Minimax detection of localized signals in statistical inverse problems

TL;DR

This paper studies the fundamental limits of detecting small localized signals in inverse problems using minimax testing, providing bounds and characterizations of detection thresholds in noisy Gaussian models.

Contribution

It introduces new minimax bounds and detection thresholds for local signals in inverse problems, with applications to classical inverse tasks.

Findings

Derived upper and lower bounds for signal detection in Gaussian noise

Characterized asymptotic minimax detection boundaries in certain inverse problems

Applied theoretical results to practical inverse problems like deconvolution and Radon transform inversion

Abstract

We investigate minimax testing for detecting local signals or linear combinations of such signals when only indirect data is available. Naturally, in the presence of noise, signals that are too small cannot be reliably detected. In a Gaussian white noise model, we discuss upper and lower bounds for the minimal size of the signal such that testing with small error probabilities is possible. In certain situations we are able to characterize the asymptotic minimax detection boundary. Our results are applied to inverse problems such as numerical differentiation, deconvolution and the inversion of the Radon transform.