This paper is concerned with signal detection in Gaussian noise under quadratically convex orthosymmetric (QCO) constraints. Specifically the null hypothesis assumes no signal, whereas the alternative considers signal which is separated in Euclidean norm from zero, and belongs to the QCO constraint. Our main result establishes the minimax rate of the separation radius between the null and alternative purely in terms of the geometry of the QCO constraint -- we argue that the Kolmogorov widths of the constraint determine the critical radius. This is similar to the estimation problem with QCO constraints, which was first established by Donoho et al. (1990); however, as expected, the critical separation radius is smaller compared to the minimax optimal estimation rate. Thus signals may be detectable even when they cannot be reliably estimated.

翻译：本文研究高斯噪声中二次凸正交对称约束下的信号检测问题。具体而言，原假设假设无信号存在，而备择假设考虑信号在欧几里得范数下与零点相分离，且属于二次凸正交对称约束。我们建立的主要结果给出了原假设与备择假设之间分离半径的极小极大速率，该速率完全由二次凸正交对称约束的几何特性决定——我们论证了该约束的柯尔莫哥洛夫宽度决定了临界半径。这与Donoho等人(1990)首次建立的二次凸正交对称约束估计问题类似；然而正如预期，临界分离半径小于极小极大最优估计速率。因此，即使信号无法被可靠估计，仍可能被检测到。