Improved Fixed-Rank Nystr\"om Approximation via QR Decomposition: Practical and Theoretical Aspects

The Nystr\"om 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 Nystr\"om method uses a sub-optimal procedure for rank reduction mainly due to its simplicity. In this paper, we highlight the drawbacks of standard Nystr\"om in terms of poor performance and lack of theoretical guarantees. To address these issues, we present an efficient method for generating improved fixed-rank Nystr\"om approximations. Theoretical analysis and numerical experiments are provided to demonstrate the advantages of the modified method over the standard Nystr\"om 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.