Robust Signal Detection with Quadratically Convex Orthosymmetric Constraints

TL;DR

This paper develops a minimax framework for robust signal detection under quadratically convex orthosymmetric constraints in Gaussian noise, revealing phase transitions and providing computationally feasible algorithms that match theoretical limits.

Contribution

It introduces a new analysis linking Kolmogorov widths to detection limits and proposes a polynomial-time algorithm for robust detection under QCO constraints, extending to  norms.

Findings

Minimax separation radius depends on constraint geometry, sample size, noise, and corruption.

Phase transitions occur with respect to corruption rate, affecting detection difficulty.

A polynomial-time algorithm nearly achieves the minimax lower bound, handling arbitrary Euclidean lengths.

Abstract

This paper studies the problem of robust signal detection in Gaussian noise under quadratically convex orthosymmetric (QCO) constraints. We consider a minimax testing framework where the signal belongs to a QCO set and is separated from zero in Euclidean norm, while an adversary is allowed to arbitrarily corrupt a fraction $\epsilon $ of the samples. We establish the minimax separation radius between the null and alternative purely in terms of the constraint geometry, sample size, corruption rate, and noise scale. Our analysis argues that the Kolmogorov widths of the constraint set play a central role in determining the detection limits, paralleling to classic results in estimation problem. The derived lower bounds exhibit phase transitions with respect to the corruption rate and confirm that robust testing is statistically easier than robust estimation. While the information-theoretic upper bound is achieved by a computationally intractable test, we develop a polynomial-time algorithm that achieves the minimax lower bound up to logarithmic factors. Unlike prior work, our algorithm handles signals of arbitrary Euclidean length while respecting the QCO constraints. Finally, we extend these results to the robust $\ell _{p}$ norm testing for $1 \le p < 2$.