Speaker: David Bindel (Cornell)

Title: Scalable kernel methods

Abstract: Kernel methods are used throughout statistical modeling, data science, and approximation theory. Depending on the community, they may be introduced in many different ways: through dot products of feature maps, through data-adapted basis functions in an interpolation space, through the natural structure of a reproducing kernel Hilbert space, or through the covariance structure of a Gaussian process. We describe these various interpretations and their relation to each other, and then turn to the key computational bottleneck for all kernel methods: the solution of linear systems and the computation of (log) determinants for dense matrices whose size scales with the number of examples. Recent developments in linear algebra make it increasingly feasible to solve these problems efficiently even with millions of data points. We discuss some of these techniques, including rank-structured factorization, structured kernel interpolation, and stochastic estimators for determinants and their derivatives. We also give a perspective on some open problems and on approaches to addressing the constant challenge posed by the curse of dimensionality.