Related papers: Fractional Diffusion Maps
Heat-kernel expansion and zeta function regularisation are discussed for Laplace type operators with discrete spectrum in non compact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several…
We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the…
We propose a simple, training-free mechanism which explains the generalization behaviour of diffusion models. By comparing pre-trained diffusion models to their theoretically optimal empirical counterparts, we identify a shared local…
Diffusion models cannot enforce hard constraints, yet applications in the physical sciences demand exact satisfaction of conservation laws, boundary conditions, and observational consistency. In this work, we identify a corrector kernel…
Diffusion maps are a commonly used kernel-based method for manifold learning, which can reveal intrinsic structures in data and embed them in low dimensions. However, as with most kernel methods, its implementation requires a heavy…
In this paper we discuss non-local operators with killing potentials, which may not be in the standard Kato class. We first discuss factorization of their Dirichlet heat kernels in metric measure spaces. Then we establish explicit estimates…
We study model spaces, in the sense of Hairer, for stochastic partial differential equations involving the fractional Laplacian. We prove that the fractional Laplacian is a singular kernel suitable to apply the theory of regularity…
In this work, we propose an unsupervised method for learning dense correspondences between shapes using a recent deep functional map framework. Instead of depending on ground-truth correspondences or the computationally expensive geodesic…
Exceptionally elegant formulae exist for the fractional Laplacian operator applied to weighted classical orthogonal polynomials. We utilize these results to construct a solver, based on frame properties, for equations involving the…
We present algorithms for solving spatially nonlocal diffusion models on the unit sphere with spectral accuracy in space. Our algorithms are based on the diagonalizability of nonlocal diffusion operators in the basis of spherical harmonics,…
Segmentation masks of pathological areas are useful in many medical applications, such as brain tumour and stroke management. Moreover, healthy counterfactuals of diseased images can be used to enhance radiologists' training files and to…
We obtain an upper heat kernel bound for the Laplacian on metric graphs arising as one skeletons of certain polygonal tilings of the plane, which reflects the one dimensional as well as the two dimensional nature of these graphs.
Computational imaging is crucial in many disciplines from autonomous driving to life sciences. However, traditional model-driven and iterative methods consume large computational power and lack scalability for imaging. Deep learning (DL) is…
Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood,…
This paper develops a nonparametric density estimator with parametric overtones. Suppose $f(x,\theta)$ is some family of densities, indexed by a vector of parameters $\theta$. We define a local kernel smoothed likelihood function which for…
Purpose: To introduce, develop, and evaluate a novel denoising technique for diffusion MRI that leverages non-linear redundancy in the data to boost the SNR while preserving signal information. Methods: We exploit non-linear redundancy of…
A fundamental drawback of kernel-based statistical models is their limited scalability to large data sets, which requires resorting to approximations. In this work, we focus on the popular Gaussian kernel and on techniques to linearize…
Nonparametric kernel density and local polynomial regression estimators are very popular in Statistics, Economics, and many other disciplines. They are routinely employed in applied work, either as part of the main empirical analysis or as…
The existing research on spectral algorithms, applied within a Reproducing Kernel Hilbert Space (RKHS), has primarily focused on general kernel functions, often neglecting the inherent structure of the input feature space. Our paper…
We introduce in this paper the mechanism of graph random features (GRFs). GRFs can be used to construct unbiased randomized estimators of several important kernels defined on graphs' nodes, in particular the regularized Laplacian kernel. As…