# Lower bounds for adaptive sparse recovery

## Abstract

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 ∈ ℝn from m linear measurements A 1x,..., Amx. One may choose each vector Ai based on A1x,..., Ai-1x, and must output x̂ satisfying ∥x̂ - x∥p ≤ (1 + ε) k-sparse x′min ∥x - x′∥p with probability at least 1-δ > 2/3, for somep p ∈ {1, 2}. For p = 2, it was recently shown that this is possible with m = O(1/εk log log(n/k)), while nonadaptively it requires Θ(1/εk log(n/k)). It is also known that even adaptively, it takes m = Ω(k/ε) for p = 2. For p = 1, there is a non-adaptive upper bound of Õ(1/√εk log n). We show: • For p = 2, m = Ω(log log n). This is tight for k = O(1) and constant ε, and shows that the log log n dependence is correct. • If the measurement vectors are chosen in R "rounds", then m = Ω(R log1/R n). For constant ε, this matches the previously known upper bound up to an O(1) factor in R. • For p = 1, m = Ω(k/(√ε·log k/ε)). This shows that adaptivity cannot improve more than logarithmic factors, providing the analogue of the m = Ω(k/ε) bound for p = 2. Copyright © SIAM.