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# Sampling-based Nyström Approximation and Kernel Quadrature

*Proceedings of the 40th International Conference on Machine Learning*, PMLR 202:12678-12699, 2023.

#### Abstract

We analyze the Nyström approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nyström approximation with i.i.d. sampling and singular-value decomposition in the continuous regime; the proof techniques are borrowed from statistical learning theory. We further introduce a refined selection of subspaces in Nyström approximation with theoretical guarantees that is applicable to non-i.i.d. landmark points. Finally, we discuss their application to convex kernel quadrature and give novel theoretical guarantees as well as numerical observations.