Non-Convex Multi-species Hopfield models

In this work we introduce a multi-species generalization of the Hopfield model for associative memory, where neurons are divided into groups and both inter-groups and intra-groups pair-wise interactions are considered, with different intensities. Thus, this system contains two of the main ingredients of modern Deep neural network architectures: Hebbian interactions to store patterns of information and multiple layers coding different levels of correlations. The model is completely solvable in the low-load regime with a suitable generalization of the Hamilton-Jacobi technique, despite the Hamiltonian can be a non-definite quadratic form of the magnetizations. The family of multi-species Hopfield model includes, as special cases, the 3-layers Restricted Boltzmann Machine (RBM) with Gaussian hidden layer and the Bidirectional Associative Memory (BAM) model.Comment: This is a pre-print of an article published in J. Stat. Phy

Neural Networks retrieving Boolean patterns in a sea of Gaussian ones

Restricted Boltzmann Machines are key tools in Machine Learning and are described by the energy function of bipartite spin-glasses. From a statistical mechanical perspective, they share the same Gibbs measure of Hopfield networks for associative memory. In this equivalence, weights in the former play as patterns in the latter. As Boltzmann machines usually require real weights to be trained with gradient descent like methods, while Hopfield networks typically store binary patterns to be able to retrieve, the investigation of a mixed Hebbian network, equipped with both real (e.g., Gaussian) and discrete (e.g., Boolean) patterns naturally arises. We prove that, in the challenging regime of a high storage of real patterns, where retrieval is forbidden, an extra load of Boolean patterns can still be retrieved, as long as the ratio among the overall load and the network size does not exceed a critical threshold, that turns out to be the same of the standard Amit-Gutfreund-Sompolinsky theory. Assuming replica symmetry, we study the case of a low load of Boolean patterns combining the stochastic stability and Hamilton-Jacobi interpolating techniques. The result can be extended to the high load by a non rigorous but standard replica computation argument.Comment: 16 pages, 1 figur

A Deterministic and Generalized Framework for Unsupervised Learning with Restricted Boltzmann Machines

Restricted Boltzmann machines (RBMs) are energy-based neural-networks which are commonly used as the building blocks for deep architectures neural architectures. In this work, we derive a deterministic framework for the training, evaluation, and use of RBMs based upon the Thouless-Anderson-Palmer (TAP) mean-field approximation of widely-connected systems with weak interactions coming from spin-glass theory. While the TAP approach has been extensively studied for fully-visible binary spin systems, our construction is generalized to latent-variable models, as well as to arbitrarily distributed real-valued spin systems with bounded support. In our numerical experiments, we demonstrate the effective deterministic training of our proposed models and are able to show interesting features of unsupervised learning which could not be directly observed with sampling. Additionally, we demonstrate how to utilize our TAP-based framework for leveraging trained RBMs as joint priors in denoising problems