arXiv research

New blurring diffusion models bridge heat dissipation and denoising.

Study shows how heat leaks from material sets in low diffusivity scenarios.

A new method uses heat diffusion to efficiently solve combinatorial optimization problems.

New model generates images by reversing heat equation, revealing disentanglement.

New method recovers curvature from heat diffusion data.

Paper develops novel privacy mechanism for Riemannian manifold data using geometric analysis and heat diffusion.

We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or h…

Establishes a link between heat diffusion and manifold distances in data.

A new graph generator uses heat diffusion on graph Laplacians to create new graph structures.

Study on uniquely determining thermal properties from boundary temperature and heat flux measurements.

Let M be a complete Riemannian manifold with a free cocompact Z^k-action. Let k(t,x,y) be the heat kernel on M. We compute the asymptotics of k(t,x,y) in the limit in which t goes to infinity and d(x,y) is comparable to sqrt{t}. We show that in this limit, the heat diffusion is governed by an effective Euclidean metric…

Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.

Generative model on manifolds reduces divergence computation and improves scalability.

Graph neural networks are explained through heat diffusion analogy.

A new clustering algorithm fuses heat diffusion and turning angle for robustness.

Study quantum diffusion on spectral triples and spinor bundles.

A new method embeds data using Gaussian processes based on the heat kernel.

Researchers calculate entropy of heat kernel on manifolds for very small times.

We study the fast diffusion equation (FDE) with a linear forcing term under the Ricci flow on complete manifolds with bounded curvature and nonnegative curvature operator. We prove Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE. In addition, we use similar meth…

Efficient diffusion model for symmetric manifolds reduces training and computation costs.

We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We describe the possible small time heat kernel expansion on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the dri…

Graph diffusion processes approximate manifold heat semigroups using graph transition matrices.

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $Ω_λ$ almost fully contains a ball of radius $\sim \frac{1}{\sqrtλ}$. …

We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov-Fokker-Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants o…

We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian…

In this note we present some gradient estimates for the diffusion equation $\partial_t u=Δu-\nabla φ\cdot \nabla u$ 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.

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…

New method uses entropy dissipation to prove isoperimetric inequalities.

Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Emery's curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distribut…

The paper studies harmonic map heat flow stability and decay rates.

We provide a short proof for the theorem that two compact Riemannian manifolds are isomorphic if and only there exists an order isomorphism which intertwines between the heat semigroups on the manifolds.

New method learns discrete graph diffusion via free-energy gradient flows.

Study mass transport in low-diffusivity using Lagrangian coordinates.

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on an Euclidean space and a compact manifold. We study the "cut locus" case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assum…

In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particu…

Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector machine compared to other competing similarity measures. Specifically, the idea of usi…

We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher diffusion driven algorithm and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L. C. Evans in the…

In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove …

Generative model for hypergraphs captures complex interactions without pairwise reductions.

New method learns under latent group sparsity using network dynamics.

The paper analyzes the score field of diffusion models using Burgers dynamics.

We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…

The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.

Heat flow on lens spaces settles into Morse functions with four critical points.

In the compagnion paper [Marginal density expansions for diffusions and stochastic volatility, part I] we discussed density expansions for multidimensional diffusions $(X^1,...,X^d)$, at fixed time $T$ and projected to their first $l$ coordinates, in the small noise regime. Global conditions were found which replace th…

Study decision boundaries using heat diffusion and probabilistic techniques.

EBMs trained on discrete data using heat equations on graph structures.

New discretization scheme for Wasserstein gradient flows using Schrödinger bridges.