Electronic Journal of Statistics

Detecting a vector based on linear measurements

Ery Arias-Castro

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We consider a situation where the state of a system is represented by a real-valued vector xn. 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 x0 — 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 an has norm bounded by 1 and $z\in \mathcal{N}(0,1)$. 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.

Article information

Electron. J. Statist., Volume 6 (2012), 547-558.

First available in Project Euclid: 10 April 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C20: Minimax procedures 62G10: Hypothesis testing 62H15: Hypothesis testing

Signal detection compressed sensing adaptive measurements normal mean model sparsity high-dimensional data


Arias-Castro, Ery. Detecting a vector based on linear measurements. Electron. J. Statist. 6 (2012), 547--558. doi:10.1214/12-EJS686. https://projecteuclid.org/euclid.ejs/1334065321

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