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

## TL;DR

This paper proposes a neural network-based method for decomposing positive semidefinite matrices into low-rank and sparse components, with theoretical convergence guarantees and applications in portfolio optimization.

## Contribution

It introduces a neural network parametrization for the low-rank component and analyzes its convergence rate, advancing matrix decomposition techniques.

## Key findings

- Convergence rate grows polynomially with input and network dimensions.
- Method effectively decomposes covariance matrices in portfolio optimization.
- Provides theoretical analysis of gradient descent convergence.

## Abstract

This paper revisits the problem of decomposing a positive semidefinite matrix as a sum of a matrix with a given rank plus a sparse matrix. An immediate application can be found in portfolio optimization, when the matrix to be decomposed is the covariance between the different assets in the portfolio. Our approach consists in representing the low-rank part of the solution as the product $MM^{T}$, where $M$ is a rectangular matrix of appropriate size, parametrized by the coefficients of a deep neural network. We then use a gradient descent algorithm to minimize an appropriate loss function over the parameters of the network. We deduce its convergence rate to a local optimum from the Lipschitz smoothness of our loss function. We show that the rate of convergence grows polynomially in the dimensions of the input, output, and the size of each of the hidden layers.