Regularized Computation of Approximate Pseudoinverse of Large Matrices Using Low-Rank Tensor Train Decompositions

TL;DR

This paper introduces a novel tensor network-based method for efficiently computing regularized low-rank approximations of Moore-Penrose pseudoinverses of large matrices, enhancing solutions to large linear systems.

Contribution

It develops a stable, efficient algorithm using tensor train decompositions and regularization for approximating pseudoinverses of large matrices, with proven logarithmic computational complexity.

Findings

Regularized pseudoinverses admit good low-rank TT representations.

The method's computational cost scales logarithmically with matrix size.

Preconditioners computed by the method effectively solve convection-diffusion problems.

Abstract

We propose a new method for low-rank approximation of Moore-Penrose pseudoinverses (MPPs) of large-scale matrices using tensor networks. The computed pseudoinverses can be useful for solving or preconditioning of large-scale overdetermined or underdetermined systems of linear equations. The computation is performed efficiently and stably based on the modified alternating least squares (MALS) scheme using low-rank tensor train (TT) decompositions and tensor network contractions. The formulated large-scale optimization problem is reduced to sequential smaller-scale problems for which any standard and stable algorithms can be applied. Regularization technique is incorporated in order to alleviate ill-posedness and obtain robust low-rank approximations. Numerical simulation results illustrate that the regularized pseudoinverses of a wide class of non-square or nonsymmetric matrices admit good approximate low-rank TT representations. Moreover, we demonstrated that the computational cost of the proposed method is only logarithmic in the matrix size given that the TT-ranks of a data matrix and its approximate pseudoinverse are bounded. It is illustrated that a strongly nonsymmetric convection-diffusion problem can be efficiently solved by using the preconditioners computed by the proposed method.