A generalization of the randomized singular value decomposition
This work provides a method for improving matrix and operator approximations in computational mathematics, though it appears incremental as it builds directly on existing randomized SVD techniques.
The authors generalized the randomized singular value decomposition (SVD) to incorporate prior knowledge using multivariate Gaussian vectors and extended it to Hilbert-Schmidt operators with Gaussian processes, constructing a new covariance kernel based on weighted Jacobi polynomials for efficient sampling and smoothness control.
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.