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 particularly large reduction in the number of samples needed for reliable signal support recovery. Starting with a lower bound, we show any sequential sampling procedure fails in the high dimensional limit provided the average number of measurements per dimension is less then D(P0||P1) 1 log s, where s is the level of sparsity and D(P0||P1) the Kullback-Leibler divergence between the underlying distributions. An extension of the Sequential Probability Ratio Test (SPRT) which requires complete knowledge of the underlying distributions is shown to achieve this bound. We introduce a simple procedure termed Sequential Thresholding which can be implemented with limited knowledge of the underlying distributions, and guarantees exact support recovery provided the average number of measurements per dimension grows faster than D(P0||P1) 1 log s, achieving the lower bound. For comparison, we show any non-sequential procedure fails provided the number of measurements grows at a rate less than D(P1||P0) 1 log n, where n is the total dimension of the problem. I
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