Related papers: Highly localized kernels on space of homogeneous t…
In this paper, we investigate the trigonometric Heckman-Opdam polynomials of type $A_1$. We establish connections with ultraspherical polynomials and derive an explicit expression for the associated Poisson kernel. Using the product…
A class of Cantor-type spaces and related geometric structures are discussed.
This paper presents an algorithm, Voted Kernel Regularization , that provides the flexibility of using potentially very complex kernel functions such as predictors based on much higher-degree polynomial kernels, while benefitting from…
In this note we apply heat kernels to derive some localization formula in sympletcic geometry, to study moduli spaces of flat connections on a Riemann surface, to obtain the push-forward measures for certain maps between Lie groups and to…
The problem of classification in machine learning has often been approached in terms of function approximation. In this paper, we propose an alternative approach for classification in arbitrary compact metric spaces which, in theory, yields…
We prove that under some purely algebraic conditions every locally homogeneous structure modelled on some homogeneous space is induced by a locally homogeneous structure modelled on a different homogeneous space.
Recent advances in operator learning theory have improved our knowledge about learning maps between infinite dimensional spaces. However, for large-scale engineering problems such as concurrent multiscale simulation for mechanical…
Applying machine learning to biological sequences - DNA, RNA and protein - has enormous potential to advance human health, environmental sustainability, and fundamental biological understanding. However, many existing machine learning…
Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex…
This paper studies the hierarchy of local minimums of a polynomial in the space. For this purpose, we first compute H-minimums, for which the first and second order optimality conditions are satisfied. To compute each H-minimum, we…
Lyubeznik conjectured that local cohomology modules of regular rings have finitely many associated primes. We examine this conjecture for polynomial rings over the integers, and record some equational identities that arise from studying…
In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice…
We study classes of reproducing kernels $K$ on general domains; these are kernels which arise commonly in machine learning models; models based on certain families of reproducing kernel Hilbert spaces. They are the positive definite kernels…
We implement an all-optical setup demonstrating kernel-based quantum machine learning for two-dimensional classification problems. In this hybrid approach, kernel evaluations are outsourced to projective measurements on suitably designed…
Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator…
Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and…
In this paper, we present a kernel subspace clustering method that can handle non-linear models. In contrast to recent kernel subspace clustering methods which use predefined kernels, we propose to learn a low-rank kernel matrix, with which…
A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$,…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
We consider the problem of high-dimensional non-linear variable selection for supervised learning. Our approach is based on performing linear selection among exponentially many appropriately defined positive definite kernels that…