On the Limits of Sequential Testing in High Dimensions

This paper presents results pertaining to sequential methods for support recovery of sparse signals in noise. Specifically, we show that any sequential measurement procedure fails provided the average number of measurements per dimension grows slower then log s / D(f0||f1) where s is the level of sparsity, and D(f0||f1) the Kullback-Leibler divergence between the underlying distributions. For comparison, we show any non-sequential procedure fails provided the number of measurements grows at a rate less than log n / D(f1||f0), where n is the total dimension of the problem. Lastly, we show that a simple procedure termed sequential thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than (log s + log log n) / D(f0||f1), a mere additive factor more than the lower bound.

Sequential Analysis in High Dimensional Multiple Testing and Sparse Recovery

This paper studies the problem of high-dimensional multiple testing and sparse recovery from the perspective of sequential analysis. In this setting, the probability of error is a function of the dimension of the problem. A simple sequential testing procedure is proposed. We derive necessary conditions for reliable recovery in the non-sequential setting and contrast them with sufficient conditions for reliable recovery using the proposed sequential testing procedure. Applications of the main results to several commonly encountered models show that sequential testing can be exponentially more sensitive to the difference between the null and alternative distributions (in terms of the dependence on dimension), implying that subtle cases can be much more reliably determined using sequential methods.