Recent Advances in Positive Semidefinite Matrix Approximation

This talk examines randomized sampling methods for approximating positive semidefinite matrices. Fast leverage-score approximation combined with the Nystrom method yields provably accurate, linear-time algorithms for kernel ridge regression and kernel principal-component analysis, avoiding the usual quadratic-time cost. Related sampling techniques give relative-error low-rank approximations of positive semidefinite matrices in sublinear time without assumptions about incoherence or condition number. The results illustrate how randomized algorithms can exploit positive semidefinite structure beyond the capabilities of traditional methods. The talk concludes with open questions, particularly about oblivious sketches of kernel matrices.