Lower Bounds for Adaptive Sparse Recovery

We give lower bounds for the problem of stable sparse recovery from adaptive linear measurements. In this problem, one would like to estimate a vector x ∈ R from m linear measurements A1x, . . . , Amx. One may choose each vector Ai based on A1x, . . . , Ai−1x, and must output x̂ satisfying ‖x̂− x‖p ≤ (1 + ) min k-sparse x′ ‖x− x‖p with probability at least 1−δ > 2/3, for some p ∈ {1, 2}. For p = 2, it was recently shown that this is possible with m = O( 1 k log log(n/k)), while nonadaptively it requires Θ( 1 k log(n/k)). It is also known that even adaptively, it takes m = Ω(k/ ) for p = 2. For p = 1, there is a non-adaptive upper bound of Õ( 1 √ k log n).