ALGORITHMS FOR APPROXIMATE SPARSE REGRESSION AND NEAREST INDUCED HULLS

The sparse affine regression problem can be cast as follows: Given a set S of n k points of S that is nearest to y. We describe an Od,epsilon(nk-1 logd-k +1 n)-time randomized (1 + epsilon)-approximation algorithm for this problem with d and epsilon constant. This is the first algorithm for this problem running in time o(nk). Its running time is similar to the query time of a data structure recently proposed by Har-Peled, Indyk, and Mahabadi (ICALP'18), while not requiring any preprocessing. Up to polylogarithmic factors, it matches a conditional lower bound relying on a conjecture about affine degeneracy testing. In the special case where k = d = O(1), we also provide a simple O(nd-1+delta)-time deterministic exact algorithm, for any delta > 0. Finally, we also show how to adapt the approximation algorithm for the sparse linear or convex regression problems with the same running time, up to polylogarithmic factors.