Geometry of a set $Γ$ is a set of rules that describe the relationship between its elements. Intrinsic geometry (本征几何) is one that does not refer to a superset or the structures therein. Extrinsic geometry (表征几何) is one that is induced from the geometry of a superset. The geometry of a set can be understood through the geometry of the space of functions on the set (and spaces of functionals and operators). This is a dual perspective in functional analysis, from inverse problems such as inverse scattering, potential theory, and spectral geometry. [@Lafon2004] uses the language of functional analysis.

## Symbols

• $\Gamma$: a set, either finite (graph) or continuous (manifold).
• $\mathrm{d}x$: the Lebesgue measure ($\mathbb{R}^n$); the Riemannian volume (Riemannian manifold).
• $(\Gamma, \mathrm{d}\mu)$: a measure space with point set $\Gamma$ and measure $\mathrm{d}\mu$.
• $f$, $F$: a function (restricted to) a manifold $\Gamma$; a function (extended to) the ambient space $\mathbb{R}^n$;
• $L^2(\Gamma, \mathrm{d}\mu)$: the space of square integrable functions on $\Gamma$. $f \in L^2(\Gamma, \mathrm{d}\mu) \Leftrightarrow \int_\Gamma |f(x)|^2~\mathrm{d}\mu(x) < \infty$.
• $H_s(\Gamma, \mathrm{d}\mu)$: a Sobolev space of functions on $\Gamma$. $f \in H_s(\Gamma, \mathrm{d}\mu) \Leftrightarrow \forall |r| \le s, \partial^r f \in L^2(\Gamma,\mathrm{d}\mu)$. Here $|r| = \sum_i r_i, r_i \in \mathbb{N}$.
• $\langle f, g \rangle_\Gamma$: inner product of functions in $L^2(\Gamma, \mathrm{d}\mu)$. $\langle f,g \rangle_\Gamma = \int_\Gamma \overline{f(x)} g(x)~\mathrm{d}\mu(x)$.
• $\|f\|_s$: norm of a function in $H_s(\Gamma, \mathrm{d}\mu)$. $\|f\|_s^2 = \sum_{|r| \le s} \int_\Gamma |\partial^r f(x)|^2~\mathrm{d}\mu(x)$.
• $k(x, y)$, $p_t(x, y)$: an integral kernel; the Neumann heat kernel on a manifold.
• $v^2(x)$: integral transform of 1 with kernel $k(x, y)$.
• $\tilde{a}(x, y)$: a Markov kernel, transition probability of a diffusion process.
• $a(x, y)$: a diffusion kernel, the symmetric conjugate to $\tilde{a}$ by $v(x)$.
• $K$, $\tilde{A}$, $A$: an integral operator; an averaging operator; a diffusion operator.
• $\Delta$: the Laplacian ($\mathbb{R}^n$); the Laplace-Beltrami operator (Riemannian manifold). Signs are defined so that $\Delta$ is positive semi-definite.
• $\{\phi_i\}$, $\{\nu_i^2\}$: eigenfunctions and eigenvalues of $\Delta$, i.e. $\Delta \phi_i = \nu_i^2 \phi_i$.
• $\Phi$, $\Phi_k$: diffusion map; diffusion map truncated to dimension $k$.

## Diffusion Process on Manifolds

Integral kernel $k(x,y)$ defines an integral transform or integral operator $K$ for functions on measure space $(\Gamma, \mathrm{d}\mu)$, such that $K f(x) = \int_\Gamma k(x, y) f(y)~\mathrm{d}\mu(y)$. Here we call an integral kernel admissible, if it is symmetric, positivity-preserving, and positive semi-definite: $k(x, y) = k(y, x)$; $k \ge 0$; $\langle f, K f \rangle \ge 0$. Admissible kernels can be designed to represent a local metric (e.g. degree of similarity) or exhibit a local behavior.

Markov kernel or stochastic kernel $\tilde{a}(x, y)$ defines a Markov process (an averaging operator) $\tilde{A}$, where $\tilde{a}(x, \cdot)$ is a probability measure. Every positivity-preserving integral kernel can be transformed into a Markov kernel $\tilde{a}(x, y)= k(x, y) / v^2(x)$, where $v^2(x) = K 1$. Assume the Markov kernel has eigendecomposition $\tilde{a}(x, y) = \sum_{i \in \mathbb{N}} \lambda_i \psi_i(x) \hat{\psi}_i(y)$. The right eigenvectors $\{\psi_i\}$ (and left eigenvectors $\{\hat{\psi}_i\}$) need not be orthogonal.

Diffusion kernel $a(x, y)$ is the symmetric conjugate of a Markov kernel $\tilde{a}$ by $v$, $a(x, y) = v(x) \tilde{a}(x, y) / v(y)$. Diffusion operator $A$ with kernel $a$ is positive semi-definite, bounded, with the largest eigenvalue 1 and a corresponding eigenfunction $v(x)$. Assume that $A$ is compact (not just bounded), then its spectrum is discrete and its eigendecomposition can be written as $a(x, y) = \sum_{i \in \mathbb{N}} \lambda_i \phi_i(x) \phi_i(y)$, where $A \phi_i(x) = \lambda_i \phi_i(x)$. The diffusion operator $A$ and the averaging operator $\tilde{A}$ have the same spectrum $\lambda$, and the eigenfunctions of $A$ and the right/left eigenfunctions of $\tilde{A}$ are related by conjugation by $v$: $\psi_i = \mathrm{diag}(v^{-1}) \phi_i$, $\hat{\psi}_i = \mathrm{diag}(v) \phi_i$. t-step diffusion operator $A^t$ aggregates the local information, and its kernel, the t-step diffusion kernel $a^{(t)}$, has eigendecomposition $a^{(t)}(x, y) = \sum_{i \in \mathbb{N}} \lambda_i^t \phi_i(x) \phi_i(y)$.

Given $\Gamma$ is a compact smooth submanifold of dimension $d$ in $\mathbb{R}^n$, i.e. a Riemannian manifold with Riemannian metric induced from the ambient space $\mathbb{R}^n$. Let $\mu$ be a probability measure on $\Gamma$ and $\mathrm{d}x$ the Riemannian measure on $\Gamma$, then probability density $p(x)$ is given by $p(x) = \mathrm{d}\mu(x) / \mathrm{d}x$. Let $\{e_i\}_{i=1}^d$ be an orthonormal basis of the tangent plane $T_x$ to $Γ$ at $x$, and the exponential map $\mathrm{exp}_x$ of the coordinates of the tangent plane forms a chart on $Γ$ around $x$, which provides normal coordinates $(s_i)_{i=1}^d$. Laplace-Beltrami operator $\Delta$ on the Riemannian manifold can be written as $\Delta f = - \sum_{i=1}^d \partial^2 f / \partial s_i^2$, where $f \in C^\infty(\Gamma)$ is a smooth function on the manifold. Neumann heat operator on the manifold (heat diffusion on the manifold with the Neumann boundary condition) is $e^{−t \Delta}$, which corresponds to the Neumann heat kernel $p_t (x, y)$ when seen as an integral operator.

### Heat Kernels on Manifolds

For the Euclidean n-space, the heat kernel is the Gaussian probability density function with scalar variance twice the diffusion time: $\bar{p}_t (d) = (4 \pi t)^{-n/2} e^{- d^2 / (4t)}$, where distance $d = |x - y|$; in short, $\bar{p}_t = \phi(0, 2 t I_n)$, where $\phi$ is the PDF of a Gaussian random vector.

For the n-sphere, the heat kernel is a function of the spherical distance $\theta = \arccos \langle x, y \rangle$. It does not have a closed form, but can be written in series approximations. The heat kernel can be bounded as [@Nowak2019]: given $T > 0$, $\forall \theta \in [0, \pi]$, $\forall t \in (0, T]$, $\mathring{p}_t(\theta) \asymp \mathcal{O}\left((t+ \pi - \theta)^{-(n-1)/2} t^{-n/2} e^{-\theta^2 / (4t)}\right)$, where the constants depend only on $n$ and $T$; in short, $\mathring{p}_t(\theta) \asymp \mathcal{O}\left((t+ \pi - \theta)^{-(n-1)/2} \bar{p}_t (\theta) \right)$. Note that $\asymp$ denotes order equivalence.

For the hyperbolic n-space, the heat kernel is [@Davies1988]: (1) if $n = 2m + 1$, $\breve p_t (d) = \frac{(-1)^m e^{-(n-1)^2 t / 4}}{\sqrt{(2\pi)^{n} (2 t)}} \left(\frac{1}{\sinh d} \frac{\partial}{\partial d}\right)^m e^{- d^2 / (4t)}$; in short, $\breve p_t (d) = (-2 t)^{m} e^{-m^2 t} \left(\frac{1}{\sinh d} \frac{\partial}{\partial d}\right)^m \bar p_t (d)$; (2) if $n = 2m + 2$, $\breve p_t (d) = \frac{(-1)^m e^{-(n-1)^2 t / 4}}{\sqrt{(2\pi)^{n+1} (4 t^3)}} \left(\frac{1}{\sinh d} \frac{\partial}{\partial d}\right)^m \int_d^\infty \frac{s e^{- s^2 / (4t)}}{\sqrt{\cosh s - \cosh d}} \text{d}s$. The heat kernels satisfy the recurrence relation: $\breve p_{n+2} = \frac{-e^{-nt}}{2\pi} \left(\frac{1}{\sinh d} \frac{\partial}{\partial d}\right) \breve p_n$.

## Diffusion Maps

Diffusion processes (or random walks, Markov processes) can be useful for the geometric descriptions of data sets. [@Coifman2006a] approximates (anisotropic) diffusion processes $\exp(-t H_\alpha)$ on submanifolds, where $H_\alpha = \Delta - \Delta(p^{1-\alpha}) / p^{1-\alpha}$ is a symmetric Schrödinger operator and parameter $\alpha \in \mathbb{R}$, with special cases: $\alpha = 1$, heat diffusion; $\alpha = 1/2$, a Fokker–Planck diffusion; $\alpha = 0$, normalized graph Laplacian with isotropic weights. In comparison, "kernel eigenmaps" are diffusion processes $\exp(-t Q_2^{-1} Q_1)$.

Diffusion distance or diffusion metric $D_t(x, y)$ at time step $t$ is the semi-metric defined as: $D_t^2 (x, y) \equiv a^{(t)}(x, x) + a^{(t)}(y, y) - 2 a^{(t)}(x, y)$, which is equivalent to $D_t^2 (x, y) = \sum_{i \in \mathbb{N}} \lambda_i^t [\phi_i(x) - \phi_i(y)]^2$ and $D_t^2(x,y) = \| a^{(t/2)}(x,\cdot) - a^{(t/2)}(y,\cdot) \|^2$. The diffusion distance becomes a metric if the diffusion kernel $a$ is positive definite. Unlike the geodesic distance, the diffusion distance is an average over all paths connecting two points, and thus robust to noise and topological short-circuits.

Diffusion map $\Phi: \Gamma \to l^2(\mathbb{N})$ is a mapping of the data to a Euclidean space, based on the eigenfunctions of the diffusion kernel: $\Phi(x) = \left( \phi_i(x) \right)_{i \in \mathbb{N}}$. Diffusion map $\Phi$ defines an embedding of the data that preserves the diffusion distance $D_t$ via a weighted Euclidean metric $D_t^2(x,y) = \langle \Phi(x) - \Phi(y), \Lambda^t (\Phi(x) - \Phi(y)) \rangle$. A truncated diffusion map $\Phi_k: \Gamma \to \mathbb{R}^k$ with $\Phi_k(x) = \left( \phi_i(x) \right)_{i=1}^k$ preserves the diffusion distance with certain accuracy. The dimension $k$ of the new representation depends on the diffusion process, and is in general greater than the dimension $d$ of the set/manifold.

(Inverse mapping.)

### Heat Diffusion

Spectral graph theory analyzes the spectrum (eigenvalues and eigenvectors) of a matrix representing a graph. For example, spectral clustering or spectral partitioning: construct a matrix representation of the graph; eigendecompose the matrix and embed the vertices to a Euclidean space using one or more eigenvectors; group the points based on their coordinates. Laplacian matrix of a graph, aka graph Laplacian, is its degree matrix minus its adjacency matrix, $L = D - A$, where $D = \mathrm{diag}\{d_i\}_{i=1}^n$. The Laplacian matrix is a discrete analog of the Laplacian operator $Δ$ in multivariable calculus. The Laplacian matrix is symmetric, so its eigenvectors can form an orthogonal basis; additionally, all eigenvalues are non-negative. Symmetric normalized graph Laplacian is $L' = D^{-1/2} L D^{-1/2} = I - D^{-1/2} A D^{-1/2}$. Consider rotation-invariant kernels $k_\varepsilon (x, y) = h(\|x-y\|^2 / \varepsilon)$ with scale parameter $\varepsilon$, where $h$ is smooth and decays exponentially, and $\|\cdot\|$ is the Euclidean metric. As $\varepsilon$ goes to 0, the local geometry specified by $k_\varepsilon$ coincide with that of the manifold. Given an averaging operator $\tilde{A}_\varepsilon f(x) = K_\varepsilon f(x) / v_\varepsilon^2 (x)$ where $v_\varepsilon^2 (x) = K_\varepsilon 1$, the weighted graph Laplacian operator is $\tilde{\Delta}_\varepsilon = (I - \tilde{A}_\varepsilon) / \varepsilon$. On certain subspaces $E_K$ of $L^2(\Gamma)$, as $\varepsilon$ goes to 0, the limit operator of $\tilde{\Delta}_\varepsilon$ is $H f = c (\Delta f + 2 \langle \nabla p / p , \nabla f \rangle)$, where $c = m_2 / (2 m_0)$, $m_0 = \int_{\mathbb{R}^d} h(\|u\|^2)~\mathrm{d}u$, and $m_2 = \int_{\mathbb{R}^d} u_i^2 h(\|u\|^2)~\mathrm{d}u$. Under a conjugation by the density $p$, we get $p H(g/p) = c (\Delta g - g \nabla p / p)$, where $g = p f$. Thus, when the density $p$ is uniform, the weighted graph Laplacian $\tilde{\Delta}_\varepsilon$ converges to (a multiple of) the Laplace-Beltrami operator $\Delta$ on the manifold [@Belkin2003]; but when the density is non-uniform, such reconstruction does not hold.

To separate the distribution $p$ on the manifold from its intrinsic geometry $k_\varepsilon$, use kernel $\tilde{k}_\varepsilon(x, y) = k_\varepsilon(x, y)/(p_\varepsilon(x) p_\varepsilon(y))$ where $p_\varepsilon(x) = K_\varepsilon 1$. This leads to an averaging operator $\bar{A}_\varepsilon f(x) = \tilde{K}_\varepsilon f(x) / \tilde{v}_\varepsilon^2 (x)$ where $\tilde{v}_\varepsilon^2 (x) = \tilde{K}_\varepsilon 1$. Define a Laplace operator $\Delta_\varepsilon = (I - \bar{A}_\varepsilon) / \varepsilon$ acting on the subspaces $E_K$ of $L^2(\Gamma)$, its limit operator $\lim_{\varepsilon \to 0} \Delta_\varepsilon \equiv \Delta_0 = \Delta / c$ is a multiple of the Laplace-Beltrami operator. The limit operator of $\bar{A}_\varepsilon^{t/\varepsilon}$ is the Neumann heat operator $e^{−t \Delta_0}$; in other words, Markov kernel $\bar{a}_\varepsilon^{(t/\varepsilon)}(x, y)$ with small $\varepsilon$, close to a fine scale Gaussian kernel, approximates the (symmetric) Neumann heat kernel $p_t (x, y)$, regardless of knowledge of the boundary. The averaging operator $\bar{A}_\varepsilon$ is compact and its eigendecomposition can be written as $\bar{A}_\varepsilon = \sum_{i \in \mathbb{N}} \lambda_{\varepsilon, i} P_{\varepsilon, i}$, where $P_{\varepsilon, i}$ is the orthogonal projector on the eigenspace corresponding to eigenvalue $\lambda_{\varepsilon, i}$. If the eigendecomposition of the Laplace-Beltrami operator is $\Delta = \sum_{i \in \mathbb{N}} \nu_i^2 P_i$, where $P_i$ is the orthogonal projector on the eigenspace corresponding to eigenvalue $\nu_i^2$. Then the eigenfunctions and eigenvalues of $\bar{A}_\varepsilon^{t/\varepsilon}$ converge to those of the heat operator $e^{-t\Delta}$: $\lim_{\varepsilon \to 0} \lambda_{\varepsilon, i}^{t/\varepsilon} = e^{-t\nu_i^2}$, $\lim_{\varepsilon \to 0} P_{\varepsilon, i} = P_i$. Let $\{\phi_i\}_{i \in \mathbb{N}}$ be the orthonormal eigenfunctions of the Laplace-Beltrami operator (also of the heat kernel) on the manifold $\Gamma$, then the eigendecomposition of the heat kernel is $p_t(x, y) = \sum_{i \in \mathbb{N}} e^{-t \nu_i^2} \phi_i(x) \phi_i(y)$. Although $\{\phi_i\}_{i \in \mathbb{N}}$ is usually viewed as a Hilbert basis of $L^2(Γ)$, it also forms a set of coordinates on the submanifold $Γ$ via diffusion maps $\Phi$, which accurately preserves the heat diffusion distance with dimension $k \ge d$.

Numerical procedure for approximate Neumann heat diffusion: (Note that skipping steps 3-4 provides the weighted graph Laplacian.)

1. Choose scale parameter $\varepsilon$ to be (on the order of) the larger of: (1) the mean square distance within the set, $\sum_x m(x) / N \min_{y \ne x} \|x - y\|^2$, where $m(x)$ is the number of observations identical to $x$ in the data; and (2) the square of the perturbation size $\|\eta\|^2$, i.e. trace of the covariance matrix of the perturbation. Note that small $\varepsilon$ may result in a disconnected graph.
2. Construct operator $[K_\varepsilon]$ with entries $[K_\varepsilon]_{xy} = k_\varepsilon(x, y)$, where $k_\varepsilon(x, y) = e^{-\|x-y\|^2/\varepsilon}$ is a Gaussian kernel.
3. Compute approximate density $p_\varepsilon = [K_\varepsilon] \mathbf{1}$ by substituting true density $p(x)$ with the empirical measure.
4. Compute density-neutralizing operator $[\tilde{K}_\varepsilon] = \mathrm{diag}(p_\varepsilon^{-1}) [K_\varepsilon] \mathrm{diag}(p_\varepsilon^{-1})$. It can be implemented as $[K_\varepsilon] / [p_\varepsilon p_\varepsilon^T]$, where $/$ is element-wise division.
5. Compute normalizing vector $\tilde{v}_\varepsilon^2 = [\tilde{K}_\varepsilon] \mathbf{1}$. Take a square root to get $\tilde{v}_\varepsilon$.
6. Compute diffusion operator $[A_\varepsilon] = \mathrm{diag}(\tilde{v}_\varepsilon^{-1}) [\tilde{K}_\varepsilon] \mathrm{diag}(\tilde{v}_\varepsilon^{-1})$. It can be implemented as $[\tilde{K}_\varepsilon] / [\tilde{v}_\varepsilon \tilde{v}_\varepsilon^T]$.
7. Eigendecompose $[A_\varepsilon] = \Phi \Lambda_\varepsilon \Phi^T$. The Neumann heat operator has approximate spectrum $\lambda_\varepsilon$ and eigenvectors $\{\phi_i / \phi_1\}_{i=1}^N$. Note that $A_\varepsilon \tilde{v}_\varepsilon = \tilde{v}_\varepsilon$, so $\phi_1$ and $\tilde{v}_\varepsilon$ are proportional.

### Anisotropic Diffusion

For differential and dynamical systems.

## Geometric Harmonics

Diffusion processes are also useful for harmonic analysis of functions on data sets. Geometric harmonics is a set of functions that allows out-of-sample extension of empirical functions on the data set.

### Bi-Lipschitz Parametrization

Geometric harmonics provide a simple solution to the relaxed distortion problem, an embedding $\Psi$ that is bi-Lipschitz with a small distortion.

Atlas Computation:

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Eigenfunction selection:

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### Extension and Restriction

Intrinsic geometry, Fourier analysis of the manifold (eigenfunctions of the Laplace-Beltrami operator). Extrinsic geometry, Fourier analysis of the ambient space.

Restriction.

Extension (a version of the Heisenberg uncertainty principle): extend the eigenfunctions to a band-limited function of band $\mathcal{O}(\nu_i)$, localized in a tube of radius $\mathcal{O}(1/\nu_i)$ around the manifold.

(Extension of f = 1 for discription of the manifold?)

Multiscale extension: empirical functions are decomposed into frequency bands, and each band is extended to a certain distance so that it satisfies some version of the Heisenberg principle.

(Intrinsic and extrinsic diffusion.)

Multiscale extension scheme:

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