Related papers: Universality, Characteristic Kernels and RKHS Embe…
We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two…
In this paper we deal with the problem of testing for the equality of $k$ probability distributions defined on $(\mathcal{X},\mathcal{B})$, where $\mathcal{X}$ is a metric space and $\mathcal{B}$ is the corresponding Borel $\sigma$-field.…
We present accurate upper and lower bounds for the covering numbers, with explicit constants, of the unit ball for two general classes of Reproducing Kernel Hilbert Space (RKHS) on the unit sphere of $\mathbb{R}^{d+1}$. In both classes, the…
Proximities are at the heart of almost all machine learning methods. If the input data are given as numerical vectors of equal lengths, euclidean distance, or a Hilbertian inner product is frequently used in modeling algorithms. In a more…
This article presents a quantum computing approach to designing of similarity measures and kernels for classification of stochastic symbolic time series. In the area of machine learning, kernels are important components of various…
Quantum kernels quantify similarity between data points by measuring the inner product between quantum states, computed through quantum circuit measurements. By embedding data into quantum systems, quantum kernel feature maps, that may be…
Recently, some works have suggested methods to combine variational probabilistic inference with Monte Carlo sampling. One promising approach is via local optimal transport. In this approach, a gradient steepest descent method based on local…
The ability to measure differences in collected data is of fundamental importance for quantitative science and machine learning, motivating the establishment of metrics grounded in physical principles. In this study, we focus on the…
We study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems. By developing a tool on simultaneous diagonalization of two positive…
Kernel methods give powerful, flexible, and theoretically grounded approaches to solving many problems in machine learning. The standard approach, however, requires pairwise evaluations of a kernel function, which can lead to scalability…
In supervised learning using kernel methods, we often encounter a large-scale finite-sum minimization over a reproducing kernel Hilbert space (RKHS). Large-scale finite-sum problems can be solved using efficient variants of Newton method,…
Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that…
The fields of signal processing and information theory have evolved with the goal of developing formulations to extract intrinsic information from limited amount of data. When one considers the modeling of unpredictably varying processes…
This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as…
This article studies global testing of the slope function in functional linear regression model in the framework of reproducing kernel Hilbert space. We propose a new testing statistic based on smoothness regularization estimators. The…
The X-ray transform is one of the most fundamental integral operators in image processing and reconstruction. In this article, we revisit the formalism of the X-ray transform by considering it as an operator between Reproducing Kernel…
This monograph studies the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely…
Supervised learning in reproducing kernel Hilbert space (RKHS) and vector-valued RKHS (vvRKHS) has been investigated for more than 30 years. In this paper, we provide a new twist to this rich literature by generalizing supervised learning…
Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals). In this paper, we…
If $\Lambda $ is a measure space, $u:\Lambda ^{m}\rightarrow \Bbb{R}$ is a given function and $N\geq m,$ the function $U(x_{1},...,x_{N})=\left( \begin{array}{l} N \\ m \end{array} \right) ^{-1}\sum_{1\leq i_{1}<\cdots <i_{m}\leq…