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.

翻译：在复杂参数空间下，鲁棒统计推断常面临严峻的计算-统计差距。我们研究高斯序列模型中强ε-污染下的极小极大信号检测问题，其中信号属于一般先验约束集K。现有最优检验需计算K的精确Kolmogorov k-宽度，对于一般非平凡集合而言，该任务在计算上不可行。为弥合这一差距，我们提出一种多项式时间检验框架，普适适用于平衡约束、第二类约束及严格二阶凸约束。通过利用半定规划松弛及配备近似次梯度预言的改进椭球算法，我们高效近似了Kolmogorov宽度。值得注意的是，我们所提出的无条件高效算法实现的鲁棒检测边界与现有上界仅相差一个多对数因子。这为无需预先获知信号精确几何复杂度的一类广泛结构化信号，提供了一种计算上可处理的检验方案。