提出一种可解析求解的非线性降维框架，兼顾灵活性与可解释性。
A Variational Manifold Embedding Framework for Nonlinear Dimensionality Reduction
- 将降维问题建模为变分最优流形嵌入，支持非线性结构建模。
- 解满足偏微分方程，能反映目标函数对称性，提升可解释性。
- 特殊情形精确恢复PCA，统一了线性与非线性方法的理论基础。
主成分分析（PCA）等降维算法在机器学习和神经科学中广泛应用，但各有局限。基于PCA的变体虽简单易懂，却难以捕捉非线性数据流形结构；而更灵活的方法如自编码器通常难以解释，图嵌入方法则可能产生病态的几何失真。为此，我们提出一种变分框架，将降维算法视为最优流形嵌入问题的解。该框架天然支持非线性嵌入，使解比PCA更具灵活性。更重要的是，其变分性质带来良好可解释性：每个解均满足一组偏微分方程，并反映嵌入目标函数的对称性。我们详细讨论这些特性，并证明在某些情况下解可解析表征。有趣的是，一个特例恰好恢复标准PCA。
原文摘要 · Abstract (English)
Dimensionality reduction algorithms like principal component analysis (PCA) are workhorses of machine learning and neuroscience, but each has well-known limitations. Variants of PCA are simple and interpretable, but not flexible enough to capture nonlinear data manifold structure. More flexible approaches have other problems: autoencoders are generally difficult to interpret, and graph-embedding-based methods can produce pathological distortions in manifold geometry. Motivated by these shortcomings, we propose a variational framework that casts dimensionality reduction algorithms as solutions to an optimal manifold embedding problem. By construction, this framework permits nonlinear embeddings, allowing its solutions to be more flexible than PCA. Moreover, the variational nature of the framework has useful consequences for interpretability: each solution satisfies a set of partial differential equations, and can be shown to reflect symmetries of the embedding objective. We discuss these features in detail and show that solutions can be analytically characterized in some cases. Interestingly, one special case exactly recovers PCA.
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