On the Limits of Sequential Testing in High Dimensions

TL;DR

This paper investigates the fundamental limits of sequential testing for support recovery in high-dimensional sparse signals, establishing bounds on measurement efficiency and proposing a simple method that nearly achieves these limits.

Contribution

It derives theoretical bounds on the measurement requirements for sequential and non-sequential procedures and introduces sequential thresholding as an effective support recovery method.

Findings

Sequential methods fail if measurements grow slower than log s / D(f0||f1).

Non-sequential methods fail if measurements grow slower than log n / D(f1||f0).

Sequential thresholding nearly attains the lower bound, requiring measurements proportional to (log s + log log n) / D(f0||f1).

Abstract

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.