A generalization of the randomized singular value decomposition

Nicolas Boulle, Alex Townsend

Keywords: Low rank approximation, Randomized SVD, Hilbert--Schmidt operators, Gaussian processes

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 theory of 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.

One-sentence Summary: The randomized SVD is generalized to multivariate Gaussian input vectors and Hilbert-Schmidt operators.