扩散映射非降维，而是揭示流形内在几何结构。
Diffusion Maps is not Dimensionality Reduction
- 通过多模式扩散坐标组合，更准确还原流形真实坐标
- 在瑞士卷数据上，标准扩散映射重建误差最高
- 适合研究流形几何本质的研究者阅读
扩散映射（DMAP）常被当作降维工具使用，但其本质是提供内在几何的谱表示，而非完整的坐标映射方法。为阐明这一区别，我们在已知等距坐标的瑞士卷数据上，比较了DMAP、Isomap与UMAP在不同潜在维度下的表现。对每种表示，均用最优仿射解码器拟合真实坐标并测量重建误差：Isomap最高效地恢复了低维坐标，UMAP表现居中，而标准DMAP仅在结合多个扩散模式后才趋于准确。因此，真实坐标位于扩散坐标张成的空间内，但标准DMAP本身无法自动识别正确的组合。
原文摘要 · Abstract (English)
Diffusion maps (DMAP) are often used as a dimensionality-reduction tool, but more precisely they provide a spectral representation of the intrinsic geometry rather than a complete charting method. To illustrate this distinction, we study a Swiss roll with known isometric coordinates and compare DMAP, Isomap, and UMAP across latent dimensions. For each representation, we fit an oracle affine readout to the ground-truth chart and measure reconstruction error. Isomap most efficiently recovers the low-dimensional chart, UMAP provides an intermediate tradeoff, and DMAP becomes accurate only after combining multiple diffusion modes. Thus the correct chart lies in the span of diffusion coordinates, but standard DMAP do not by themselves identify the appropriate combination.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。