Reconstructs Riemannian geometry from diffusion properties.
problem Recovering Riemannian geometry from diffusion data.
method Intrinsic reconstruction from diffusion semigroup and calculus.
result Reveals Riemannian structure from diffusion properties.
Study large deviations for hypoelliptic diffusion on sub-Riemannian manifolds.
problem Large deviations for hypoelliptic diffusion measures on sub-Riemannian manifolds.
method Rough path theory and manifold-valued Malliavin calculus.
result Proved a large deviation principle for pinned hypoelliptic diffusion measures.
New method synthesizes data on curved spaces for better interpolation.
problem Synthesizing data on curved spaces for better interpolation.
method Riemannian Diffusion Schrödinger Bridge
result Generalizes Diffusion Schrödinger Bridge to curved spaces for better interpolation.
New method for sampling diffusion bridges on sub-Riemannian manifolds.
problem Sampling conditioned diffusion processes on sub-Riemannian manifolds is challenging.
method Score matching for machine learning, adapted to non-holonomic frames.
result Demonstrated method works on Heisenberg group and other sub-Riemannian manifolds.
We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and t…
Develops Stein's method for Riemannian manifolds using diffusion.
problem Bounding integral metrics on probability measures on Riemannian manifolds.
method Exploits the relationship between diffusion generators and Stein operators to derive Stein factors.
result Derives curvature-dependent Stein factors that generalize existing results for Euclidean spaces.
Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.
problem Improving convergence rate of Riemannian stochastic gradient descent.
method Using stochastic differential geometry, the paper shows RSGD can be approximated by the Riemannian stochastic modified flow (RSMF).
result RSGD can be approximated by the solution to the RSMF driven by an infinite-dimensional Wiener process, increasing the order of approximation.
This work proposes a geometric approach to equivariant message passing on Riemannian manifolds.
problem Efficiently processing data on Riemannian manifolds with equivariance.
method Geometric insight into equivariant message passing on Riemannian manifolds, using an equivariant embedding and diffusion process.
result A new class of equivariant GNNs on Riemannian manifolds.
Improved diffusion models for manifold learning.
problem Learning distributions on general manifolds with geometric complexity.
method Revised approximations for score matching on symmetric spaces.
result Improved performance and scalability to high dimensions.
The paper develops stochastic methods on geometric spaces for transformations.
problem Existence and uniqueness of stochastic processes on geometric spaces.
method Stochastic parallel transport and equivariant diffusions on the group of diffeomorphisms.
result Existence and uniqueness of stochastic parallel transport and equivariant diffusions.
SSDMs generate quantum states directly, outperforming classical methods.
problem Generating pure-state quantum representations efficiently.
method Score-based generative model on complex projective manifold.
result SSDMs match target pure-state ensembles by orders of magnitude.
We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-R…
The paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.
problem Analyzing diffusion processes on Riemannian manifolds with weighted metrics.
method Proving a Laplacian comparison theorem and applying it to various geometric and analytic properties of diffusion processes.
result Optimal conditions on m-Bakry-Émery Ricci tensor for various geometric and analytic properties to hold on weighted complete Riemannian manifolds. Paper develops novel privacy mechanism for Riemannian manifold data using geometric analysis and heat diffusion.
problem Privacy-preserving estimation of generalized Frechet mean on Riemannian manifolds.
method Characterizes Renyi divergence via Harnack inequalities, introduces mechanisms based on heat diffusion and Langevin process.
result Proposes mechanisms for nonnegative and general Riemannian manifolds with detailed utility analyses.
Generative model on manifolds reduces divergence computation and improves scalability.
problem Difficulties in modeling data on non-Euclidean spaces due to expensive divergence computation and approximations of heat kernel.
method Riemannian Diffusion Mixture, a principled framework using a mixture of bridge processes.
result Achieves superior performance on diverse manifolds with reduced simulation steps.
Graphs approximate semigroups for diffusion on Riemannian manifolds.
problem Approximating semigroups for diffusion on Riemannian manifolds.
method Discretized approximation using random walks on proximity graphs.
result Quantitative error estimates for convergence of discrete semigroups to continuous semigroups.
In this note we present some gradient estimates for the diffusion equation ∂tu=Δu−∇φ⋅∇u on Riemannian manifolds, where φ is a C^2 function, which generalize estimates of R. Hamilton's and Qi S. Zhang's on the heat equation.
New diffusion models handle constrained domains, improving generative tasks.
problem Diffusion models struggle with manifolds defined by inequality constraints.
method Developed two noising processes: logarithmic barrier metric and reflected Brownian motion.
result Demonstrated practical utility on synthetic and real-world tasks.
RSGMs extend SGMs to Riemannian manifolds for better data modeling.
problem Current SGMs are limited to Euclidean spaces; RSGMs handle Riemannian manifolds.
method RSGMs use a noising stage with a diffusion process and a denoising model approximating the time-reversal of the diffusion on Riemannian manifolds.
result RSGMs improve generative modeling for data on Riemannian manifolds.
The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.
problem Analyzing Riemannian Gaussian distributions on symmetric spaces.
method Analytical computation of marginals using orthogonal and skew orthogonal polynomials, and diffusion kernels.
result Riemannian Gaussian distributions are random matrix types, and their probability density functions can be computed analytically.
Paper uses diffusion models to design electric aircraft quickly.
problem Designing electric aircraft efficiently and accurately.
method Simulation-based inference with hierarchical diffusion models.
result Rediscovers known aircraft design trends and laws.
New framework for diffusion geometry simplifies complex calculations.
problem Challenges in applying calculus and geometry to real data.
method Reformulates calculus and geometry via diffusion processes.
result Improves precision, robustness, and computational efficiency.
Finite-time extinction and smoothing effects in fractional fast diffusion on manifolds.
problem Finite-time extinction and smoothing effects in fractional fast diffusion equations.
method Nonlinear semigroups techniques, weighted Lp spaces, fractional Green function. result Sharp extinction rates and pointwise lower bounds for solutions.
Study phase transitions with prescribed mean curvature in Riemannian manifolds.
problem Understanding phase transitions with prescribed mean curvature in geometric settings.
method Analyzing solutions to inhomogeneous semilinear elliptic PDEs, establishing bounds and asymptotics.
result Established upper and lower bounds for eigenvalues of phase transition problems.
New method recovers curvature from heat diffusion data.
problem Recovering Riemannian curvature from heat diffusion.
method Information-theoretic approach using relative entropy.
result Local curvature determined by heat diffusion.
Geodesic walks converge to Brownian motion on Finsler manifolds.
problem Understanding random walks on Finsler manifolds.
method Analyzing convergence of geodesic random walks to diffusion processes.
result The Brownian motion on a Riemannian metric is a key result.
TDS provides exact samples for conditional distributions in diffusion models.
problem Lack of exact sampling methods for diffusion models.
method Sequential Monte Carlo (SMC) algorithm with twisting technique.
result TDS offers more accurate approximations with fewer particles compared to heuristics.
VDWs enhance graph neural networks for analyzing complex data.
problem Analyzing data on non-Euclidean geometries.
method Incorporating vector diffusion wavelets into geometric graph neural networks.
result VDW-GNNs effectively analyze synthetic and real-world data.
We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: ut=Δφ(up) associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the m-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which…
New method samples from multi-modal distributions on Riemannian manifolds without training.
problem Sampling from multi-modal distributions on Riemannian manifolds is challenging.
method Simulation of a non-equilibrium deterministic dynamics to transport noise toward target distributions.
result Method is entirely training-free and effective on various multi-modal problems.
A new image completion method inspired by brain cells.
problem Image restoration from corrupted data.
method Biologically-inspired sub-Riemannian model with frequency and phase.
result Completion of two-dimensional images using cortical cell responses.
Global solutions and smoothing effects for reaction-diffusion equations on manifolds.
problem Global existence and smoothing effects for reaction-diffusion equations on Riemannian manifolds.
method Functional analytic methods, Sobolev and Poincaré inequalities.
result Existence of global solutions under certain conditions on the manifold.
The paper calculates how random changes affect paths on a complex geometric space.
problem Computing the evolution of paths on a manifold of Riemannian metrics.
method Using diffusion processes and stochastic kinetic energy functional.
result Computed the evolution equation for the Lagrangian.
Optimizes data-driven design problems on implicit manifolds using score functions.
problem Optimizing over implicit low-dimensional manifolds in high-dimensional data.
method Introduces a link function connecting data distribution to manifold operations, enabling efficient optimization.
result Establishes theoretical guarantees for feasibility and optimality of proposed algorithms.
Constructs stochastic processes on sub-Riemannian manifolds using Cartan connections.
problem Developing stochastic processes on sub-Riemannian manifolds.
method Introduces stochastic development using Cartan connections, derives generator, and provides conditions for existence.
result Derives a general expression for the generator of the stochastic process and provides conditions for the existence of a Cartan connection.
Proposes graph neural network layers for manifold-valued graphs.
problem Graphs with features in a Riemannian manifold.
method Diffusion layer and tangent multilayer perceptron.
result Outperforms state-of-the-art networks on Alzheimer's classification.
The paper proves the concavity of entropy power for diffusion equations and applies it to new inequalities.
problem Proving concavity of p-Rényi entropy power for diffusion equations. method Analyzing positive solutions to doubly nonlinear diffusion equations and applying Lp-Sobolev and Gagliardo-Nirenberg inequalities. result New proofs and improvements of Lp-Gagliardo-Nirenberg inequalities. Researchers calculate entropy of heat kernel on manifolds for very small times.
problem Estimating entropy of heat kernel on compact Riemannian manifolds for small times.
method Asymptotic expansion, polynomial expressions in curvature tensor components.
result First three coefficients of entropy expansion computed and expressed as polynomials.
We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with…
New methods estimate curvature, tangent spaces, and dimension of noisy data.
problem Estimating geometric properties of noisy or sparse data.
method Diffusion geometry tools for Riemannian manifold analysis.
result Significantly outperforms existing methods in noisy or sparse data.
By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and…
Riemannian metric matching learns the geometry of high-dimensional datasets using neural networks.
problem Estimating the geometry of high-dimensional datasets from samples
method Riemannian metric matching using neural networks
result Riemannian metric matching rivals or improves k-NN-based diffusion geometry estimators Establishes a link between heat diffusion and manifold distances in data.
problem No theoretical link between diffusion-based manifold learning and geodesic distances.
method Formulates heat geodesic embeddings based on Riemannian geometry.
result Method outperforms state-of-the-art in preserving manifold distances and cluster structure.
By further developing the generalized Γ-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose …
For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditi…
This work interprets diffusion score matching using normalizing flows for better model training and evaluations.
problem Limitations of diffusion score matching when dealing with certain types of distributions.
method The approach involves interpreting the diffusion matrix using normalizing flows to provide better interpretation and usage of diffusion score matching.
result Diffusion score matching is equivalent to the original score matching evaluated in the transformed space defined by the normalizing flow.
Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.
problem Analyzing sub-Riemannian heat kernels and their derivatives on incomplete manifolds.
method Localized asymptotic analysis, focusing on minimizing geodesics and the non-abnormal cut locus.
result Uniform bounds and expansions for heat kernels and their derivatives on compacts, including the diffusion bridge measure.
A new method de-randomizes MCMC dynamics using the Stein operator.
problem Estimating complex target distributions in Bayesian inference.
method De-randomized kernel-based particle samplers that discretize the fiber-gradient Hamiltonian flow.
result GSVGD de-randomizes complex MCMC dynamics, maintaining high sample quality.