# Improved Algorithms for Adaptive Compressed Sensing

In the problem of adaptive compressed sensing, one wants to estimate an approximately k-sparse vector x∈R^n from m linear measurements A_1 x, A_2 x,..., A_m x, where A_i can be chosen based on the outcomes A_1 x,..., A_i-1 x of previous measurements. The goal is to output a vector x̂ for which x-x̂_p < C ·_k-sparse x'x-x'_q with probability at least 2/3, where C > 0 is an approximation factor. Indyk, Price and Woodruff (FOCS'11) gave an algorithm for p=q=2 for C = 1+ϵ with ((k/ϵ) (n/k)) measurements and (^*(k) (n)) rounds of adaptivity. We first improve their bounds, obtaining a scheme with (k · (n/k) +(k/ϵ) ·(1/ϵ)) measurements and (^*(k) (n)) rounds, as well as a scheme with ((k/ϵ) · (n (n/k))) measurements and an optimal ( (n)) rounds. We then provide novel adaptive compressed sensing schemes with improved bounds for (p,p) for every 0 < p < 2. We show that the improvement from O(k (n/k)) measurements to O(k (n/k)) measurements in the adaptive setting can persist with a better ϵ-dependence for other values of p and q. For example, when (p,q) = (1,1), we obtain O(k/√(ϵ)· n ^3 (1/ϵ)) measurements.

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