\({g} = [P]^κ {ψ}\) is the "diffusion maps basis",
\([P] = \mathrm{diag}\{[K] 1\}^{−1} [K]\) is a transition matrix with right eigenvectors {ψ},
\([K]_{ij} = k_ε(η^i, η^j)\) are transition likelihood,
\(k_ε(x, y)\) is a kernel (symmetric, non-negative function) with a smoothing parameter ε,
such as the Gaussian kernel \(k_ε(x, y) = \exp(− (x − y)^2 / (4ε))\),
κ is the analysis scale of the local geometric structure of the dataset;
if only the first m diffusion maps basis vectors \([g](m)\) are retained,
then \([η](m) = [η] P_{[g](m)}\) is a reduced-order representation that projects \([η]\) onto \([g](m)\);
sample by solving a reduced-order Itô stochastic differential equation (ISDE) numerically:
\([η^l] = [Z(lρ)] [g](m)^T\), \(l = 1, 2, ...\) and \(ρ = M_0 Δt\), where
m satisfies mean-square convergence criterion \(|[c](m) - [c]|_F < ε_0 |[c]|_F\) given \(ε_0\);
\([Z]\) satisfies the reduced-order ISDE (so that \([Z] [g](m)^T\) admits \(p_{[H](m)}(η)\)):
\[\begin{cases}
d[Z] = [Y] dr \\
d[Y] = [L]([Z] [g](m)^T) [a](m) dr − f_0/2 [Y] dr + \sqrt{f_0} d[W] [a](m) \\
[Z](0) = [H] [a](m);\quad [Y](0) = [N] [a](m)
\end{cases}\],
\(Δt = 2 \pi s' / \text{Fac}\) is the sampling step of the integration scheme (oversampled if Fac > 1),
\(M_0\) is a multiplier such that \(ρ \gg 4 / f_0\), the relaxation time of the dynamical system;
The Störmer–Verlet discretization scheme preserves energy for non-dissipative Hamiltonian dynamical systems:
\[\begin{cases}
[Z_{l+1/2}] = [Z_l] + Δt/2 [Y_l] \\
[Y_{l+1}] = (1-b)/(1+b) [Y_l] + Δt/(1+b) [L_{l+1/2}] [a](m) + \sqrt{f_0}/(1+b) [ΔW_{l+1}] [a](m) \\
[Z_{l+1}] = [Z_{l+1/2}] + Δt/2 [Y_{l+1}]
\end{cases}\],
where \([L_{l+1/2}] = [L]([Z_{l+1/2}] [g](m)^T)\) and \(b = f_0 Δt/4\).
Markov stochastic process of a nonlinear second-order dynamical system (dissipative Hamiltonian system)
\[\begin{cases}
d[U] = [V] dr \\
d[V] = [L]([U]) dr − f_0/2 [V] dr + \sqrt{f_0} d[W] \\
[U](0) = [H];\quad [V](0) = [N]
\end{cases}\], where
\([L]([u]) = ( -∇ν(u^j) )_j\) and \(ν(u) = - \text{LogSumExp}\{ -(u - s'/s η^i)^2 / (2 s'^2) \}\),
\([W]\) are N independent v-dimensional normalized Wiener process (increments are standard Gaussian),
\([N]\) are N independent v-dimensional normalized Gaussian vector,
\([H]\) is a random matrix with a realization \([η]\),
\(f_0\) is a dissipation parameter such that the transient response of the ISDE are rapidly killed.
The ISDE has a unique invariant measure and a unique solution
that is a second-order diffusion stochastic process, which is stationary and ergodic,
and such that the probability distribution of random matrix \([U]\) is \(p_{[H]}(η)\);
Parameters: \((ε, κ, m, f_0, Δt, M_0)\) (or replace m with \(ε_0\));