Adaptive sensing performance lower bounds for sparse signal detection and support estimation

TL;DR

This paper establishes fundamental limits for adaptive sensing in detecting and estimating sparse signals, showing that certain signal magnitude thresholds are necessary for reliable detection and support recovery, independent of ambient dimension.

Contribution

It provides precise lower bounds on signal magnitude for detection and support estimation in adaptive sensing, confirming the near-optimality of existing methods.

Findings

Detection requires non-zero components to exceed √(2/s) in magnitude.

Support can be exactly identified if non-zero components exceed √(2 log s).

Results are independent of the ambient signal dimension n.

Abstract

This paper gives a precise characterization of the fundamental limits of adaptive sensing for diverse estimation and testing problems concerning sparse signals. We consider in particular the setting introduced in (IEEE Trans. Inform. Theory 57 (2011) 6222-6235) and show necessary conditions on the minimum signal magnitude for both detection and estimation: if ${\mathbf {x}}\in \mathbb{R}^n$ is a sparse vector with $s$ non-zero components then it can be reliably detected in noise provided the magnitude of the non-zero components exceeds $\sqrt{2/s}$. Furthermore, the signal support can be exactly identified provided the minimum magnitude exceeds $\sqrt{2\log s}$. Notably there is no dependence on $n$, the extrinsic signal dimension. These results show that the adaptive sensing methodologies proposed previously in the literature are essentially optimal, and cannot be substantially improved. In addition, these results provide further insights on the limits of adaptive compressive sensing.