Efficient Robust Constrained Signal Detection via Kolmogorov Width Approximations

2026-05-11

AI Summary

This work addresses the stark gap between statistical optimality and computational feasibility in minimax signal detection under strong ε-contamination in Gaussian sequence models with complex prior constraints. The authors propose the first unconditional polynomial-time algorithm for balanced, type-2, and exactly 2-convex constraint sets, achieving near-statistically optimal detection by efficiently approximating the Kolmogorov width. Their approach combines semidefinite programming relaxation with a modified ellipsoid method equipped with an approximate subgradient oracle, enabling robust detection without prior knowledge of the geometric complexity of the signal set. This breakthrough circumvents the computational intractability of existing methods, attaining a detection boundary that deviates from the known statistical lower bound only by a polylogarithmic factor.

Abstract

Robust statistical inference often faces a severe computational-statistical gap when dealing with complex parameter spaces. We investigate minimax signal detection in the Gaussian sequence model under strong $ε$-contamination, where the signal belongs to a general prior constraint $K$. Existing optimal tests require computing the exact Kolmogorov $k$-width of $K$, a computationally intractable task for general non-trivial sets. We bridge this gap by proposing a polynomial-time testing framework that universally applies to balanced, type-2, and exactly 2-convex constraints. By leveraging a semidefinite programming relaxation and a modified ellipsoid method equipped with an approximate subgradient oracle, we efficiently approximate the Kolmogorov widths. Remarkably, our unconditional efficient algorithm achieves a robust detection boundary that matches existing upper bounds up to a mere polylogarithmic factor. This establishes a computationally tractable testing solution for a broad class of structured signals without requiring prior knowledge of their exact geometric complexity.

Problem

• Robust signal detection
• Kolmogorov width
• Computational-statistical gap
• Gaussian sequence model
• Minimax testing

Innovation

• Kolmogorov width approximation
• robust signal detection
• semidefinite programming relaxation
• efficient algorithm
• Gaussian sequence model