Sketching Outside the Hilbert Space for the Infinite-Dimensional Randomised SVD
Abstract
In randomised numerical linear algebra, common choices for distributions of sketch vectors ω∈ ℝn are isotropic, so that 𝔼[ωωT]=In. Unfortunately, the natural extension of an isotropic vector in (ℝn, |· |2) to an infinite-dimensional separable Hilbert space 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 ω to be a continuous Gaussian process with trace-class covariance operators, or leave ω implicit by instead sampling Aω as if ω 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.
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BibTeX
@misc{lynchdavies2026sketching,
title = {{Sketching Outside the Hilbert Space for the Infinite-Dimensional Randomised SVD}},
author = {Morgan Lynch-Davies and Marcus Webb},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31333},
eprint = {2610.31333},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31333},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sketching Outside the Hilbert Space for the Infinite-Dimensional Randomised SVD
AU - Morgan Lynch-Davies
AU - Marcus Webb
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31333
UR - https://arxiv.org/abs/2610.31333
AB - In randomised numerical linear algebra, common choices for distributions of sketch vectors ω∈ ℝ^n are isotropic, so that 𝔼[ωω^T]=I_n. Unfortunately, the natural extension of an isotropic vector in (ℝ^n, |· |_2) to an infinite-dimensional separable Hilbert space 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 ω to be a continuous Gaussian process with trace-class covariance operators, or leave ω implicit by instead sampling Aω as if ω 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.
ER -