Improved fixed-rank Nyström approximation via QR decomposition: Practical and theoretical aspects

Abstract

The Nyström method is a popular technique that uses a small number of landmark points to compute a fixed-rank approximation of large kernel matrices that arise in machine learning problems. In practice, to ensure high quality approximations, the number of landmark points is chosen to be greater than the target rank. However, for simplicity the standard Nyström method uses a sub-optimal procedure for rank reduction. In this paper, we examine the drawbacks of the standard Nyström method in terms of poor performance and lack of theoretical guarantees. To address these issues, we present an efficient modification 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, has greatly improved accuracy in many cases, and is optimal in a sense that we make precise.