# A generalization of the randomized singular value decomposition

arXiv: https://arxiv.org/abs/2105.13052 · HTML: https://mlsanity.com/paper/2105.13052/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 44 references that are in the map.
- First posted on arXiv 27 May 2021.

## Builds on (most cited first, 5 of 5)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [On the folded normal distribution](https://mlsanity.com/paper/1402.3559/) (2014)
- [Learning elliptic partial differential equations with randomized linear algebra](https://mlsanity.com/paper/2102.00491/) (2021)

## Built on by (most cited first, 6 of 6)

- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [Elliptic PDE learning is provably data-efficient](https://mlsanity.com/paper/2302.12888/) (2023)
- [Operator learning without the adjoint](https://mlsanity.com/paper/2401.17739/) (2024)
- [chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data](https://mlsanity.com/paper/2501.18715/) (2025)
- [A unified error analysis for randomized low-rank approximation with application to data assimilation](https://mlsanity.com/paper/2405.04811/) (2024)

## Abstract (arXiv)

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.