Randomized Least-Squares with Minimal Oversampling and Interpolation in General Spaces

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1515-1538, August 2024.
Abstract. In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected [math] sense, as soon as the sample size [math] is larger than the dimension [math] of the approximation space by a constant multiplicative ratio. On the other hand, for [math], we obtain an interpolation strategy with a stability factor of order [math]. The proposed sampling algorithms are greedy procedures based on [Batson, Spielman, and Srivastava, Twice-Ramanujan sparsifiers, in Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, 2009, pp. 255–262] and [Lee and Sun, SIAM J. Comput., 47 (2018), pp. 2315–2336], with polynomial computational complexity.