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We study a relaxed version of the column-sampling problem for the Nystr\"omapproximation of kernel matrices, where approximations are defined frommultisets of landmark points in the ambient space; such multisets are referredto as Nystr\"om samples. We consider an unweighted variation of the radialsquared-kernel discrepancy (SKD) criterion as a surrogate for the classicalcriteria used to assess the Nystr\"om approximation accuracy; in this setting,we discuss how Nystr\"om samples can be efficiently optimised throughstochastic gradient descent. We perform numerical experiments which demonstratethat the local minimisation of the radial SKD yields Nystr\"om samples withimproved Nystr\"om approximation accuracy.

Local optimisation of Nystr\"om samples through stochastic gradient descent