Matrix factorization with neural networks

TL;DR

This paper presents a novel decimation scheme that transforms matrix factorization into neural network models, enabling efficient factorization and denoising of large matrices with theoretical and empirical validation.

Contribution

It introduces a new decimation algorithm that maps matrix factorization to neural networks and provides a comprehensive theoretical analysis of its effectiveness.

Findings

Decimation effectively factorizes extensive-rank matrices.

The neural network-based method denoises matrices efficiently.

Performance matches theoretical predictions.

Abstract

Matrix factorization is an important mathematical problem encountered in the context of dictionary learning, recommendation systems and machine learning. We introduce a new `decimation' scheme that maps it to neural network models of associative memory and provide a detailed theoretical analysis of its performance, showing that decimation is able to factorize extensive-rank matrices and to denoise them efficiently. We introduce a decimation algorithm based on ground-state search of the neural network, which shows performances that match the theoretical prediction.