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LIT-1250: Diffusion maps, spectral clustering and reaction coordinates of dynamical systems

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StatusActive — read 2026-10-20 (NOTE-986) in full, from arXiv:math/0503445v1; worth reading for the bridge from a diffusion map to the dynamics behind the data. For samples from p = e^{−U}, the ordinary normalized graph Laplacian (α = 0) tends to a backward Fokker–Planck generator with potential 2U, and the anisotropic normalization α = 1/2 to the generator of the Langevin dynamics with potential U itself, whose slow eigenfunctions the paper proposes as reaction coordinates. It also has worked examples: Ornstein–Uhlenbeck (the finite-ε spectrum λ_k = (τ/(τ+ε))^k), a double well whose first eigenfunction is ±1 on the two wells, and a three-well potential with two pathways. Read it knowing that the generators as printed are twice what the paper's own kernel and time step give (its Ornstein–Uhlenbeck lemma and its equation (32) disagree by that factor, which this reading checked); that "converge" means a formal small-ε expansion of the operator, with the large-sample limit cited and no spectral convergence shown; that its sufficient condition for the first eigenfunctions to depend only on the slow variable is printed the wrong way round; that its double-well example is separable, so the slow coordinate is closed by construction; and that from equilibrium samples the eigenfunctions are recovered well while the slow eigenvalue, the inter-well rate, is not. In what sense slow eigenfunctions are closed coarse variables is the record's addition (THEORY-762). The published ACHA version was not reached.
Version2
Tagsmathematics, representation-learning, probabilistic-modeling, anthology-candidate
Cited as “paper” byNOTE-986 — Nadler, Lafon, Coifman and Kevrekidis — Diffusion maps and reaction coordinates
Cited as “source” byTHEORY-762 — For a reversible diffusion dX = −∇U dt + √2 dW in equilibrium, an eigenfunction ψ of its generator with eigenvalue −λ is closed in conditional mean with no memory: E[ψ(X_{t+s}) | the whole past] = e^{−λs}ψ(X_t), and so the same given ψ's own past; but ψ(X_t) is a Markov process in its own right only when |∇ψ|² is a function of ψ, since otherwise its own path reveals a noise intensity that its present value does not fix

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Nadler, B., Lafon, S., Coifman, R. R. & Kevrekidis, I. G. (2006). Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. Applied and Computational Harmonic Analysis 21(1): 113–127. doi:10.1016/j.acha.2005.07.004. arXiv:math/0503445 (22 March 2005).

Read in full in NOTE-986, from the arXiv copy. THEORY-762 states in what sense the slow eigenfunctions it recovers are closed coarse variables.

Why this is here

The companion of Coifman and Lafon (LIT-1246, NOTE-985) that turns diffusion maps toward dynamics. It is the record’s source for reading the slow eigenfunctions of a sampled Fokker–Planck operator as the slow variables of the system that produced the data. That bears on QUESTION-222 (are they closed coarse variables?) and on CLAIM-164 and CLAIM-157 (spectral gaps and metastable clusters).

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