## Bernoulli

• Bernoulli
• Volume 20, Number 4 (2014), 2217-2246.

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

Rui M. Castro

#### 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.

#### Article information

Source
Bernoulli, Volume 20, Number 4 (2014), 2217-2246.

Dates
First available in Project Euclid: 19 September 2014

https://projecteuclid.org/euclid.bj/1411134458

Digital Object Identifier
doi:10.3150/13-BEJ555

Mathematical Reviews number (MathSciNet)
MR3263103

Zentralblatt MATH identifier
1357.94030

#### Citation

Castro, Rui M. Adaptive sensing performance lower bounds for sparse signal detection and support estimation. Bernoulli 20 (2014), no. 4, 2217--2246. doi:10.3150/13-BEJ555. https://projecteuclid.org/euclid.bj/1411134458

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