Detecting a Vector Based on Linear Measurements

We consider a situation where the state of a system is represented by a real-valued vector x ∈ Rn. Under normal circumstances, the vector x is zero, while an event manifests as non-zero entries in x, possibly few. Our interest is in designing algorithms that can reliably detect events — i.e., test whether x = 0 or x 6= 0 — with the least amount of information. We place ourselves in a situation, now common in the signal processing literature, where information on x comes in the form of noisy linear measurements y = 〈a,x〉 + z, where a ∈ Rn has norm bounded by 1 and z ∈ N (0, 1). We derive information bounds in an active learning setup and exhibit some simple near-optimal algorithms. In particular, our results show that the task of detection within this setting is at once much easier, simpler and different than the tasks of estimation and support recovery.