The randomized singular value decomposition (SVD) has become a popular approach to computing cheap, yet accurate, low-rank approximations to matrices due to its efficiency and strong theoretical guarantees. Recent work by Boull\'e and Townsend (FoCM, 2023) presents an infinite-dimensional analog of the randomized SVD to approximate Hilbert-Schmidt operators. However, many applications involve computing low-rank approximations to symmetric positive semi-definite matrices. In this setting, it is well-established that the randomized Nystr\"om approximation is usually preferred over the randomized SVD. This paper explores an infinite-dimensional analog of the Nystr\"om approximation to compute low-rank approximations to non-negative self-adjoint trace-class operators. We present an analysis of the method and, along the way, improve the existing infinite-dimensional bounds for the randomized SVD. Our analysis yields bounds on the expected value and tail bounds for the Nystr\"om approximation error in the operator, trace, and Hilbert-Schmidt norms. Numerical experiments on integral operators arising from Gaussian process sampling and Bayesian inverse problems are used to validate the proposed infinite-dimensional Nystr\"om algorithm.

翻译：随机化奇异值分解（SVD）因其高效性和强大的理论保证，已成为计算廉价且精确的低秩矩阵逼近的流行方法。Boullé 和 Townsend（FoCM, 2023）最近的工作提出了随机化 SVD 的无限维类比，用于逼近 Hilbert-Schmidt 算子。然而，许多应用涉及计算对称半正定矩阵的低秩逼近。在此背景下，公认的是随机化 Nyström 逼近通常优于随机化 SVD。本文探讨了 Nyström 逼近的无限维类比，用于计算非负自伴迹类算子的低秩逼近。我们对该方法进行了分析，并在此过程中改进了现有的随机化 SVD 无限维误差界。我们的分析给出了 Nyström 逼近误差在算子范数、迹范数和 Hilbert-Schmidt 范数下的期望值界和尾部界。通过高斯过程采样和贝叶斯反问题中产生的积分算子进行数值实验，以验证所提出的无限维 Nyström 算法。