Asymptotic error bounds for kernel-based Nyström low-rank approximation matrices

• Many kernel-based learning algorithms have the computational load.
• The Nyström low-rank approximation is designed for reducing the computation.
• We propose the spectrum decomposition condition with a theoretical justification.
• Asymptotic error bounds on eigenvalues and eigenvectors are derived.
• Numerical experiments are provided for covariance kernel and Wishart matrix.

Many kernel-based learning algorithms have the computational load scaled with the sample size nn due to the column size of a full kernel Gram matrix K. This article considers the Nyström low-rank approximation. It uses a reduced kernel K̂, which is n×mn×m, consisting of mm columns (say columns i1,i2,⋯,imi1,i2,⋯,im) randomly drawn from K. This approximation takes the form K≈K̂U−1K̂T, where U is the reduced m×mm×m matrix formed by rows i1,i2,⋯,imi1,i2,⋯,im of K̂. Often mm is much smaller than the sample size nn resulting in a thin rectangular reduced kernel, and it leads to learning algorithms scaled with the column size mm. The quality of matrix approximations can be assessed by the closeness of their eigenvalues and eigenvectors. In this article, asymptotic error bounds on eigenvalues and eigenvectors are derived for the Nyström low-rank approximation matrix.