Cameron Musco (Microsoft Research New England)
In this talk, I will discuss recent advances in sampling methods for positive semidefinite (PSD) matrix approximation. I will start by discussing how fast leverage score approximation schemes can be combined with the popular Nystrom method for PSD kernel matrix approximation. These techniques yield the first provably accurate, linear time approximation algorithms for fundamental kernel methods like kernel ridge regression and kernel PCA, avoiding a traditional quadratic time bottleneck. I will then discuss how the same sampling techniques can be leveraged to give a surprising result: a relative error low-rank approximation to any positive semidefinite matrix can be computed in sublinear time, without any assumptions on incoherence or condition number. This result demonstrates the ability of randomized methods to exploit the structure of PSD matrices and go well beyond what is possible with traditional algorithmic techniques. I will conclude with a discussion of open questions, especially focused on oblivious sketching methods for kernel matrices.