Reviews: Recursive Sampling for the Nystrom Method

The authors provide an algorithm which learns a provably accurate low-rank approximation to a PSD matrix in sublinear time. Specifically, it learns an approximation that has low additive error with high probability by sampling columns from the matrix according to a certain importance measure, then forming a Nystrom approximation using these kernels. The importance measure used is an estimate of the ridge leverage scores, which are expensive to compute exactly ( O(n 3)). Their algorithm recursively estimates these leverage score by starting from a set of columns, using those to estimate the leverage scores, sampling a set of columns according to those probabilities, and repeating ... the authors show that when this process is done carefully, the leverage score estimates are accurate enough that they can be used to get almost as good an approximation as using the true ridge leverage scores. The cost of producing the final approximation is O(ns 2) computation time and O(ns) computations of entries in the PSD matrix.