# Minimal model of permutation symmetry in unsupervised learning

## TL;DR

This paper introduces a minimal two-unit model in a restricted Boltzmann machine to analyze how permutation symmetry influences learning, revealing that correlation between hidden units reduces the data needed for concept formation and that symmetry breaking occurs spontaneously.

## Contribution

The study provides a semi-rigorous proof that the critical data size for symmetry breaking is independent of the number of hidden units and introduces an efficient algorithm for inferring hidden unit receptive fields.

## Key findings

- Correlation between hidden units reduces critical data size.
- Weakly-correlated receptive fields significantly lower the data threshold.
- Permutation symmetry can be spontaneously broken during learning.

## Abstract

Permutation of any two hidden units yields invariant properties in typical deep generative neural networks. This permutation symmetry plays an important role in understanding the computation performance of a broad class of neural networks with two or more hidden units. However, a theoretical study of the permutation symmetry is still lacking. Here, we propose a minimal model with only two hidden units in a restricted Boltzmann machine, which aims to address how the permutation symmetry affects the critical learning data size at which the concept-formation (or spontaneous symmetry breaking in physics language) starts, and moreover semi-rigorously prove a conjecture that the critical data size is independent of the number of hidden units once this number is finite. Remarkably, we find that the embedded correlation between two receptive fields of hidden units reduces the critical data size. In particular, the weakly-correlated receptive fields have the benefit of significantly reducing the minimal data size that triggers the transition, given less noisy data. Inspired by the theory, we also propose an efficient fully-distributed algorithm to infer the receptive fields of hidden units. Furthermore, our minimal model reveals that the permutation symmetry can also be spontaneously broken following the spontaneous symmetry breaking. Overall, our results demonstrate that the unsupervised learning is a progressive combination of spontaneous symmetry breaking and permutation symmetry breaking which are both spontaneous processes driven by data streams (observations). All these effects can be analytically probed based on the minimal model, providing theoretical insights towards understanding unsupervised learning in a more general context.