Minimax Signal Detection Under Weak Noise Assumptions

TL;DR

This paper establishes minimax signal detection bounds in a sequence model with weak noise assumptions, showing that the detection rates match classical Gaussian noise results even under minimal noise conditions.

Contribution

It provides non-asymptotic bounds for minimax separation radius under very weak noise assumptions without Gaussianity or independence.

Findings

Minimax separation rates are comparable to classical Gaussian noise models.

Results hold under bounded fourth moments, with no independence or Gaussianity assumptions.

Classical rates are recovered under additional noise conditions in inverse problems.

Abstract

We consider minimax signal detection in the sequence model. Working with certain ellipsoids in the space of square-summable sequences of real numbers, with a ball of positive radius removed, we obtain upper and lower bounds for the minimax separation radius in the non-asymptotic framework, i.e., for a fixed value of the involved noise level. We use very weak assumptions on the noise (i.e., fourth moments are assumed to be uniformly bounded). In particular, we do not use any kind of Gaussianity or independence assumption on the noise. It is shown that the established minimax separation rates are not faster than the ones obtained in the classical sequence model (i.e., independent standard Gaussian noise) but, surprisingly, are of the same order as the minimax estimation rates in the classical setting. Under an additional condition on the noise, the classical minimax separation rates are also retrieved in benchmark well-posed and ill-posed inverse problems.