Probabilistic approaches to machine learning, or probabilistic learning [@Ghahramani2015], include probabilistic modeling, defining likelihoods, parameter estimation using likelihood and Bayesian techniques, probabilistic approaches to classification, clustering, and regression, and related topics such as model selection and bias/variance tradeoffs.

Probabilistic learning on manifolds concerns a probability distribution concentrated on a (measure-zero) subset of the Euclidean space, which is viewed as a manifold.

Move points toward the "principal curve" [@Hastie, Stuetzle, 1989]; A point is on the "density ridge" if the maximum negative curvature of the PDF at the point is perpendicular to the gradient of the PDF at the point [@Scott1991b].

Other references (directional statistics): Statistical Analysis of Spherical Data [@Fisher, Lewis, Embleton, 1987]; Statistics on Spheres [@Watson1985];

## Density estimation and approximation on Manifold

Probability density estimation and approximation on manifolds. Considerations: computational cost, convergence rate, assumption applicability. See [@Brigant2019] for a survey.

Parametric families of probability densities on manifolds: MISE is usually $\mathcal{O}(N^{-1})$ by parametric estimation.

1. heat kernels: wrapped Gaussian for 1-sphere;
2. maximum entropy distributions with moment constraints [@Pennec2006]:
• matrix manifolds: von Mises-Fisher (vMF) for n-spheres, matrix vMF for Stiefel manifolds, etc.;
• hyperbolic spaces: hyperbolic vMF;
• symplectic manifolds: group-invariant natural exponential families;
3. based on geodesic distance:
• Riemannian symmetric spaces: Gaussian-like distribution $f(p) \propto \exp[- d^2(p, \mu) / (2 \sigma^2)]$ [@Said2018], e.g. positive definite matrices [@Said2017] with a certain structure, e.g. complex, Toeplitz, or block-Toeplitz;

Non-parametric density estimation:

1. projection of the density function onto eigenfunctions of the Laplace–Beltrami operator [@Hendriks1990]: $\text{MISE} = \mathcal{O}(Q^{-s} + N^{-1} Q^{d/2})$, optimal case $\mathcal{O}(N^{-2s/(2s+d)})$, where $s$ is smoothness class of the density;
2. kernel density estimation (local linearization) [@Pelletier2005]: function of geodesic distance, support smaller than injectivity radius, volume density bounded away from 0; $\text{MISE} = \mathcal{O}(r^4 + N^{-1} r^{-d})$, optimal case $\mathcal{O}(N^{-4/(4+d)})$.

Quantization: approximate a random variable by a quantized/discretized version; optimal quantization minimizes the $L^p$ distance to the true measure from a discrete measure of $n$ supporting points; any quantization defines a clustering by Voronoï cells; the optimal centers are asymptotically distributed as $h^{d/(d+p)}$; Competitive Learning Vector Quantization is a classical algorithm for quadratic ($p = 2$) vector quantization.

## Manifold Sampling

Sampling density estimates on (high dimensional) manifolds defined by limited data, done by a synthesis of methods [@Soize2016]:

1. (Implicit): Kernel density estimation (KDE) for the probability distribution of the sample matrix;
2. Diffusion maps for the "local" geometric structure, aka manifold, of the dataset: top eigenvectors of the diffusion map is used for a reduced-order representation of the sample matrix;
3. Markov chain Monte Carlo (MCMC) method based on Ito stochastic differential equation (ISDE) for generating realizations of the sample matrix;

### Preliminary: Sampling a Gaussian KDE

The Gaussian KDE of a $v×N$ sample matrix $[η]$, modified as in [@Soize2015]: $$p_H(η) = 1/N \sum_i π(η; η^i s'/s, s')$$

• $π(η; m, σ) = \exp\{ - \|(η - m)/σ\|^2 / 2 \} / (\sqrt{2 π} σ)^v$ is the Gaussian kernel;
• $s = (4 / ((v + 2)N))^{1 / (v + 4)}$ is the optimal Silverman bandwidth;
• $s' = s / \sqrt{s^2 + N / (N − 1)}$;

Sampling the Gaussian KDE of the random vector by solving an ISDE [@Soize1994, pp. 211-216, Thm. 4-7]. The following Markov stochastic process of a nonlinear second-order dissipative Hamiltonian dynamical system has a unique invariant measure and a unique solution that is a second-order diffusion stochastic process, which is stationary, ergodic, and $U(t) \sim p_{H}(η)$: $$\begin{cases} dU = V dt \\ dV = L(U) dt − 1/2 f_0 V dt + \sqrt{f_0} dW \\ U(0) = H;\quad V(0) = N \end{cases}$$

• $L(u) = -∇ν(u)$ is the conservative force;
• $ν(u) = -\text{LogSumExp}\{ -\|(u - η^i s'/s) / s'\|^2 / 2\}$ is the potential (Hamiltonian);
• $f_0$ is a dissipation parameter such that the transient response of the ISDE are rapidly killed;
• $W$ is the v-dimensional normalized Wiener processes (increments are standard Gaussian);
• $H \sim p_{H}(η)$ is a random vector with realizations $[η]$;
• $N$ is the v-dimensional normalized Gaussian vector;

### Procedure: Sampling a manifold-reduced Gaussian KDE

1. Shift and scale the data $[ x ]$, a matrix of $p$ attributes by $N$ observations, to $(ϵ, 1)$;
2. Normalize the data $[x_0]$ by principal component analysis (PCA): $[η] = \mathrm{diag}(μ)^{−1/2} [φ]^T [x_0]$
• $μ$ are the $v \le p$ positive eigenvalues of the empirical covariance matrix $[c] = [x_0] [x_0]^T /(N-1)$;
• $φ$ the corresponding $v$ orthonormal eigenvectors;
3. Characterize the manifold using a diffusion maps basis: $[g] = [P]^κ [ψ] = [ψ] [\Lambda]^κ$
• $[P] = \mathrm{diag}\{[K] 1\}^{−1} [K]$ is the diffusion map (a transition matrix), $\{ψ\}$ is the right eigenvectors of $[P]$, $[\Lambda]$ is the diagonal matrix of the corresponding eigenvalues, and $κ$ is the "analysis scale" of the local geometric structure of the dataset;
• $[K]_{ij} = k_ε(η^i, η^j)$ are transition likelihood, $k_ε(x,y)=\exp\{− \|x − y\|^2 / (4ε)\}$ is the Gaussian kernel with smoothing parameter $ε$, the kernel may be set to any decreasing function of a metric $f(d(x, y))$ with $f(0) = 1, f(\infty) = 0$;
• $[η] = [z] [g]^T$, where $[z] = [η] [a]$ and $[a] = [g] ([g]^T [g])^{-1}$, because full projection $P_{[g]} = [g] ([g]^T [g])^{-1} [g]^T = I$;
• $[η](m) = [η] P_{[g](m)}$ is a reduced-order representation of $[η]$ that projects $[η]$ on $[g](m)$, the first $m$ vectors of $[g]$;
4. Sample the reduced-order sample matrix by solving an ISDE: $[η](t) = [Z](t) [g](m)^T$, $t = l M_0 Δt$, $l = 1, 2, ...$
• $m$ satisfies mean-square convergence criterion $\|[c](m) - [c]\|_F < ε_0 \|[c]\|_F$ for some $ε_0 = \mathcal{O}(10^{-3})$;
• $[Z](t)$ satisfies the following ISDE where $[L](u) = (-∇ν(u^j))_j$: $$\begin{cases} d[Z] = [Y] dt \\ d[Y] = [L]([Z] [g](m)^T) [a](m) dt − 1/2 f_0 [Y] dt + \sqrt{f_0} d[W] [a](m) \\ [Z](0) = [H] [a](m);\quad [Y](0) = [N] [a](m) \end{cases}$$ If exists $[L'] = \nabla \log \rho'$ such that $[L']([Z]) = [L]([Z] [g](m)^T) [a](m)$, then $[Z] \sim \rho'$.
• The Störmer–Verlet discretization scheme of the ISDE (preserves energy for non-dissipative Hamiltonian dynamical systems): $$\begin{cases} [Z_{l+1/2}] = [Z_l] + [Y_l] Δt/2 \\ [Y_{l+1}] = ((1-b) [Y_l] + [L_{l+1/2}] [a](m) Δt + \sqrt{f_0} [ΔW_{l+1}] [a](m)) / (1+b)\\ [Z_{l+1}] = [Z_{l+1/2}] + [Y_{l+1}] Δt/2 \end{cases}$$ where $[L_{l+1/2}] = [L]([Z_{l+1/2}] [g](m)^T)$ and $b = f_0 Δt/4$.
• $Δt = 2πs' / \text{Fac}$ is the sampling step of the integration scheme (oversampled if Fac>1);
• $M_0 Δt > 4 / f_0$, the relaxation time of the dynamical system, so samples are approximately independent, e.g. $M_0 > 2 \log(100) \text{Fac} / (\pi f_0 s')$;

Parameters: $(ε, κ = 1, ε_0, f_0 = 1.5, Δt, M_0 = 110)$, or replace $Δt$ with Fac.

The computational cost is no greater than the direct MCMC in the Preliminary section. But the main advantage is a probability distribution concentrated on manifold.

## Misc

Extensions:

• Polynomial chaos expansion (PCE) [@Soize2017] around a manifold;
• Inference: estimate geological structural properties using well logs [@Thimmisetty2018];
• Stochastic optimization [@Ghanem2018] using PLoM as a surrogate:
• well placement [@Ghanem2018a] in a reservoir for maximum oil production;
• scramjet design [@Ghanem2019], maximize the combustion efficiency, subject to 3 inequality constratins (on the burned equivalence ratio, the stagnation pressure loss across the combustor, and the maximum pressure RMS), with 5 control variables (global equivalence ratio, primary-secondary ratio, primary/secondary injector locations, primary injector angle);
• additive manufacturing [@Marmarelis2020], infer the relation between build parameters and material properties, minimize the chance of failure and learn the Pareto frontier;
• Maximum-entropy selection of diffusion kernel bandwidth [@Soize2019]: applications to simulations of a ScramJet engine/combustion and a climate model;
• Physical or statistical constraints [@Soize2020], applied to a stochastic (elliptic) PDE;
• Inverse problems: computational mechanics; ultrasonic wave propagation to identify elasticity field in human bones [@Soize2020a];

Applications: optimization (cheap sample on unknown feasible set, a submanifold): stochastic optimization [@Kingma2015]; maximum likelihood; Bayesian inference [@ChenYC2020]; dynamical system on manifolds [@Talmon2013; @Talmon2015];