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 vectorx for which 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 + with O((k/ )loglog(n/k)) measurements and O(log * (k)loglog(n)) rounds of adaptivity. We first improve their bounds, obtaining a scheme with ) measurements and O(log * (k)loglog(n)) rounds, as well as a scheme with O((k/ ) · loglog(n log(n/k))) measurements and an optimal O(loglog(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 log(n/k)) measurements to O(k log log(n/k)) measurements in the adaptive setting can persist with a better -dependence for other values of p and q. For example, when (p, q) = (1, 1), we obtain O( k √ · log log n log 3 ( 1 )) measurements. We obtain nearly matching lower bounds, showing our algorithms are close to optimal. Along the way, we also obtain the first nearly-optimal bounds for (p, p) schemes for every 0 < p < 2 even in the non-adaptive setting.
This was improved by REF in two incomparable ways: either R can be improved to O(log log n) or the sample complexity can be improved to O( log log 1 k + k log log n), splitting n and in the sample complexity.
Improved Algorithms for Adaptive Compressed Sensing
ArXiv
ArXiv
Mathematics Computer Science