用特征值分解提升关键点嵌入效果，让数据降维更稳定高效。
Neumann eigenmaps for landmark embedding
- 基于子图的诺伊曼拉普拉斯矩阵进行特征分解，融合关键点信息
- 计算效率高，能准确还原反射随机游走的扩散距离
- 天然支持尼尔斯通外推，适合处理重要点被移除的情况
我们提出诺伊曼特征映射（Neumann eigenmaps, NeuMaps），一种利用数据集中关键样本（即地标点）改进标准扩散映射嵌入的新方法。将这些地标点视为原始数据图的子图，通过重构的诺伊曼拉普拉斯矩阵的特征分解得到NeuMaps。结果表明，NeuMaps具有两大优势：(1) 提供计算高效的嵌入，可准确恢复子图上反射随机游走对应的扩散距离；(2) 在扩散映射框架内自然集成尼尔斯通外推，得益于离散诺伊曼边界条件。在数字分类与分子动力学实例中，我们证明NeuMaps不仅优于现有基于地标点的嵌入方法，还显著提升了扩散映射对重要点移除的稳定性。
原文摘要 · Abstract (English)
We present Neumann eigenmaps (NeuMaps), a novel approach for enhancing the standard diffusion map embedding using landmarks, i.e distinguished samples within the dataset. By interpreting these landmarks as a subgraph of the larger data graph, NeuMaps are obtained via the eigendecomposition of a renormalized Neumann Laplacian. We show that NeuMaps offer two key advantages: (1) they provide a computationally efficient embedding that accurately recovers the diffusion distance associated with the reflecting random walk on the subgraph, and (2) they naturally incorporate the Nyström extension within the diffusion map framework through the discrete Neumann boundary condition. Through examples in digit classification and molecular dynamics, we demonstrate that NeuMaps not only improve upon existing landmark-based embedding methods but also enhance the stability of diffusion map embeddings to the removal of highly significant points.
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