Improved Fixed-Rank Nyström Approximation via QR Decomposition: Practical and Theoretical Aspects

The Nyström method is a popular technique for computing fixed-rank approximations of large kernel matrices using a small number of landmark points. In practice, to ensure high quality approximations, the number of landmark points is chosen to be greater than the target rank. However, the standard Nyström method uses a sub-optimal procedure for rank reduction mainly due to its simplicity. In this paper, we highlight the drawbacks of standard Nyström in terms of poor performance and lack of theoretical guarantees. To address these issues, we present an efficient method for generating improved fixed-rank Nyström approximations. Theoretical analysis and numerical experiments are provided to demonstrate the advantages of the modified method over the standard Nyström method. Overall, the aim of this paper is to convince researchers to use the modified method, as it has nearly identical computational complexity, is easy to code, and has greatly improved accuracy in many cases.