PCA++: 均匀性如何在对比学习中诱导对背景噪声的鲁棒性
PCA++: How Uniformity Induces Robustness to Background Noise in Contrastive Learning
AI总结：
针对标准PCA在结构化背景噪声下恢复低维信号的局限，提出带硬均匀性约束的对比PCA方法PCA++，通过广义特征问题有闭式解，在高维强噪声场景稳定，实验优于PCA+和标准PCA，阐明均匀性对鲁棒性的作用。
摘要：
高维数据常包含被结构化背景噪声掩盖的低维信号，这限制了标准PCA的有效性。受对比学习启发，我们解决从正样本对中恢复共享信号子空间的问题，正样本对是共享相同信号但背景不同的配对观测。我们的基线方法PCA+仅使用对齐对比学习，在背景变化温和时成功，但在强噪声或高维场景下失效。为此，我们引入PCA++，一种硬均匀性约束的对比PCA，对投影特征强制身份协方差。PCA++通过广义特征问题有闭式解，在高维中保持稳定，并可证明能正则化背景干扰。我们在固定纵横比和增长尖峰场景下提供精确高维渐近性，展示均匀性在鲁棒信号恢复中的作用。实验上，PCA++在模拟数据、受损MNIST和单细胞转录组学上优于标准PCA和仅对齐的PCA+，可靠恢复条件不变结构。更广泛地说，我们阐明均匀性在对比学习中的作用，表明显式特征分散可抵御结构化噪声并增强鲁棒性。
英文摘要：
High-dimensional data often contain low-dimensional signals obscured by structured background noise, which limits the effectiveness of standard PCA. Motivated by contrastive learning, we address the problem of recovering shared signal subspaces from positive pairs, paired observations sharing the same signal but differing in background. Our baseline, PCA+, uses alignment-only contrastive learning and succeeds when background variation is mild, but fails under strong noise or high-dimensional regimes. To address this, we introduce PCA++, a hard uniformity-constrained contrastive PCA that enforces identity covariance on projected features. PCA++ has a closed-form solution via a generalized eigenproblem, remains stable in high dimensions, and provably regularizes against background interference. We provide exact high-dimensional asymptotics in both fixed-aspect-ratio and growing-spike regimes, showing uniformity's role in robust signal recovery. Empirically, PCA++ outperforms standard PCA and alignment-only PCA+ on simulations, corrupted-MNIST, and single-cell transcriptomics, reliably recovering condition-invariant structure. More broadly, we clarify uniformity's role in contrastive learning, showing that explicit feature dispersion defends against structured noise and enhances robustness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
