Adaptive sparse recovery with limited adaptivity

Abstract

The goal of adaptive sparse recovery is to estimate an approximately sparse vector x from a series of linear measurements A1x, A2x, . . ., ARx, where each matrix Aimay depend on the previous observations. With an unlimited number of rounds R, it is known that O(klog log n) measurements suffice for O(1)-approximate k-sparse recovery in Rn, and that Ω(k + log log n) measurements are necessary. We initiate the study of what happens with a constant number of rounds of adaptivity. Previous techniques could not give nontrivial bounds using less than 5 rounds of adaptivity, and were inefficient for any constant R. We give nearly matching upper and lower bounds for any constant number of rounds R. Our lower bound shows that Ω(k(log nk)1/R) measurements are necessary for any k < 2(log n k )1/R; significantly, this is the first lower bound that combines k and n in an adaptive setting. Our upper bound shows that O(k(log nk)1/R•log*k) measurements suffice. The O(log*k) gap between the two bounds comes from a similar gap for nonadaptive sparse recovery in the high-SNR regime, and would be reduced to constant factors with improvements to nonadaptive high-SNR sparse recovery.