Identifying the Support of Rectangular Signals in Gaussian Noise

TL;DR

This paper develops a statistically optimal and computationally efficient method for identifying the support of rectangular signals in Gaussian noise, extending to multivariate, multiple, and robust scenarios with polynomially growing signal lengths.

Contribution

It introduces a new boundary for support identification, proves its optimality, and extends the framework to multiple signals and non-Gaussian noise settings.

Findings

Support identification is as statistically hard as detection.

Proposed method is asymptotically optimal and computationally efficient.

Extensions include multiple signals and robustness to noise distribution.

Abstract

We consider the problem of identifying the support of the block signal in a sequence when both the length and the location of the block signal are unknown. The multivariate version of this problem is also considered, in which we try to identify the support of the rectangular signal in the hyper- rectangle. We allow the length of the block signal to grow polynomially with the length of the sequence, which greatly generalizes the previous results in [16]. A statistical boundary above which the identification is possible is presented and an asymptotically optimal and computationally efficient procedure is proposed under Gaussian white noise in both the univariate and multivariate settings. The problem of block signal identification is shown to have the same statistical difficulty as the corresponding problem of detection in both the univariate and multivariate cases, in the sense that whenever we can detect the signal, we can identify the support of the signal. Some generalizations are also considered here: (1) We ex- tend our theory to the case of multiple block signals. (2) We also discuss about the robust identification problem when the noise distribution is un- specified and the block signal identification problem under the exponential family setting.