Diffusion Maps is not Dimensionality Reduction

arXiv cs.LG / 3/31/2026

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Key Points

The paper argues that diffusion maps (DMAP) are frequently mischaracterized as dimensionality reduction, but they more accurately yield a spectral representation of intrinsic geometry rather than a full coordinate charting method.
Using a Swiss roll with known isometric coordinates, the authors compare DMAP to Isomap and UMAP by training an “oracle” affine readout and evaluating reconstruction error.
Results show Isomap most efficiently recovers the correct low-dimensional chart, UMAP offers an intermediate accuracy–tradeoff, and DMAP only becomes accurate after combining multiple diffusion modes.
The study concludes that the true chart can lie in the span of diffusion coordinates, but standard DMAP outputs do not, by themselves, determine the correct linear combination to recover coordinates effectively.

Abstract

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.