Related papers: Fractional Diffusion Maps

Invariance to nuisance transformations is one of the desirable properties of effective representations. We consider transformations that form a \emph{group} and propose an approach based on kernel methods to derive local group invariant…

Kernel approximation via nonlinear random feature maps is widely used in speeding up kernel machines. There are two main challenges for the conventional kernel approximation methods. First, before performing kernel approximation, a good…

Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph associated with it changes depending on…

We recover the Riemannian gradient of a given function defined on interior points of a Riemannian submanifold in the Euclidean space based on a sample of function evaluations at points in the submanifold. This approach is based on the…

We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the…

Diffusion maps approximate the generator of Langevin dynamics from simulation data. They afford a means of identifying the slowly-evolving principal modes of high-dimensional molecular systems. When combined with a biasing mechanism,…

For $d\ge 2$ and $0<\beta<\alpha<2$, consider a family of non-local operators $\mathcal{L}^{b}=\Delta^{\alpha/2}+\mathcal{S}^{b}$ on $\mathbb{R}^d$, where $$ \mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta)\int_{ \{z\in…

Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves…

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously…

A novel, fast and practical way of enhancing images is introduced in this paper. Our approach builds on Laplacian operators of well-known edge-aware kernels, such as bilateral and nonlocal means, and extends these filter's capabilities to…

We give a short proof of a strong version of the short time asymptotic expansion of heat kernels associated to Laplace type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the…

Diffusion Maps framework is a kernel based method for manifold learning and data analysis that defines diffusion similarities by imposing a Markovian process on the given dataset. Analysis by this process uncovers the intrinsic geometric…

To generate data from trained diffusion models, most inference algorithms, such as DDPM, DDIM, and other variants, rely on discretizing the reverse SDEs or their equivalent ODEs. In this paper, we view such approaches as decomposing the…

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These type of operators are used to model anomalous diffusion and, for a special…

Finding a suitable density function is essential for density-based clustering algorithms such as DBSCAN and DPC. A naive density corresponding to the indicator function of a unit $d$-dimensional Euclidean ball is commonly used in these…

We introduce the concept of Hypoelliptic Diffusion Maps (HDM), a framework generalizing Diffusion Maps in the context of manifold learning and dimensionality reduction. Standard non-linear dimensionality reduction methods (e.g., LLE,…

Interpolation and approximation of functionals with conditionally positive definite kernels is considered on sets of centers that are not determining for polynomials. It is shown that polynomial consistency is sufficient in order to define…

We derive estimates of the derivatives of the heat kernel on noncompact symmetric spaces and on locally symmetric spaces. Applying these estimates we study the $L^{p}$-boundedness of Littlewood-Paley-Stein operators and the Laplacian of the…

We consider the problem of non-rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to…

We study the heat kernel for a Laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group $G\times U(1)$ over a Riemannian manifold $M$ without boundary. The total connection on…