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{\"o}m approximation is usually preferred over the randomized SVD. This paper explores an infinite-dimensional analog of the Nystr{\"o}m 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{\"o}m approximation error in the operator, trace, and Hilbert-Schmidt norms. Numerical experiments for simple integral operators validate the proposed framework.

翻译：随机奇异值分解（SVD）因其高效性和强理论保证，已成为计算矩阵廉价且精确的低秩近似的主流方法。Boullé与Townsend（FoCM, 2023）近期的工作提出了随机SVD的无穷维类比，用于逼近希尔伯特-施密特算子。然而，许多实际应用涉及对称半正定矩阵的低秩近似计算。在此场景下，已有充分证据表明随机Nyström近似通常优于随机SVD。本文探索了Nyström近似的无穷维类比，用于计算非负自伴迹类算子的低秩近似。我们对该方法进行了分析，并在此过程中改进了现有随机SVD的无穷维界。我们的分析给出了算子范数、迹范数和希尔伯特-施密特范数下Nyström近似误差的期望界与尾部界。针对简单积分算子的数值实验验证了所提出框架的有效性。