Robust Signal Detection with Quadratically Convex Orthosymmetric Constraints

TL;DR

The paper advances robust signal detection in Gaussian noise under quadratically convex orthosymmetric (QCO) constraints by deriving minimax characterizations for the separation radius $\rho_{critical}$ in terms of the constraint geometry (via Kolmogorov widths), contamination level $\varepsilon$, and noise level $\sigma$. It develops three lower bounds (geometric, differential-privacy-based, and mixture/estimation-based) and two upper bounds (an exponential-time and a polynomial-time algorithm) that nearly close the minimax gap, and extends the results to $\ell_p$ norms for $1\le p<2$. Through experiments using ellipsoidal/hyperrectangular constraints and two adversarial strategies, the work demonstrates phase transitions in $\varepsilon$ and shows the polynomial-time detector nearly attains minimax performance across regimes. The results provide both theoretical insight and practical procedures for robust testing under complex prior information and adversarial data contamination, with implications for high-dimensional hypothesis testing and robust statistics.

Abstract

This paper is concerned with robust signal detection in Gaussian noise under quadratically convex orthosymmetric (QCO) constraints. Specifically, the null hypothesis $H _{0}$ assumes no signal, whereas the alternative $H _{1}$ considers signal that is separated in Euclidean norm from zero and belongs to a set $K$ satisfying the QCO constraints. In addition, an adversary is allowed to inspect all the $N$ samples and replace up to $εN$ of them with arbitrary values, where $0 < ε< c_0 < \frac{1}{2}$ is the corruption rate. Our main results establish the minimax rate of the separation radius $ρ_{\text{critical}}$ between $H _{0}$ and $H _{1}$ purely in terms of the geometry of $K$, the corruption rate $ε$ (up to logarithmic factors in $\frac{1}{ε}$) and the scale of the noise $σ$. We argue that the Kolmogorov widths of the constraint play a central role in determining the minimax rate. This indicates similarity with the (uncorrupted) estimation problem under QCO constraints, which was first established by Donoho et al. (1990). Moreover, the minimax lower bound reveals interesting phase transitions of the testing problem regarding $ε$. Consistent with classic belief about testing and estimation, the testing problem is ``easier'' even when one compares the results with recent papers studying the constrained robust estimation problem. In addition to the main results above, where the upper bound is achieved with an intractable algorithm, inspired by Canonne et al. (2023), we develop a polynomial time algorithm which also nearly (up to logarithmic factors) achieves the minimax lower bound. In contrast to Canonne et al. (2023), our algorithm works for signals of arbitrary Euclidean length, and respects the QCO constraint. Finally, all the results above are naturally extended to the $\ell _{p}$ norm testing problem for $1 \le p<2$.

Figures (7)

• Figure 1: Testing $H_0: \mu = 0$ vs $H_1: \|\mu\|_2 \ge \rho, \mu \in K$ for the axis aligned ellipsoid (which is a special case of a QCO set) $K$ based on a single observation $X$. Upon applying a rotation to the data, one can use the test we develop in Section \ref{['subsection: upper bounds']} to solve the problems optimally for rotated (non-axis aligned) ellipsoids or more generally QCO sets (which are still centered around the origin). See our first remark in the discussion section for more details.
• Figure 2: An example of an ellipse and its square-set in $\mathbb{R}^{2}$. The square-set is a triangle in $\mathbb{R}^{2}$, which is convex.
• Figure 3: plot of rate of correct reject versus $\ell _{2}$ norm of the mean under $H _{1}$ and classic setting with $N=200$.
• Figure 4: plot of rate of correct reject versus $\ell _{2}$ norm of the mean under $H _{1}$ and classic setting with $N=2000$.
• Figure 5: plot of rate of correct reject versus $\ell _{2}$ norm of the mean under $H _{1}$ and high-dimensional setting with $N=100$.

Theorems & Definitions (71)

• Definition 2.1: Quadratically convex orthosymmetric (QCO) set
• Remark
• Definition 2.2: Kolmogorov $k$-width
• Remark
• Definition 2.3: First optimal dimension and projection
• Definition 2.4: Second optimal dimension and projection
• Example 2.5: $\mathbb{R}^{d}$
• Example 2.6: Hyperrectangles in $\mathbb{R}^{d}$
• Example 2.7: Ellipses in $\mathbb{R}^{d}$
• Remark