Diffusion Maps is not Dimensionality Reduction
This work addresses a misconception in the machine learning community about the role of diffusion maps, which is incremental but clarifies foundational concepts in manifold learning.
The paper clarifies that diffusion maps (DMAP) are not a dimensionality-reduction tool but rather a spectral representation of intrinsic geometry, and it demonstrates this by comparing DMAP, Isomap, and UMAP on a Swiss roll dataset, showing that DMAP requires combining multiple diffusion modes to accurately recover the low-dimensional chart.
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.