A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection

TL;DR

This paper unifies non-asymptotic and asymptotic methods for minimax signal detection in Gaussian models, showing how tools from one approach can inform the other, with applications to inverse problems.

Contribution

It provides a unified framework linking non-asymptotic and asymptotic minimax signal detection approaches, highlighting their interconnections and practical implications.

Findings

Unified treatment of non-asymptotic and asymptotic approaches

Tools from asymptotic analysis can inform non-asymptotic bounds

Application to mildly ill-posed inverse problems

Abstract

We are concerned with minimax signal detection. In this setting, we discuss non-asymptotic and asymptotic approaches through a unified treatment. In particular, we consider a Gaussian sequence model that contains classical models as special cases, such as, direct, well-posed inverse and ill-posed inverse problems. Working with certain ellipsoids in the space of squared-summable sequences of real numbers, with a ball of positive radius removed, we compare the construction of lower and upper bounds for the minimax separation radius (non-asymptotic approach) and the minimax separation rate (asymptotic approach) that have been proposed in the literature. Some additional contributions, bringing into light links between non-asymptotic and asymptotic approaches to minimax signal, are also presented. An example of a mildly ill-posed inverse problem is used for illustrative purposes. In particular, it is shown that tools used to derive `asymptotic' results can be exploited to draw `non-asymptotic' conclusions, and vice-versa.