Diffusion maps reveal intrinsic geometry of high-dimensional data
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Diffusion maps (Coifman & Lafon 2006): Build a kernel matrix K(x,y)=exp(−‖x−y‖²/σ²), normalize to form a Markov chain P. Eigenvectors of P give coordinates that respect the geometry of the data manifold.
The embedding preserves diffusion distance — points connected by many short random-walk paths are close, regardless of ambient space geometry.
Left: original 3D data. Right: diffusion map (ψ₂, ψ₃) colored by intrinsic parameter.