Eigenvalues of a diffusion operator reveal the intrinsic geometry of high-dimensional data
Diffusion maps (Coifman & Lafon 2006): Build a Markov chain on the data via kernel K(x,y)=exp(−‖x−y‖²/ε).
The eigenvectors of the transition matrix define diffusion coordinates that capture geodesic (manifold) distances,
not Euclidean ones. Left: original data colored by diffusion coordinate φ₁. Right: data plotted in (φ₁, φ₂) space — the intrinsic geometry is revealed.