arXiv subjects

James Tian

Publications and source records attributed to James Tian.

43 records · Page 3Linked to original sources

arXiv2024

Operator-Valued Kernels, Machine Learning, and Dynamical Systems

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 Gaussian processes, as they arise in applications to dynamics and to machine learning. Further applications are given in non-commutative probability theory, including a new non-commutative Radon--Nikodym theorem.

math.OA↗

arXiv2024

Gaussian processes in Non-commutative probability

Motivated by questions in quantum theory, we study Hilbert space valued Gaussian processes, and operator-valued kernels, i.e., kernels taking values in B(H) (= all bounded linear operators in a fixed Hilbert space H). We begin with a systematic study of p.d. B(H)-valued kernels and the associated of H-valued Gaussian processes, together with their correlation and transfer operators. In our consideration of B(H)-valued kernels, we drop the p.d. assumption. We show that input-output models can be computed for systems of signed kernels taking the precise form of realizability via associated transfer block matrices (of operators analogous to the realization transforms in systems theory), i.e., represented via 2\times2 operator valued block matrices. In the context of B(H)-valued kernels we present new results on regression with H-valued Gaussian processes.

math.FA↗

arXiv2024

Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure

Motivated by applications, we introduce a general and new framework for operator valued positive definite kernels. We further give applications both to operator theory and to stochastic processes. The first one yields several dilation constructions in operator theory, and the second to general classes of stochastic processes. For the latter, we apply our operator valued kernel-results in order to build new Hilbert space-valued Gaussian processes, and to analyze their structures of covariance configurations.

math.FA↗

arXiv2023

Conditional mean embeddings and optimal feature selection via positive definite kernels

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 constructive learning algorithms. For initially given non-linear data, we consider optimization-based feature selections. This entails the use of convex sets of positive definite (p.d.) kernels in a construction of optimal feature selection via regression algorithms from learning models. Thus, with initial inputs of training data (for a suitable learning algorithm,) each choice of p.d. kernel $K$ in turn yields a variety of Hilbert spaces and realizations of features. A novel idea here is that we shall allow an optimization over selected sets of kernels $K$ from a convex set $C$ of positive definite kernels $K$. Hence our \textquotedblleft optimal\textquotedblright{} choices of feature representations will depend on a secondary optimization over p.d. kernels $K$ within a specified convex set $C$.

cs.LG↗

arXiv2023

Operator theory, kernels, and Feedforward Neural Networks

In this paper we show how specific families of positive definite kernels serve as powerful tools in analyses of iteration algorithms for multiple layer feedforward Neural Network models. Our focus is on particular kernels that adapt well to learning algorithms for data-sets/features which display intrinsic self-similarities at feedforward iterations of scaling.

cs.LG↗

arXiv2022

Dual pairs of operators, harmonic analysis of singular non-atomic measures and Krein-Feller diffusion

We show that a Krein-Feller operator is naturally associated to a fixed measure $μ$, assumed positive, $σ$-finite, and non-atomic. Dual pairs of operators are introduced, carried by the two Hilbert spaces, $L^{2}\left(μ\right)$ and $L^{2}\left(λ\right)$, where $λ$ denotes Lebesgue measure. An associated operator pair consists of two specific densely defined (unbounded) operators, each one contained in the adjoint of the other. This then yields a rigorous analysis of the corresponding $μ$-Krein-Feller operator as a closable quadratic form. As an application, for a given measure $μ$, including the case of fractal measures, we compute the associated diffusion, semigroup, Dirichlet forms, and $μ$-generalized heat equation.

math.FA↗

arXiv2021

Positive Definite Kernels, Algorithms, Frames, and Approximations

The main purpose of our paper is a new approach to design of algorithms of Kaczmarz type in the framework of operators in Hilbert space. Our applications include a diverse list of optimization problems, new Karhunen-Loève transforms, and Principal Component Analysis (PCA) for digital images. A key feature of our algorithms is our use of recursive systems of projection operators. Specifically, we apply our recursive projection algorithms for new computations of PCA probabilities and of variance data. For this we also make use of specific reproducing kernel Hilbert spaces, factorization for kernels, and finite-dimensional approximations. Our projection algorithms are designed with view to maximum likelihood solutions, minimization of "cost" problems, identification of principal components, and data-dimension reduction.

math.FA↗