Detecting a Vector Based on Linear Measurements

TL;DR

This paper develops algorithms for reliably detecting non-zero vectors from noisy linear measurements, demonstrating that detection is simpler and more efficient than estimation or support recovery in such systems.

Contribution

The paper introduces near-optimal algorithms for event detection in linear measurement systems and establishes fundamental information bounds for this task.

Findings

Detection is easier and simpler than estimation.

Proposed algorithms are near-optimal in performance.

Established fundamental information bounds for detection.

Abstract

We consider a situation where the state of a system is represented by a real-valued vector. Under normal circumstances, the vector is zero, while an event manifests as non-zero entries in this vector, possibly few. Our interest is in the design of algorithms that can reliably detect events (i.e., test whether the vector is zero or not) with the least amount of information. We place ourselves in a situation, now common in the signal processing literature, where information about the vector comes in the form of noisy linear measurements. 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.