Unsupervised learning of binary vectors: a Gaussian scenario

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.