Compressed learning of high-dimensional sparse functions

This paper presents a simple randomised algorithm for recovering high-dimensional sparse functions, i.e. functions ƒ : [0, 1]^d → ℝ which depend effectively only on k out of d variables, meaning ƒ(x<inf>1</inf>, …, x<inf>d</inf>) = g(x<inf>i1</inf>, …, x<inf>ik</inf> ), where the indices 1 ≤ i<inf>1</inf> &#60; i<inf>2</inf> &#60; … &#60; i<inf>k</inf> ≤ d are unknown. It is shown that (under certain conditions on g) this algorithm recovers the k unknown coordinates with probability at least 1–6 exp(−L) using only O(k(L+log k)(L+log d)) samples of ƒ.