AI 中文总结
研究非线性高斯PCA中KL基的最优性，通过建立Mallat-Zeitouni猜想保留能量形式的近似版本，证明KL基在一定因子内最优，此无维度比较依赖保留坐标数，随d增长优化优势消失，为算法分析提供合适比较。
AI 中文摘要
主成分分析（PCA）对于高斯数据的线性重构是最优的，这是其在算法和信号处理中核心作用的基础属性。然而，其非线性类似物却非常微妙：2011年，Mallat和Zeitouni推测即使每个样本自适应选择保留坐标，Karhunen-Loève（KL）基仍保持最优，这一属性从理论上证明了PCA后接稀疏阈值处理这一普遍流程的合理性。本文建立了Mallat-Zeitouni猜想保留能量形式的1 + O(1/√d)近似版本，表明KL基在该因子内是最优基。这种无维度比较仅取决于保留坐标的数量，并表明随着d增长，在所有正交基上优化的可能优势消失。它补充了Litvak和Tikhomirov（2018年《应用概率年鉴》）的通用常数重构误差比较，同时提供了适合算法分析的比较。我们的证明基于一个简洁的概念简化：通过Schur-Horn优超将任意旋转松弛到确定性阈值界，并将剩余损失与高斯水平集上秩为d的均匀拟阵的相关间隙相识别。
英文摘要
Principal component analysis (PCA) is optimal for the linear reconstruction of Gaussian data, a foundational property underlying its central role in algorithms and signal processing. Its nonlinear analogue, however, is notoriously subtle: in 2011, Mallat and Zeitouni conjectured that the Karhunen--Loève (KL) basis remains optimal even when the retained coordinates are chosen adaptively per sample, a property that would theoretically justify the ubiquitous pipeline of PCA followed by sparse thresholding. In this paper, we establish a $1+O(1/\sqrt{d})$-approximate version of the retained-energy form of the Mallat--Zeitouni conjecture, showing that the KL basis is within this factor of the optimal basis. This dimension-free comparison depends only on the number of retained coordinates and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows. It complements the universal-constant reconstruction-error comparison of Litvak and Tikhomirov (Ann. Appl. Probab., 2018), while providing a comparison naturally suited for algorithmic analysis. Our proof rests on a clean, conceptual reduction: we relax arbitrary rotations to a deterministic threshold bound via Schur--Horn majorization, and identify the remaining loss with the correlation gap of the rank-$d$ uniform matroid over Gaussian level sets.
