Sequential Testing for Sparse Recovery

Abstract: This paper studies sequential methods for recovery of sparse signals in high dimensions. When compared to fixed sample size procedures, in the sparse setting, sequential methods can result in a large reduction in the number of samples needed for reliable signal support recovery. Starting with a lower bound, we show any coordinate-wise sequential sampling procedure fails in the high dimensional limit provided the average number of measurements per dimension is less then log s/D(P_0||P_1) where s is the level of sparsity and D(P_0||P_1) the Kullback-Leibler divergence between the underlying distributions. A series of Sequential Probability Ratio Tests (SPRT) which require complete knowledge of the underlying distributions is shown to achieve this bound. Motivated by real world experiments and recent work in adaptive sensing, we introduce a simple procedure termed Sequential Thresholding which can be implemented when the underlying testing problem satisfies a monotone likelihood ratio assumption. Sequential Thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than log s/ D(P_0||P_1), achieving the lower bound. For comparison, we show any non-sequential procedure fails provided the number of measurements grows at a rate less than log n/D(P_1||P_0), where n is the total dimension of the problem.