Sketching Outside the Hilbert Space for the Infinite-Dimensional Randomised SVD

Abstract

In randomised numerical linear algebra, common choices for distributions of sketch vectors $\omega \in {\mathbb{R}}^n$ are isotropic, so that $\mathbb{E}[\omega {\omega }^T]=I_n$. Unfortunately, the natural extension of an isotropic vector in $({\mathbb{R}}^n,\Vert \cdot {\Vert }_2)$ to an infinite-dimensional separable Hilbert space $\mathcal{H}$ may not exist as an element of the space, posing an obstacle for any attempt to lift the randomised SVD to infinite-dimensional target operators $A$. Current workarounds either forego isotropy and take $\omega$ to be a continuous Gaussian process with trace-class covariance operators, or leave $\omega$ implicit by instead sampling $A\omega$ as if $\omega$ were isotropic Gaussian. We unify their approaches by introducing "sketch functionals", an example of cylindrical random variables (Vakhania--Tarieladze--Chobanyan 1987) which allow us to treat general, non-Gaussian distributions with bounded covariance operators. To demonstrate the utility of the new theory, we apply it to develop novel bounds for the infinite-dimensional randomised SVD with a general class of sub-Gaussian sketch distributions.