetric cases.
- Solutions likely organized in a frozen-1RSB structure.
- Replica method estimates capacity threshold for wide $u$-function cases.

## Abstract

We study the problem of determining the capacity of the binary perceptron for two variants of the problem where the corresponding constraint is symmetric. We call these variants the rectangle-binary-perceptron (RPB) and the $u-$function-binary-perceptron (UBP). We show that, unlike for the usual step-function-binary-perceptron, the critical capacity in these symmetric cases is given by the annealed computation in a large region of parameter space (for all rectangular constraints and for narrow enough $u-$function constraints, $K<K^*$). We prove this fact (under two natural assumptions) using the first and second moment methods. We further use the second moment method to conjecture that solutions of the symmetric binary perceptrons are organized in a so-called frozen-1RSB structure, without using the replica method. We then use the replica method to estimate the capacity threshold for the UBP case when the $u-$function is wide $K>K^*$. We conclude that full-step-replica-symmetry breaking would have to be evaluated in order to obtain the exact capacity in this case.