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The use of second order boundary kernels for distribution function estimation was recently addressed in the literature (C. Tenreiro, 2013, Boundary kernels for distribution function estimation, REVSTAT-Statistical Journal, 11, 169-190). In…
We prove that kernel covariance embeddings lead to information-theoretically perfect separation of distinct continuous probability distributions. In statistical terms, we establish that testing for the \emph{equality} of two non-atomic…
The fragmentation equation is commonly expressed in terms of two functions, the rate of fragmentation and the mean number of fragments. In the case of binary fragmentation an alternative description is possible based on the fragmentation…
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 survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or…
Low-rank approximation of kernels is a fundamental mathematical problem with widespread algorithmic applications. Often the kernel is restricted to an algebraic variety, e.g., in problems involving sparse or low-rank data. We show that…
The discrete kernel method was developed to estimate count data distributions, distinguishing discrete associated kernels based on their asymptotic behaviour. This study investigates the class of discrete asymmetric kernels and their…
We present in this work a new family of kernels to compare positive measures on arbitrary spaces $\Xcal$ endowed with a positive kernel $\kappa$, which translates naturally into kernels between histograms or clouds of points. We first cover…
In a recent paper we used a basic decomposition property of polyanalytic functions of order $2$ in one complex variable to characterize solutions of the classical $\overline{\partial}$-problem for given analytic and polyanalytic data. Our…
This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer…
It is often said that control and estimation problems are in duality. Recently, in (Aubin-Frankowski,2021), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the Hilbert space of controlled trajectories,…
Kernel techniques are among the most popular and flexible approaches in data science allowing to represent probability measures without loss of information under mild conditions. The resulting mapping called mean embedding gives rise to a…
Permutation-valued features arise in a variety of applications, either in a direct way when preferences are elicited over a collection of items, or an indirect way in which numerical ratings are converted to a ranking. To date, there has…
The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and…
In this note, we develop some of the basic theory of s-finite (measures and) kernels, a little-studied class that Staton has recently argued convincingly to be precisely the semantic counterpart of (first-order) probabilistic programs. We…
Quantum computing, with its potential to enhance various machine learning tasks, allows significant advancements in kernel calculation and model precision. Utilizing the one-class Support Vector Machine alongside a quantum kernel, known for…
In modern data analysis, nonparametric measures of discrepancies between random variables are particularly important. The subject is well-studied in the frequentist literature, while the development in the Bayesian setting is limited where…
The paper discusses a series of results concerning reproducing kernel Hilbert spaces, related to the factorization of their kernels. In particular, it is proved that for a large class of spaces isometric multipliers are trivial. One also…
Many scientific problems require identifying a small set of covariates that are associated with a target response and estimating their effects. Often, these effects are nonlinear and include interactions, so linear and additive methods can…
The kernel trick concept, formulated as an inner product in a feature space, facilitates powerful extensions to many well-known algorithms. While the kernel matrix involves inner products in the feature space, the sample covariance matrix…