Efficient Robust Constrained Signal Detection via Kolmogorov Width Approximations

TL;DR

This paper introduces a polynomial-time testing framework for robust signal detection in Gaussian models, efficiently approximating Kolmogorov widths for complex sets, and achieving near-optimal detection boundaries.

Contribution

It develops a computationally efficient method using semidefinite programming to approximate Kolmogorov widths, enabling robust detection for broad structured signal classes.

Findings

Achieves detection boundaries matching upper bounds up to polylog factors.

Provides a universal polynomial-time testing framework for various convex constraints.

Bridges the computational-statistical gap in robust signal detection.

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 $\epsilon$-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.