Related papers: Sparse Solutions for Inverse Problems in Reproduci…
We consider an incremental approximation method for solving variational problems in infinite-dimensional Hilbert spaces, where in each step a randomly and independently selected subproblem from an infinite collection of subproblems is…
By applying methods of Duhamel algebra and reproducing kernels, we prove that every linear bounded operator on the Hardy-Hilbert space H^{2}(D) has a nontrivial invariant subspace. This solves affirmatively the Invariant Subspace Problem in…
Motivated by the growing interest in representation learning approaches that uncover the latent structure of high-dimensional data, this work proposes new algorithms for reconstruction-based manifold learning within Reproducing-Kernel…
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 \cite{Lions}, J. L. Lions considered a reproducing kernel Hilbert space (RKHS) of harmonic functions on a regular domain with Sobolev traces and obtained a formula that expresses the kernel of this space as an integral on the boundary of…
We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional…
Studies of sparse representation of deterministic signals have been well developed. Amongst there exists one called adaptive Fourier decomposition (AFD) established through adaptive selections of the parameters defining a Takenaka-Malmquist…
Traditional machine learning models, particularly neural networks, are rooted in finite-dimensional parameter spaces and nonlinear function approximations. This report explores an alternative formulation where learning tasks are expressed…
Sliced inverse regression is a popular tool for sufficient dimension reduction, which replaces covariates with a minimal set of their linear combinations without loss of information on the conditional distribution of the response given the…
This paper proposes a new method for estimating sparse precision matrices in the high dimensional setting. It has been popular to study fast computation and adaptive procedures for this problem. We propose a novel approach, called Sparse…
In this paper we combine the theory of reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with special emphasis on reproducing property of kernels. From the reproducing property of…
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,…
Recent advances suggest that a wide range of computer vision problems can be addressed more appropriately by considering non-Euclidean geometry. This paper tackles the problem of sparse coding and dictionary learning in the space of…
We develop a stochastic approximation framework for learning nonlinear operators between infinite-dimensional spaces utilizing general Mercer operator-valued kernels. Our framework encompasses two key classes: (i) compact kernels, which…
We study the estimation and prediction of functional autoregressive~(FAR) processes, a statistical tool for modeling functional time series data. Due to the infinite-dimensional nature of FAR processes, the existing literature addresses its…
Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However,…
We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by…
In this work, we develop and study an empirical projection operator scheme for solving nonparametric regression problems. This scheme is based on an approximate projection of the regression function over a suitable reproducing kernel…
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…
We extend the diffusion-map formalism to data sets that are induced by asymmetric kernels. Analytical convergence results of the resulting expansion are proved, and an algorithm is proposed to perform the dimensional reduction. In this work…