A generalization of the randomized singular value decomposition

TL;DR

This paper extends the randomized SVD to multivariate Gaussian vectors and Gaussian processes, enabling incorporation of prior knowledge and efficient sampling for matrix and operator approximations.

Contribution

It introduces a generalized randomized SVD framework using multivariate Gaussians and Gaussian processes, with a new covariance kernel based on weighted Jacobi polynomials.

Findings

Demonstrates the algorithm's effectiveness on matrices and Hilbert--Schmidt operators.

Provides a new covariance kernel for Gaussian processes that controls smoothness.

Shows rapid sampling capabilities for the proposed Gaussian process.

Abstract

The randomized singular value decomposition (SVD) is a popular and effective algorithm for computing a near-best rank $k$ approximation of a matrix $A$ using matrix-vector products with standard Gaussian vectors. Here, we generalize the randomized SVD to multivariate Gaussian vectors, allowing one to incorporate prior knowledge of $A$ into the algorithm. This enables us to explore the continuous analogue of the randomized SVD for Hilbert--Schmidt (HS) operators using operator-function products with functions drawn from a Gaussian process (GP). We then construct a new covariance kernel for GPs, based on weighted Jacobi polynomials, which allows us to rapidly sample the GP and control the smoothness of the randomly generated functions. Numerical examples on matrices and HS operators demonstrate the applicability of the algorithm.