Related papers: Positive Definite Kernels, Algorithms, Frames, and…
We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are…
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…
We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present ageneral positive definite kernel setting using bilinear forms, and we provide new…
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…
We study non-linear data-dimension reduction. We are motivated by the classical linear framework of Principal Component Analysis. In nonlinear case, we introduce instead a new kernel-Principal Component Analysis, manifold and feature space…
With view to applications in stochastic analysis and geometry, we introduce a new correspondence for positive definite kernels (p.d.) $K$ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of…
Kernel methods approximate nonlinear maps in a data-driven manner by projecting the target map onto a finite-dimensional Hilbert space called the solution space. Traditionally, this space is a subspace of a fixed ambient reproducing kernel…
Motivated by practical applications, I present a novel and comprehensive framework for operator-valued positive definite kernels. This framework is applied to both operator theory and stochastic processes. The first application focuses on…
In this article we study the field of Hilbertian metrics and positive definit (pd) kernels on probability measures, they have a real interest in kernel methods. Firstly we will make a study based on the Alpha-Beta-divergence to have a…
We review machine learning methods employing positive definite kernels. These methods formulate learning and estimation problems in a reproducing kernel Hilbert space (RKHS) of functions defined on the data domain, expanded in terms of a…
Motivated by applications, we consider here new operator theoretic approaches to Conditional mean embeddings (CME). Our present results combine a spectral analysis-based optimization scheme with the use of kernels, stochastic processes, and…
Kernel methods are powerful learning methodologies that allow to perform non-linear data analysis. Despite their popularity, they suffer from poor scalability in big data scenarios. Various approximation methods, including random feature…
In a general context of positive definite kernels $k$, we develop tools and algorithms for sampling in reproducing kernel Hilbert space $\mathscr{H}$ (RKHS). With reference to these RKHSs, our results allow inference from samples; more…
In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape…
Matrix approximations are a key element in large-scale algebraic machine learning approaches. The recently proposed method MEKA (Si et al., 2014) effectively employs two common assumptions in Hilbert spaces: the low-rank property of an…
We propose a stable version of Principal Component Analysis (PCA) in the general framework of a separable Hilbert space. It consists in interpreting the projection on the first eigenvectors as a step function applied to the spectrum of the…
Data-driven approximations of the infinite-dimensional Koopman operator rely on finite-dimensional projections, where the predictive accuracy of the resulting models hinges heavily on the invariance of the chosen subspace. Subspace pruning…
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…
In the context of kernel optimization, we prove a result that yields new factorizations and realizations. Our initial context is that of general positive operator-valued kernels. We further present implications for Hilbert space-valued…
We consider frames F in a given Hilbert space, and we show that every F may be obtained in a constructive way from a reproducing kernel and an orthonormal basis in an ambient Hilbert space. The construction is operator-theoretic, building…