Generalized low rank approximation to the symmetric positive semidefinite matrix

TL;DR

This paper proposes a method for approximating symmetric positive semidefinite matrices with low rank, transforming the problem into an unconstrained optimization and solving it using nonlinear conjugate gradient, demonstrated through a numerical example.

Contribution

It introduces a novel approach to low rank approximation of PSD matrices by converting the problem into an unconstrained optimization and applying nonlinear conjugate gradient methods.

Findings

The method effectively approximates PSD matrices with low rank.

The approach is feasible as demonstrated by numerical experiments.

Abstract

In this paper, we investigate the generalized low rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $$\underset{ rank(X)\leq k}{\min} \sum^m_{i=1}\left \Vert A_i - B_i XB_i^T \right \Vert^2_F,$$ where $X$ is an unknown symmetric positive semidefinite matrix and $k$ is a positive integer. We firstly use the property of a symmetric positive semidefinite matrix $X=YY^T$, $Y$ with order $n\times k$, to convert the generalized low rank approximation into unconstraint generalized optimization problem. Then we apply the nonlinear conjugate gradient method to solve the generalized optimization problem. We give a numerical example to illustrate the numerical algorithm is feasible.