Related papers: Fractional Diffusion Maps
Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of dimension $\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the random operators $$…
Kernel approximation using randomized feature maps has recently gained a lot of interest. In this work, we identify that previous approaches for polynomial kernel approximation create maps that are rank deficient, and therefore do not…
We present incomplete gamma kernels, a generalization of Locally Optimal Projection (LOP) operators. In particular, we reveal the relation of the classical localized $ L_1 $ estimator, used in the LOP operator for point cloud denoising, to…
A data-driven framework is presented, that enables the prediction of quantities, either observations or parameters, given sufficient partial data. The framework is illustrated via a computational model of the deposition of Cu in a Chemical…
Classical and non classical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincar\'e inequality. This leads to Heat…
In this article we study a non-local diffusion problem that involves three different fractional Laplacian operators acting on two domains. Each domain has an associated operator that governs the diffusion on it, and the third operator…
Kernel methods form a powerful, versatile, and theoretically-grounded unifying framework to solve nonlinear problems in signal processing and machine learning. The standard approach relies on the kernel trick to perform pairwise evaluations…
This paper explores the application diffusion maps as graph shift operators in understanding the underlying geometry of graph signals. The study evaluates the improvements in graph learning when using diffusion map generated filters to the…
Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper…
Let $D$ be an open set of $\mathbb{R}^d$, $\alpha\in (0, 2)$ and let $\mathcal{L}_{\alpha}^D$ be the generator of the censored $\alpha$-stable process in $D$. In this paper, we establish sharp two-sided heat kernel estimates for…
Computational fluid dynamics (CFD) is a powerful tool for modeling turbulent flow and is commonly used for urban microclimate simulations. However, traditional CFD methods are computationally intensive, requiring substantial hardware…
Current Lagrangian (particle-tracking) algorithms used to simulate diffusion-reaction equations must employ a certain number of particles to properly emulate the system dynamics---particularly for imperfectly-mixed systems. The number of…
Diffusion is a fundamental graph procedure and has been a basic building block in a wide range of theoretical and empirical applications such as graph partitioning and semi-supervised learning on graphs. In this paper, we study…
In this paper, we proceed to study the nonlocal diffusion problem proposed by Li and Wang [8], where the left boundary is fixed, while the right boundary is a nonlocal free boundary. We first give some accurate estimates on the longtime…
We consider the problem of making nonparametric inference in a class of multi-dimensional diffusions in divergence form, from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of…
In-network distributed estimation of sparse parameter vectors via diffusion LMS strategies has been studied and investigated in recent years. In all the existing works, some convex regularization approach has been used at each node of the…
We introduce a formalism of fractional diffusion on networks based on a fractional Laplacian matrix that can be constructed directly from the eigenvalues and eigenvectors of the Laplacian matrix. This fractional approach allows random walks…
We show that locally bounded, local weak solutions to certain nonlocal, nonlinear diffusion equations modeled on the fractional porous media and fast diffusion equations given by \begin{align*} \partial_t u + (-\Delta)^s(|u|^{m-1}u) = 0…
Asymptotic expansions of heat kernels and heat traces of Schr\"odinger operators on non-compact spaces are rarely explored, and even for cases as simple as $\mathbb{C}^n$ with (quasi-homogeneous) polynomials potentials, it's already very…
In this work, we propose a simple kernel ridge regression (KRR) framework with a dynamic-aware validation strategy for long-term prediction of complex dynamical systems. By employing a data-driven kernel derived from diffusion maps, the…