Related papers: Exact heat kernel on a hypersphere and its applica…

Hyperspectral imaging is a powerful technology that is plagued by large dimensionality. Herein, we explore a way to combat that hindrance via non-contiguous and contiguous (simpler to realize sensor) band grouping for dimensionality…

Let $(X,d,\mu)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,\mu)$ by applying the parabolic Harnack inequality and the…

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain $D$ in a length metric space. The length metric is the intrinsic metric…

We numerically compute the heat kernel on a square lattice torus equipped with the measure corresponding to Liouville quantum gravity (LQG). From the on-diagonal heat kernel we verify that the spectral dimension of LQG is 2. Furthermore,…

This paper introduces a novel, non-deterministic method for embedding data in low-dimensional Euclidean space based on computing realizations of a Gaussian process depending on the geometry of the data. This type of embedding first appeared…

We propose a method for the approximation of high- or even infinite-dimensional feature vectors, which play an important role in supervised learning. The goal is to reduce the size of the training data, resulting in lower storage…

These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…

A diagramatic heat kernel expansion technique is presented. The method is especially well suited to the small-derivative expansion of the heat kernel, but it can also be used to reproduce the results obtained by the approach known as…

Graph convolutional networks gain remarkable success in semi-supervised learning on graph structured data. The key to graph-based semisupervised learning is capturing the smoothness of labels or features over nodes exerted by graph…

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…

Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view…

We construct a kernel density estimator on symmetric spaces of non-compact type and establish an upper bound for its convergence rate, analogous to the minimax rate for classical kernel density estimators on Euclidean space. Symmetric…

A spectral mixture (SM) kernel is a flexible kernel used to model any stationary covariance function. Although it is useful in modeling data, the learning of the SM kernel is generally difficult because optimizing a large number of…

We propose a novel approach for pixel classification in hyperspectral images, leveraging on both the spatial and spectral information in the data. The introduced method relies on a recently proposed framework for learning on distributions…

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in…

Density-based clustering methodology has been widely considered in the statistical literature for classifying Euclidean observations. However, this approach has not been contemplated for directional data yet. In this work, directional…

We obtain sharp two-sided heat kernel estimates on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume…

It is common, in deconvolution problems, to assume that the measurement errors are identically distributed. In many real-life applications, however, this condition is not satisfied and the deconvolution estimators developed for…

Permutation invariance is fundamental in molecular point-cloud generation, yet most diffusion models enforce it indirectly via permutation-equivariant networks on an ordered space. We propose to model diffusion directly on the quotient…

In this paper, we firstly establish weighted heat kernel comparison theorems for the weighted heat equation on complete manifolds with radial curvatures bounded, and then by mainly using this conclusion, we can obtain two eigenvalue…