Emergent Geometry from Adaptive Diffusion#
Prerequisites: Latent Fractal Gas: Algorithm and Analytic Conditions, C³ Regularity of the Fractal Gas Fitness, and The Geometric Gas: Regularity, Stability, and Entropy.
TLDR#
A positive definite diffusion shape \(D\) defines a metric \(g=D^{-1}\). For the smooth Hessian construction, \(g=H+\epsilon_\Sigma I\) requires a positive spectral margin. The implementation also clips eigenvalues; its recorded metric is identified separately below. Both constructions admit explicit ellipticity and Lipschitz bounds.
The metric determines lengths and the volume density \(J=\sqrt{\det g}\). A stationary sampling law depends on the complete generator. We derive the overdamped and kinetic generators that have spatial density \(Z^{-1}J e^{-U/T}\), and distinguish those comparison models from the implemented kinetic, cloning, and killing dynamics.
Gram determinants give cell areas and volumes. Their quadrature errors, importance-sampling errors, and convergence rates follow from the regularity and probability estimates established in the analytical chapters, with the hypotheses of each estimate retained.
Introduction#
Imagine measuring a cloud of small random displacements at one point. In one direction the cloud is wide; in another it is narrow. Use that cloud as a ruler: a displacement is large when it is large compared with the noise in its direction. The inverse covariance gives precisely this ruler.
There are three calculations to keep apart. The first builds a ruler from a covariance matrix. The second asks how rulers at neighboring points fit together. That is where derivatives, connections, and curvature enter. The third asks where moving particles spend their time. That depends on forces, friction, noise, cloning, and survival.
Knowing the ruler solves the first problem. It supplies coefficients for the other two. We will carry those coefficients through the calculations, so that a volume formula, a stationary density, and a mixing estimate each have a specific reason to hold.
The Adaptive Diffusion Tensor#
Start in the coordinates used by the algorithm. A symmetric Hessian has orthogonal eigenvectors, and the adaptive noise assigns a separate scale to each one. Positive curvature of the fitness potential makes the corresponding noise smaller. Negative curvature needs a separate rule: adding a small positive number does not necessarily make a negative eigenvalue positive.
The analytical construction uses a shift larger than the negative part of the spectrum. The code also has a clipping rule. Both are simple to inspect one eigenvalue at a time, but their derivatives behave differently at the clipping threshold.
Definition 669 (Adaptive diffusion shape and metric)
Work in the declared Euclidean latent coordinates. On a fixed alive-set and companion stratum, let \(V_{\mathrm{fit}}^{(i)}(z;S,c)\) be the per-walker fitness with the indicated other coordinates and sampled companion assignment frozen. When this function is \(C^2\), set
The smooth shifted-Hessian construction is
on the set where \(g\) is positive definite. These are a metric, a noise square root, and a normalized covariance shape, respectively.
If a finite update has noise amplitude \(\Sigma_{\mathrm{step}}=c_2\Sigma_{\mathrm{reg}}\), then its conditional velocity-increment covariance is \(c_2^2D_{\mathrm{reg}}\). If a continuous SDE has amplitude \(\sqrt{2T}\Sigma_{\mathrm{reg}}\), its second-order generator is \(T D_{\mathrm{reg}}^{ij}\partial_i\partial_j\) in the coordinates on which the noise acts. The prefactor and those coordinates must be specified before identifying a generator.
Definition 670 (Expected fitness and a limiting fitness field)
Let \(c\) denote the complete companion assignment used to compute the sampled fitness, and let \(P_S(dc)\) be its actual joint law. Define
When the state varies with the query \(z\), differentiation includes the dependence of both the sampled fitness and \(P_S\). In particular, a derivative of this expectation need not equal the expectation of a frozen-companion derivative. The distinction and the joint-assignment formula are specified in Definition 594 and Lemma 206.
A mean-field fitness field \(V_{\mathrm{fit}}(z;\mu)\) is an identified limit of these finite-particle expected fields, with their prescribed normalizations, when such a limit exists. Defining its Hessian requires \(C^2\) regularity. Using it as the limit of finite-particle Hessians requires convergence of those derivatives. The geometry statements below apply to the particular sampled or expected field that has been specified.
Proposition 222 (Metric represented by the implemented Hessian branch)
For a successful eigendecomposition without an added retry jitter, the full
Hessian branch of KineticOperator._compute_diffusion_tensor uses
where \(A_+\) replaces the negative eigenvalues of a symmetric matrix by zero. The diagonal branch uses \((g_{\mathrm{clip}})_{ii}=\epsilon_\Sigma+\max(H_{ii},0)\) and zero off-diagonal entries. Thus, for \(c_2>0\),
The un-clipped formula \(g=H+\epsilon_\Sigma I\) agrees with the full branch
when \(H\) is symmetric positive semidefinite. An eigensolver retry adds a
further scalar shift before clipping; the fallback uses the clipped diagonal.
The grad_proxy, voronoi_proxy, and supplied-tensor branches require
their own covariance identification.
These statements describe the existing implementation, without changing its update.
Proof
For \(H_{\mathrm{sym}}=Q\operatorname{diag}(\lambda_j)Q^T\), the code shifts by \(\epsilon_\Sigma\), clips each shifted eigenvalue below at \(\epsilon_\Sigma\), and takes its inverse square root. The resulting metric eigenvalue is
Squaring the symmetric noise matrix gives the stated covariance and inverse. The diagonal computation is the same scalar identity.
Coordinates and the meaning of the metric#
Remark 247 (Coordinate convention)
The inverse covariance transforms as a metric when the covariance transforms as a contravariant tensor. If \(y=\psi(x)\) and \(A=D\psi(x)\), then \(D_y=A D_x A^T\) and \(g_y=A^{-T}g_xA^{-1}\).
An ordinary coordinate Hessian does not obey this law under an arbitrary nonlinear coordinate change: its transformation includes first derivatives of the potential and second derivatives of the coordinate map. The construction here therefore uses the algorithm’s fixed Euclidean coordinates. An intrinsic alternative would require a specified background metric \(G\), its covariant Hessian \(\operatorname{Hess}_G V\), and correctly raised indices in \(g^{-1}\). This alternative is a different mathematical specification.
A Hessian eigenvalue measures curvature of a scalar potential in these coordinates. Riemannian curvature depends on derivatives of the metric. A constant anisotropic \(g\) is flat, regardless of its determinant.
From fitness curvature to a metric
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Regularity and Convergence#
A matrix can be positive at every point and still become arbitrarily close to singular far away. Bounds need to hold on the whole region visited in the argument. On an unbounded state space, a local calculation is combined with the confining estimates from the convergence chapters.
The regularity proofs already give the right inputs. We retain their distinction between fixed sampled assignments, averaged assignments, and changes of alive set. This matters because a smooth function inside each region can jump when the rule selecting the region changes.
Two-sided spectral control#
Assumption 15 (Spectral bounds for the smooth construction)
On the specified region and collection of states, suppose
Write \(m=\epsilon_\Sigma-\Lambda_->0\) and \(M=\epsilon_\Sigma+\Lambda_+\). A conclusion uniform in \(N\) requires \(m\) bounded away from zero and \(M\) bounded above uniformly in \(N\).
The second-derivative bounds in Theorem 259 provide such spectral estimates where their measurement, normalization, companion-law, and stratum hypotheses hold. The third-derivative bounds in Theorem 260 provide the spatial Lipschitz estimates used below. Global or uniform bounds require the corresponding uniform input bounds from those theorems.
Theorem 326 (Uniform ellipticity from metric bounds)
Under Assumption 15,
For the clipped construction, an upper bound \(H_{\mathrm{sym}}\preceq\Lambda_+I\) instead gives
and the inverse bounds follow by reciprocation. Its positive lower metric bound holds without a lower bound on the Hessian. A positive lower diffusion bound still requires the upper Hessian bound.
For kinetic noise these inequalities concern the velocity block; the full phase-space diffusion matrix has a zero position block.
Proof
Orthogonally diagonalize \(H\). The shifted metric eigenvalues are \(\epsilon_\Sigma+\lambda_j(H)\in[m,M]\). Inversion reciprocates each eigenvalue, proving the first assertion. The clipped eigenvalues lie between \(\epsilon_\Sigma\) and \(\epsilon_\Sigma+\max(\Lambda_+,0)\) by Proposition 222. Finally, the vector fields of velocity noise have zero position component, so their phase-space covariance has the asserted zero block.
Lipschitz bounds and smoothness thresholds#
Lemma 273 (Operator-Lipschitz bound for inverse square root)
For symmetric positive definite matrices \(A,B\succeq mI\),
Proof
Along \(A_t=A+t(B-A)\), let \(X_t=A_t^{-1/2}\). Differentiating \(X_tA_tX_t=I\) gives
In an orthonormal eigenbasis of \(A_t\),
so \(\|L_t^{-1}\|_{F\to F}\leq(2\sqrt m)^{-1}\). Since \(\|X_t\|_{\mathrm{op}}\leq m^{-1/2}\),
Integrating in \(t\) proves the first bound. The resolvent identity \(A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}\) proves the second.
Proposition 223 (Lipschitz continuity of adaptive diffusion)
Let \(\mathfrak d\) be a specified distance on the field’s inputs \(u=(z,S)\). If \(g(u)\succeq mI\) and
then
For a frozen state on a convex coordinate region, \(\sup_z\|\nabla_zH\|_{\mathbb R^d\to F}\leq K_3\) gives \(L_g=K_3\) for \(\mathfrak d(z,z')=|z-z'|\). A bound for \(\mathfrak d=|z-z'|+W_1(\mu_S,\mu_{S'})\) requires, in addition, a Hessian dependence estimate in that Wasserstein distance. Spatial \(C^3\) regularity alone provides only the frozen-state bound.
For the clipped construction, \(\|H_{\mathrm{sym}}(u)-H_{\mathrm{sym}}(u')\|_F \leq L_H\mathfrak d(u,u')\) implies the same conclusions with \(m=\epsilon_\Sigma\) and \(L_g=L_H\).
Proof
The smooth assertions follow by the preceding lemma and integration of \(\nabla_zH\) along the segment.
For the clipped assertion, the map \(A\mapsto A_+\) is the orthogonal projection, in the Frobenius inner product, onto the closed convex cone of positive semidefinite matrices. To verify this directly, write \(A=A_+-A_-\) with \(A_\pm\succeq0\) and \(A_+A_-=0\). For any \(P\succeq0\),
Thus \(A_+\) minimizes the distance. The two projection variational inequalities, added together, give
Clipping is therefore nonexpansive. Apply the inverse-square-root lemma after adding \(\epsilon_\Sigma I\).
Remark 248 (Derivatives needed by geometric constructions)
A \(C^3\) fitness field gives a \(C^1\) smooth shifted metric and locally Lipschitz diffusion on a positive-margin region. Classical pointwise Riemann curvature requires a \(C^2\) metric, hence a \(C^4\) fitness field for this Hessian construction. Higher-order estimates follow from Theorem 270 under that theorem’s full hypotheses.
Clipping is Lipschitz, but is generally nondifferentiable when a Hessian eigenvalue crosses zero. Curvature formulas for the recorded clipped field require a smooth spectral region, a justified smooth reconstruction, or an explicit weak interpretation. Lipschitz bounds themselves remain valid across the clipping threshold.
For continuous dynamics, local Lipschitz coefficients give local well-posedness. Nonexplosion additionally follows from a coercive Lyapunov bound for the full drift and jump mechanism, as in Corollary 102. A finite-step update has its own transition kernel and uses the discrete convergence results.
Covariance and Geometric Descriptions#
At a single point, stretch each eigenvector by the square root of its metric eigenvalue. The noise cloud becomes round. Doing this point by point is a choice of orthonormal frame.
Now let the frame vary as a particle moves. Its variation contributes to the equations. This is why inspecting the covariance is insufficient to identify a process as Brownian motion on a manifold. The drift must be carried through the same calculation. For a kinetic process, one must also remember that the random kicks act on velocity while position is transported by velocity.
Observation 2 (Two descriptions of the same covariance)
For \(g=D^{-1}\), the columns of \(g^{-1/2}\) are an orthonormal frame for the metric:
Consequently, a noise increment with coordinate covariance \(2TD\,dt\) has isotropic covariance \(2TI\,dt\) in that frame.
For a diffusion acting on position, write \(J=\sqrt{\det g}\) and
Its coordinate generator admits the exact identity
This is a description of the same generator with a specified additional vector field. For kinetic noise, the second derivatives are instead \(\partial_{v_i}\partial_{v_j}\); the position-space Laplace–Beltrami identity does not replace those velocity derivatives.
Proof
The first assertion follows by multiplying the three matrices. Expanding the divergence in \(\Delta_g\) gives the second identity, including its first-order term.
Remark 249 (Coordinate changes and preservation of the process)
A coordinate change preserves the law of a process when its full generator is transformed, including the Itô second-derivative term or the equivalent Stratonovich vector-field transformation. A position-dependent frame is not generally the Jacobian of a coordinate change. An isometry to Euclidean space would require a flat metric locally, with additional global conditions for a global isometry.
The frame identity above therefore establishes a covariance interpretation. Identical transition laws, invariant measures, and rates require the full generator identity.
Predict the executed thermostat increment
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Kinetic Evolution and Stationary Measures#
A velocity kick does not immediately move position. In the equations \(dx=v\,dt\), the position path has finite variation between jumps. This small observation settles the Stratonovich correction when the noise coefficient depends only on position: the noise never differentiates that coefficient in a noisy direction.
There is a separate question about equilibrium. Suppose we want particles to sample volume measured by the new ruler, weighted by a potential. We can construct dynamics with exactly that stationary density and verify it by a flux calculation. Comparing those dynamics with the algorithm then tells us which terms would need to agree.
Stratonovich and Itô forms of velocity noise#
Proposition 224 (Conversion for the kinetic equations)
Between jumps, write the specified continuous model as
Let \(B_\ell(S)\) be the full phase-space noise vector fields. The equivalent Itô drift is
If all noise coefficients depend only on positions, this correction is zero. For noise field \(\ell=(i,\alpha)\) acting only on velocity \(v_i\), its velocity component is
Dependence on the complete swarm is included by the full vector-field formula when the noise fields have more general support.
Proof
For a smooth test function \(f\), the Stratonovich generator is
Every \(B_\ell\) has zero position component. It therefore differentiates a position-only coefficient to zero. Taking the velocity components gives the displayed formula. This is the calculation used in Lemma 248.
The implemented BAOAB step freezes coefficients at its prescribed substeps. Its finite-step noise covariance and its transition law are the operational objects. A continuous SDE approximation requires the consistency conditions in Remark 201 and Quantitative Error Bounds.
Geometric drift and exact Gibbs constructions#
Lemma 274 (Geometric drift for an overdamped comparison model)
Let \(g\) be a \(C^1\) positive definite spatial metric, \(D=g^{-1}\), \(J=\sqrt{\det g}\), and \(T>0\). The Itô position SDE
has generator
If \(Z=\int J e^{-U/T}\,dx<\infty\), and the conservative realization has a generator core with boundary conditions and decay that justify the corresponding integration by parts, then
is invariant and the generator is symmetric in \(L^2(\mu)\).
If \(g\succeq mI\) and \(\sup_{|e|=1}\|\partial_e g\|_{\mathrm{op}}\leq K_3\), the geometric part \(b_{\mathrm{geo}}^i=TJ^{-1}\partial_j(JD^{ij})\) satisfies
This lemma specifies an overdamped comparison model. Its drift is not automatically a term of the Fractal Gas velocity equation.
Proof
Expand the divergence in the displayed generator. The coefficient of \(\partial_i f\) is the stated drift. For smooth compactly supported \(f,h\), or functions satisfying the stated boundary conditions,
Symmetry follows, and \(h=1\) gives stationarity for the conservative realization.
Jacobi’s determinant formula gives
Thus \(|\nabla\log J|\leq dK_3/(2m)\) and \(\|\partial_eD\|_{\mathrm{op}}\leq K_3/m^2\). Since \(\operatorname{div}D=\sum_j(\partial_jD)e_j\),
Proposition 225 (A kinetic model with the Riemannian Gibbs marginal)
Let \(g\) be a \(C^2\) spatial metric and define the Hamiltonian on position and canonical momentum by
Consider the specified comparison model
When its Gibbs density is normalizable and the dynamics are conservative with a generator core and boundary behavior allowing the integrations below,
is invariant. Its position marginal is
In velocity coordinates \(v=g^{-1}p\), the same model has
The agreement of the noise shape with \(g^{-1}\) is one part of this specification. The metric force, geodesic drift, and thermostat scaling are the other parts.
Proof
The Hamiltonian vector field \((\nabla_pE,-\nabla_xE)\) has zero divergence and annihilates \(E\). Hence its transport preserves \(e^{-E/T}\,dx\,dp\). At fixed \(x\), the momentum Ornstein–Uhlenbeck part has probability flux
because \(\nabla_pE=g^{-1}p\). The sum therefore preserves the displayed density, subject to the conservative realization in the statement.
The Gaussian integral is
which proves the marginal formula.
To transform the equation, write \(p_k=g_{kj}v^j\). Since \(x\) has finite variation,
Also \(\partial_kE=\partial_kU-\tfrac12(\partial_kg_{ij})v^iv^j\), where the derivative on the left holds \(p\) fixed. Multiplication by \(g^{ak}\) and symmetrization in \(i,j\) gives the Christoffel term
The noise becomes \(g^{-1}g^{1/2}=g^{-1/2}\) and has zero Stratonovich correction by Proposition 224. Finally, \(dx\,dp=(\det g)\,dx\,dv\), so the velocity-coordinate density is
Its velocity integral gives the same spatial marginal.
The determinant in the last calculation has a concrete origin. At a fixed position, we integrate a Gaussian over momenta. The ellipsoid of allowed momenta has a volume proportional to \(\sqrt{\det g}\). That is why this particular kinetic model produces Riemannian volume in its spatial marginal.
Changing only the noise ellipsoid leaves the force and transport terms to be checked. The calculation above makes that check possible: compare each coefficient with the stated model, then include the cloning and killing terms.
Riemannian Volume and Integration#
Remark 250 (Stationary density, survival conditioning, and volume)
The exact Gibbs identities above apply to their specified conservative generators. The Fractal Gas generator includes its actual force, velocity diffusion, alignment, cloning, and killing terms. A proposed density for that system must solve the resulting stationary equation or the killed left-eigenmeasure equation.
For example, if \(\mu\) is invariant for \(L\) and soft killing has rate \(\kappa(x)\), then for \(A=L-\kappa\),
Thus the same \(\mu\) solves \(\mu A=-\alpha\mu\) precisely when \(\kappa=\alpha\) almost everywhere under \(\mu\). Boundary killing requires its own generator domain. For a general killed process a QSD \(\nu\) satisfies \(\nu Q_t=e^{-\alpha t}\nu\). The invariant law of its Doob transform is the eigenfunction-weighted law \(\eta\nu\), normalized by \(\nu(\eta)=1\), as proved in Proposition 180.
The local QSD density estimates in Mass, Hellinger, and Transport Convergence and the confining estimates do not identify that density with an exact Gibbs expression.
Volume density and the reconstructed field#
Definition 671 (Riemannian volume element)
For a positive definite spatial metric \(g(x)\) on a coordinate domain,
For an oriented \(k\)-dimensional surface parametrized by \(\Phi:U\subset\mathbb R^k\to\mathbb R^d\), its induced volume density is
In particular, a constant \(g\) changes the coordinate volume by a constant factor while its Riemannian curvature remains zero.
Remark 251 (The metric field used in a volume integral)
A volume integral uses a spatial field, not only a list of matrices at walker locations. One may specify a frozen-state query field, an identified limiting field from Definition 670, or a reconstruction from the data. Its regularity and positive margin are part of the specification.
If reconstructed Hessians converge uniformly to \(H\) and share a uniform positive margin, the inverse and inverse-square-root bounds above give uniform convergence of their covariance shapes. For the clipped branch, uniform Hessian convergence gives uniform metric convergence by nonexpansiveness. Propagation of chaos controls particle laws; identifying these derivative fields requires the regularity and consistency conditions in Corollary 116.
When the metric and the samples come from the same swarm, their dependence must be retained. The same-sample coverage estimate Lemma 282 uses a pathwise upper metric bound. An importance-sampling proof instead needs the joint conditions stated below; conditioning on a fitted metric does not make its training samples independent.
Lemma 275 (Volume stability under a metric perturbation)
Suppose \(g\succeq mI\) and \(\|\widetilde g-g\|_{\mathrm{op}}\leq\eta<m\). For any rank-\(k\) matrix \(B\),
The same inequalities hold for integrated volumes of a fixed parametrized surface when the bounds hold along that surface.
Proof
The metric bound gives
Congruence by \(B\) preserves these inequalities. Conjugate further by \((B^TgB)^{-1/2}\); the resulting \(k\) eigenvalues lie in \([1-\eta/m,1+\eta/m]\). Multiply them, take square roots, and integrate the pointwise inequalities for the final assertion.
Fan triangulation for areas#
A loop of points does not select a unique spanning surface. We can select one by joining every edge to a chosen center. The resulting collection of flat triangles is a concrete object whose area we can measure.
For a planar polygon that is star-shaped about the center, the triangles fill the polygon once. For a nonplanar loop, the fan is a chosen spanning surface. If triangles overlap, summing their unsigned areas counts the overlap with multiplicity. These are geometric choices made before the metric enters the calculation.
Algorithm 9 (Fan quadrature for Riemannian area)
Input: A cycle \(z_0,\ldots,z_{n-1},z_n=z_0\) in a coordinate domain, a metric field \(g\), and the fan surface formed from \(z_c=n^{-1}\sum_{i=0}^{n-1}z_i\). Assume the triangles lie in the domain. For the area of a planar enclosed region, require that this fan triangulates that region without overlapping interiors.
Construction: Evaluate \(g_c=g(z_c)\). For \(B_i=[z_i-z_c,\ z_{i+1}-z_c]\), set
Return \(A_g^{(c)}=\sum_i A_i^{(c)}\).
Interpretation: This is the exact area of the selected fan for the constant metric \(g_c\). For a variable metric it is a quadrature approximation to the sum of the triangle areas.
Cost: One metric evaluation and \(O(nd^2)\) arithmetic operations. If \(g\succeq mI\) and \(\|g(x)-g(z_c)\|_{\mathrm{op}}\leq K_1h\) on the fan, with \(h=\max_i|z_i-z_c|\) and \(K_1h<m\), then
Thus a family with \(A_g^{(c)}=O(h^2)\) has absolute error \(O(h^3)\). The constants retain the metric bounds and the total fan area.
Proof
The linear parametrization of each triangle has constant derivative \(B_i\). Its induced constant-metric area is the square root of its Gram determinant times the area \(1/2\) of the standard two-simplex. The perturbation estimate is Lemma 275 with \(k=2\); its two factors are exactly \(1\pm K_1h/m\). Sum over triangles.
The following implementation evaluates that specified constant-metric fan quadrature. Coordinates and the metric must use compatible units.
from collections.abc import Callable
import numpy as np
def compute_riemannian_area_fan(
vertices: np.ndarray, # [n, d], without a repeated endpoint
metric: Callable[[np.ndarray], np.ndarray],
) -> float:
"""Return the unsigned fan area with the metric frozen at its center.
The supplied cycle specifies a chosen fan; overlapping triangles are
counted with multiplicity. The metric must be symmetric positive definite.
"""
if vertices.ndim != 2 or vertices.shape[0] < 3 or vertices.shape[1] < 2:
raise ValueError("Expected at least three vertices in dimension >= 2.")
center = vertices.mean(axis=0)
g_center = np.asarray(metric(center))
if not np.allclose(g_center, g_center.T):
raise ValueError("The metric must be symmetric.")
np.linalg.cholesky(g_center)
edges = vertices - center
next_edges = np.roll(edges, -1, axis=0)
g11 = np.einsum("ni,ij,nj->n", edges, g_center, edges)
g22 = np.einsum("ni,ij,nj->n", next_edges, g_center, next_edges)
g12 = np.einsum("ni,ij,nj->n", edges, g_center, next_edges)
# Exact Gram determinants are nonnegative; clipping handles roundoff.
return float(0.5 * np.sqrt(np.maximum(g11 * g22 - g12**2, 0.0)).sum())
Tetrahedral volumes and quadrature order#
Definition 672 (Frozen-metric tetrahedron volume)
For vertices \(z_0,z_1,z_2,z_3\in\mathbb R^d\), \(d\geq3\), let
The frozen-metric volume is
The exact volume for a variable metric on the affine tetrahedron is
The first formula is exact when \(g\) is constant along the tetrahedron.
Lemma 276 (Barycentric metric quadrature)
Let \(T\) be a nondegenerate affine \(k\)-simplex of diameter \(h\), barycenter \(x_c\), and Euclidean \(k\)-volume \(V_E\). Suppose on \(T\)
with \(g\) of class \(C^2\) on a neighborhood of \(T\). Then
In particular, if \(V_E=O(h^k)\) with uniform coefficient bounds, the absolute error is \(O(h^{k+2})\). For tetrahedra this is \(O(h^5)\). Freezing every triangle at a common fan center instead of its own barycenter uses the first-order estimate above.
Proof
Choose an orthonormal matrix \(Q\in\mathbb R^{d\times k}\) spanning the simplex plane, and put \(G(x)=Q^Tg(x)Q\), \(w(x)=\sqrt{\det G(x)}\). Along a unit direction in that plane, Jacobi’s formula gives
Using \(G\succeq mI\) and \(w\leq M^{k/2}\) bounds \(|w''|\) by the coefficient in brackets times \(M^{k/2}\). Taylor expansion about \(x_c\) has a linear term whose integral over the simplex vanishes, since
The remaining term is at most one half the second-derivative bound times \(|x-x_c|^2\). Integrating and using \(|x-x_c|\leq h\) proves the result.
The tetrahedron implementation uses the same frozen-metric convention.
def compute_riemannian_volume_tetrahedron(
vertices: np.ndarray, # [4, d]
metric: Callable[[np.ndarray], np.ndarray],
) -> float:
"""Return barycentric metric quadrature for an affine tetrahedron."""
if vertices.ndim != 2 or vertices.shape[0] != 4 or vertices.shape[1] < 3:
raise ValueError("Expected four vertices in dimension >= 3.")
g_center = np.asarray(metric(vertices.mean(axis=0)))
if not np.allclose(g_center, g_center.T):
raise ValueError("The metric must be symmetric.")
np.linalg.cholesky(g_center)
edges = vertices[1:] - vertices[0] # [3, d]
gram = edges @ g_center @ edges.T
return float(np.sqrt(max(float(np.linalg.det(gram)), 0.0)) / 6.0)
Operational volume weights#
Proposition 226 (Volume weights and a frozen neighbor kernel)
For a symmetric positive definite recorded \(\Sigma_{\mathrm{step}}=c_2g^{-1/2}\) with \(c_2>0\),
The function compute_riemannian_volume_weights applies this determinant
factor to recorded Euclidean Voronoi volumes, with its numerical clamps and
interior-cell mask. Without active clamps, a retained cell has weight
This is frozen-site quadrature. Its approximation error is controlled by variation of \(J\) across that cell.
For a frozen symmetric nonnegative neighbor matrix \(w_{ij}=w_{ji}\), positive retained weights \(a_i\), and positive row sums \(Z_i=\sum_jw_{ij}a_j\), define
Then
is invariant and satisfies detailed balance for this neighbor chain. The identity applies before any additional degree, threshold, or state-dependent modifications to the weights.
Proof
Take determinants in \(\Sigma_{\mathrm{step}}=c_2g^{-1/2}\) to obtain the first identity. For the chain,
Summation over \(i\) yields invariance. Thus even this frozen chain has stationary weights \(a_iZ_i\), which reduce to weights proportional to \(a_i\) when \(Z_i\) is constant. Its detailed-balance identity concerns the specified neighbor chain; the full swarm has additional dynamics.
Monte Carlo integration with an identified sampling law#
Suppose you sample twice as often in one region as in another. An integral must compensate for that sampling frequency. The compensation is the reciprocal of the actual density.
Riemannian volume changes the quantity being integrated: its density is \(J\). It does not eliminate the need to know how the samples were drawn. If a Gibbs formula has been proved, the determinants cancel in the importance weight. If only bounds on the density are available, retain the density in the formula and use those bounds to control the error.
Proposition 227 (Normalized importance sampling and variance)
Fix a spatial metric \(g\) and a normalized coordinate sampling density \(p\). For \(p>0\) almost everywhere on the support of \(fJ\), define
Assume \(\int|f|J\,dx<\infty\).
If each \(X_i\) has marginal density \(p\), then \(\mathbb E\widehat I_N=I[f]\). If the samples are independent and \(\int f^2J^2/p\,dx<\infty\), then
\[ \mathbb E|\widehat I_N-I[f]|^2 =\frac1N\left(\int\frac{f^2J^2}{p}\,dx-I[f]^2\right). \]If the actual joint law \(\pi_N\) has these marginals and satisfies the full-gradient LSI
\[ \operatorname{Ent}_{\pi_N}(F^2) \leq2C_*\int\sum_i (|\nabla_{x_i}F|^2+|\nabla_{v_i}F|^2)\,d\pi_N, \]then for a fixed \(L\)-Lipschitz position weight \(w\),
\[ \operatorname{Var}_{\pi_N}(\widehat I_N)\leq C_*L^2/N. \]The sufficient joint-law criteria for a constant \(C_*\) uniform in \(N\) are proved in Corollary 94. For a law mixing discrete alive-status strata, the status entropy and variance terms must also be included.
More generally, let \(\pi_N^x\) be the joint position law. For bounded \(|w|\leq B\) and \(H_N=D_{\mathrm{KL}}(\pi_N^x\|p^{\otimes N})<\infty\),
\[ \operatorname{Var}_{\pi_N}(\widehat I_N) \leq\mathbb E_{\pi_N}|\widehat I_N-I[f]|^2 \leq \frac{4B^2}{N}\left(H_N+\frac12\log2\right). \]This is the established bound in Theorem 245. Its second-moment estimate also controls bias when the actual one-particle marginals differ from \(p\). A joint phase-space entropy bound supplies this positional bound by relative-entropy contraction under marginalization, when its reference has position marginal \(p^{\otimes N}\).
If an exact identity \(p=Z^{-1}J e^{-U/T}\) has been proved, then \(w=Z f e^{U/T}\). Its use for an absolute integral requires the normalizing constant \(Z\) and the stated integrability conditions. A QSD can be used when its actual normalized marginal density supplies \(p\).
Proof
Marginal integration gives
For independent samples, cross-covariances vanish, leaving \(\operatorname{Var}(w)/N\), with the displayed second moment.
The LSI implies the Poincaré inequality with constant \(C_*\): substitute \(F=1+\varepsilon h\), expand both sides through \(\varepsilon^2\), and divide by \(2\varepsilon^2\). For the empirical weight,
This proves the second assertion. The third is the bounded-observable entropy estimate cited in its statement. The Gibbs simplification is substitution into the exact normalized weight.
Remark 252 (Dependence, bias, and fitted geometry)
For any square-integrable sequence, including correlated episodes,
A temporal rate therefore needs a covariance or mixing estimate for that sequence. If \(p_i\) is the actual \(i\)th marginal, the bias of the fixed reference weight is \(N^{-1}\sum_i\int w(p_i-p)\,dx\). For Lipschitz \(w\), it is at most \(L N^{-1}\sum_iW_1(p_i,p)\), using the finite-particle estimates of Quantitative Error Bounds.
If a fitted metric \(\widehat g_N\) is computed from the same sample, \(\widehat I_N\) becomes a function of all coordinates through the metric. The Poincaré bound uses the complete gradient of that function, including those additional derivatives. Alternatively, a deterministic reference metric can be compared pathwise: if \(|\widehat J_N/J-1|\leq a_N\) on the sample locations, then
Metric error bounds supply \(a_N\) through Lemma 275. This comparison retains the dependence between fitted geometry and samples.
The exponent \(N^{-1/2}\) in a root-mean-square bound has no explicit dimension dependence. Its second moment, Lipschitz constant, and LSI constant can depend on dimension; each must be bounded for a uniform claim.
Coordinate area versus geometric area
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Hypocoercivity in Anisotropic Geometry#
Noise acts directly on velocity. Position feels its effect through \(dx=v\,dt\). A useful distance between two trajectories therefore includes a cross term between their position and velocity differences. That term records the transfer between the two parts of the motion.
The covariance bounds tell us how large the noise terms can be. A decay rate also needs restoring forces and control of the other dynamics. We can see every contribution by coupling two copies with the same noise. For a quadratic confining potential, the calculation can be completed explicitly. For nonconvex confinement, the established entropy proof supplies another complete route.
Definition 673 (Hypocoercive quadratic norm)
Use nondimensional position and velocity coordinates. For \(\lambda_v>0\) and \(|b|<2\sqrt{\lambda_v}\), define
Its matrix and eigenvalues are
Thus \(0<q_-\leq q_+\) and
For swarms with matched labels on a common state space, use \(N^{-1}\sum_i\|(\delta x_i,\delta v_i)\|_{\mathrm{hyp}}^2\). A permutation-invariant transportation distance additionally minimizes over the declared matching or coupling.
Theorem 327 (Quantified contraction for an anisotropic coupling)
Let two copies of a conservative continuous diffusion-jump process admit a coupling on a common Euclidean state space. Between coupled jumps, use the same Brownian motion:
For a constant positive definite matrix \(Q\), put \(\Delta=S-S'\) and \(\mathcal H(S,S')=\Delta^TQ\Delta\). Let \(\mathcal J_c\) denote the jump part of the coupled generator. Suppose the following bounds hold:
If \(r=r_b-r_\sigma-r_J>0\), and localization is justified by nonexplosion and the required moment bounds, then
Consequently the squared transportation cost induced by \(Q\) contracts by \(e^{-rt}\). The conclusion is uniform in \(N\) when the displayed constants and the norm comparisons to the normalized swarm distance are uniform.
Application to an actual swarm requires the actual coupled force, alignment, noise, and jump terms in these inequalities. Survival conditioning has its own normalized evolution and must be included before applying a contraction estimate.
Proof
Itô’s formula gives the continuous part of the coupled generator:
There is no state-independent noise trace in this difference equation: the coupled covariance is the square of the difference of the noise matrices. Add \(\mathcal J_c\mathcal H\) to obtain
Apply Dynkin’s formula to \(e^{rt}\mathcal H\) up to localizing stopping times. The assumed moment bounds justify removal of the stopping times, yielding the expectation estimate. Start from any coupling of the initial laws and minimize its initial cost to obtain the transportation bound.
Proposition 228 (Explicit confining example and a multiplicative-noise margin)
For the kinetic model in nondimensional coordinates
define
For constant \(\Sigma\), synchronous coupling gives
If instead \(\|\Sigma(z)-\Sigma(z')\|_F\leq L_\Sigma|z-z'|\) and \(r_0L_\Sigma^2<1\), the rate is at least
The same rate holds for the normalized sum of independent copies. The general coupling theorem quantifies additional interaction and jump contributions.
Proof
For \(A=\begin{pmatrix}0&1\\-a&-\gamma\end{pmatrix}\), direct multiplication gives
where \(Q_0\) is the displayed \(2\times2\) block. Since \(p>0\), \(Q_0\) is positive definite. The deterministic difference therefore satisfies
For constant noise its difference under synchronous coupling vanishes. For variable noise, its contribution is \(r_0\|\Sigma(z)-\Sigma(z')\|_F^2\), at most \(r_0L_\Sigma^2(|\delta x|^2+|\delta v|^2)\). This gives the second rate. Dividing \(Q_0\) by \(p\) produces the normalization of Definition 673, with \(b=2q/p\) and \(\lambda_v=r_0/p\).
Entropy convergence under the established joint-law criteria#
Proposition 229 (The analytical entropy route)
Let the particular continuous joint law \(\pi_N\) satisfy the full-gradient LSI of Corollary 94 with constant \(C_*\). For a density \(h\) relative to \(\pi_N\), write
and for a positive definite matrix \(G_N\preceq g_+I\) let
If the complete normalized evolution satisfies \(\dot\Phi_{G_N}(h_t)\leq-\delta I(h_t)\) with \(\delta>0\), then
For the constant-diffusion kinetic Gibbs model with bounded potential Hessian, Theorem 285 proves the derivative bound, with its explicit matrix \(G\) and constant \(\eta\), giving
In particular, Corollary 95 supplies a uniform rate for its nonconvex product potentials. For the complete geometric evolution, the derivative and perturbation margins are those of Proposition 186 and Corollary 105.
Proof
Apply the LSI to \(\sqrt h\) to obtain \(H_{\pi_N}(h)\leq(C_*/2)I(h)\). Since \(I_{G_N}(h)\leq g_+I(h)\),
Substitute this inequality into the assumed derivative bound and apply Grönwall. The referenced kinetic theorem establishes the derivative estimate by its full commutator calculation; the nonconvex product corollary supplies its uniform LSI and Hessian bounds. The geometric version retains all coefficient, alignment, cloning, and survival terms.
There are now two complete calculations to use. A coupling calculation compares trajectories. An entropy calculation compares a probability density with a specified target law. Each gives an explicit decay rate when its own terms have been bounded.
The regularization parameter enters those bounds in competing ways. Increasing \(\epsilon_\Sigma\) decreases the noise covariance, and it also decreases the inverse-square-root Lipschitz bound. One change may help an error term while another slows exploration. The displayed contraction or entropy margin decides the outcome for the model being studied.
Measure interacting relaxation
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Geometry, Sampling, and Convergence#
The inverse diffusion shape supplies the metric used by the subsequent spatial constructions. Matrix bounds give lengths and volumes with explicit constants. Generator calculations identify stationary laws; joint-law inequalities give sampling errors; dissipative estimates give dynamical rates.
Remark 253 (Hypotheses carried into continuum and dynamical applications)
Applications of this chapter retain the following data.
Metric identification. Specify the smooth shifted field or the recorded clipped field, the finite-step noise prefactor, and the reconstruction between observed points. The covariance identity is Proposition 222 for the corresponding implementation branch.
Regularity and geometry. Use two-sided metric bounds and the derivatives required by the construction. Spatial \(C^1\) regularity suffices for the stated first-order bounds; classical curvature uses \(C^2\). A continuum manifold and faithful causal sampling use the independent geometric and sampling hypotheses of Corollary 116.
Sampling law. Exact Gibbs weights use a proved generator identity. A QSD uses its killed eigenmeasure equation and normalized marginal. Empirical variance uses independence, the stated joint LSI, or the bounded-observable relative-entropy estimate. Fitted geometry retains its dependence on the same samples.
Dynamics. Covariance ellipticity, moment confinement, QSD convergence, and entropy dissipation are separate estimates. Theorem 327 applies to the stated conservative coupling. Killed-kernel convergence is supplied by Convergence, Survival, and Parameter Dependence; actual-law LSI and entropy rates use the precise criteria of Chapters 15 and 17.
The positive metric here is spatial. The Lorentzian slab construction, causal reachability, and proper time are specified in Scutoid Spacetime: Moving Cells and Neighbor Changes. They are not inferred from a covariance determinant.
References#
The principal proofs used in this chapter are:
C³ Regularity of the Fractal Gas Fitness: sampled and expected fitness fields, derivative bounds, and implementation branches.
Smooth and Analytic Regularity of the Fractal Gas Fitness: higher derivatives under its complete regularity hypotheses.
Exchangeability, Empirical Laws, and Functional Inequalities and Quantitative Error Bounds: joint-law fluctuations and finite-particle observable errors.
Logarithmic Sobolev inequalities and entropy convergence: proved LSI criteria, nonconvex kinetic hypocoercivity, and the Doob invariant measure.
Continuum Limits: Hypotheses and Consistency and The Geometric Gas: Regularity, Stability, and Entropy: conditional continuum consistency and the complete geometric evolution.
The determinant, Gaussian marginal, matrix perturbation, and simplex quadrature arguments are proved directly above.