Unsupervised learning of binary vectors: a Gaussian scenario

TL;DR

This paper analyzes an unsupervised learning model with Gaussian-distributed data influenced by a binary symmetry-breaking direction, revealing exponential convergence of learning, phase transitions, and optimal binary vector recovery methods.

Contribution

It introduces a detailed analysis of Gibbs learning in a binary vector model, including phase transition phenomena and optimal binary recovery strategies.

Findings

Gibbs learning approaches perfect binary match exponentially.

Both second and first order phase transitions are identified.

Sign-based methods achieve near-optimal binary performance.

Abstract

We study a model of unsupervised learning where the real-valued data vectors are isotropically distributed, except for a single symmetry breaking binary direction $\bm{B}\in\{-1,+1\}^{N}$, onto which the projections have a Gaussian distribution. We show that a candidate vector $\bm{J}$ undergoing Gibbs learning in this discrete space, approaches the perfect match $\bm{J}=\bm{B}$ exponentially. Besides the second order ``retarded learning'' phase transition for unbiased distributions, we show that first order transitions can also occur. Extending the known result that the center of mass of the Gibbs ensemble has Bayes-optimal performance, we show that taking the sign of the components of this vector leads to the vector with optimal performance in the binary space. These upper bounds are shown not to be saturated with the technique of transforming the components of a special continuous vector, except in asymptotic limits and in a special linear case. Simulations are presented which are in excellent agreement with the theoretical results.