Phase diagram of restricted Boltzmann machines and generalized Hopfield networks with arbitrary priors

TY - JOUR

T1 - Phase diagram of restricted Boltzmann machines and generalized Hopfield networks with arbitrary priors

AU - Barra, Adriano

AU - Genovese, G.

AU - Sollich, Peter

AU - Tantam, Daniele

PY - 2018/2/20

Y1 - 2018/2/20

N2 - Restricted Boltzmann Machines are described by the Gibbs measure of a bipartite spin glass, which in turn can be seen as a Generalised Hopfield network. This equivalence allows us to characterise the state of these systems in terms of their retrieval capabilities, both at low and high load, of pure states. We study the paramagnetic-spin glass and the spin glass-retrieval phase ransitions, as the pattern (i.e. weight) distribution and spin (i.e. unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region becomes larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.

AB - Restricted Boltzmann Machines are described by the Gibbs measure of a bipartite spin glass, which in turn can be seen as a Generalised Hopfield network. This equivalence allows us to characterise the state of these systems in terms of their retrieval capabilities, both at low and high load, of pure states. We study the paramagnetic-spin glass and the spin glass-retrieval phase ransitions, as the pattern (i.e. weight) distribution and spin (i.e. unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region becomes larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.

U2 - 10.1103/PhysRevE.97.022310

DO - 10.1103/PhysRevE.97.022310

M3 - Article

SN - 1539-3755

VL - 97

SP - 0223101-1 - 0223101-14

JO - PHYSICAL REVIEW E

JF - PHYSICAL REVIEW E

IS - 2

ER -