在噪声和方向偏好下实现鲁棒特征选择，无需严格对称假设。
Universal Feature Selection with Noisy Observations and Weak Symmetry Conditions
- 基于奇异值分解的依赖矩阵框架，容忍第二阶矩偏差。
- 特征选择误差指数渐近最优，受对称偏差δ与噪声η₁,η₂影响。
- 适合存在观测噪声或非对称结构的实际数据任务。
本文放宽了文献[4][5]中严格的对称性假设，将通用特征选择框架扩展至可处理噪声观测及可能具有方向偏好属性结构的情形。引入弱球面对称性概念，通过二阶矩距离量化偏离旋转不变性的程度。在此松弛条件下，我们基于从噪声数据计算的典型依赖矩阵的奇异值分解，构建了通用特征选择框架。主要结果表明，所选特征达到渐近最优误差指数，残差项依赖于对称性偏差δ及噪声水平η₁、η₂。当δ、η₁、η₂较小时，结果恢复文献[5]结论，证明精确球面对称性并非必需。整体表明该选择框架对二阶矩偏差与观测噪声具有鲁棒性，显著拓展其在各类推断任务中的适用性，并为实际场景下的通用特征选择提供了理论依据。
原文摘要 · Abstract (English)
This paper relaxes the restrictive symmetry conditions adopted in [4], [5] and extends their universal feature selection framework to accommodate noisy observations as well as attribute structures that may exhibit directional preferences. We introduce the notion of weak spherical symmetry, quantified by second-moment distances, which allows controlled deviations from rotational invariance. Under this relaxed condition, we develop a universal feature selection framework based on the singular value decomposition of the canonical dependence matrix computed from noisy data. Our main result shows that the selected features achieve asymptotically optimal error exponents up to a residual term that depends on the symmetry deviation $δ$ and the noise levels $η_1, η_2$. When $δ, η_1, η_2$ are relatively small, our result recovers that of [5], thereby demonstrating that exact spherical symmetry is unnecessary. Overall, our findings highlight the robustness of the selection framework against second-moment deviations and observation noise, thereby broadening its applicability across diverse inference tasks and providing a theoretically grounded tool for universal feature selection in practical scenarios.
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