Low-Rank plus Sparse Decomposition of Covariance Matrices using Neural\n Network Parametrization

2019/08/01 by Michel Baes, Baes, Michel, Calypso Herrera +5 · 1 citation
Computer Science · Engineering · #FOS: Mathematics #Medical Image Segmentation Techniques #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques

paper · pdf · doi:10.48550/arxiv.1908.00461

openalex publication_date 2019/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper revisits the problem of decomposing a positive semidefinite matrix\nas a sum of a matrix with a given rank plus a sparse matrix. An immediate\napplication can be found in portfolio optimization, when the matrix to be\ndecomposed is the covariance between the different assets in the portfolio. Our\napproach consists in representing the low-rank part of the solution as the\nproduct MM^T, where M is a rectangular matrix of appropriate size,\nparametrized by the coefficients of a deep neural network. We then use a\ngradient descent algorithm to minimize an appropriate loss function over the\nparameters of the network. We deduce its convergence rate to a local optimum\nfrom the Lipschitz smoothness of our loss function. We show that the rate of\nconvergence grows polynomially in the dimensions of the input, output, and the\nsize of each of the hidden layers.\n

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