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 D(f
₀
∥f
₁
)
^-1
log s where s is the level of sparsity, and D(f
₀
∥f
₁
) 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 D(f
₁
∥f
₀
)
^-1
log n, 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 D(f
₀
∥f
₁
)
^-1
(log s+log log n), a mere additive factor more than the lower bound.