Daydreaming algorithm for Biased Patterns

Abstract

The \emph{Daydreaming} algorithm has been recently proposed in Ref.~\cite{Serricchio2025} as a learning rule that simultaneously reinforces stored patterns and suppresses spurious attractors to improve the storage capacity of the Hopfield model. Its effectiveness has been reported for both uncorrelated and correlated data. However, the existing formulation has mainly assumed unbiased patterns, and the formulation for biased patterns has not yet been sufficiently established. Biased patterns are known to be much more problematic for models of associative memories. In this study, we reformulate Daydreaming for biased patterns motivated by the underlying rationale of the pseudo-inverse rule. Specifically, we introduce the retrieval dynamics and an energy function based on the centered representation, and we derive a corresponding update rule for centered Daydreaming. We compare the centered pseudo-inverse rule with centered Daydreaming for biased patterns and examine the retrieval maps and eigenvalue distributions of the coupling matrices. As a result, the centered Daydreaming yields larger basins of attraction than the centered pseudo-inverse rule, and such a beneficial property seems to be due to the broadness of the eigenvalue spectrum of the coupling matrix. To better understand this connection, we construct modified coupling matrices whose spectra interpolate between a pseudo-inverse-like spectrum and the Daydreaming one. The results clearly indicate that the broader spectrum generated by Daydreaming contributes to the enlarged basin of attraction.