Browsing by Subject "reproducing kernel Hilbert space"

Now showing 1 - 3 of 3

Access status: Open Access ,
Conditional mean embedding and optimal feature selection via positive definite kernels
(Wydawnictwa AGH, 2024) Jørgensen, Palle E.T.; Song, Myung-Sin; Tian, James
Motivated by applications, we consider new operator-theoretic approaches to conditional mean embedding (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 kernels in a construction o foptimal feature selection via regression algorithms from learning models. Thus, with initial inputs of training data (for a suitable learning algorithm), each choice of a kernel $K$ in turn yields a variety of Hilbert spaces and realizations of features. A novel aspect of our work is the inclusion of a secondary optimization process over a specified convex set of positive definite kernels, resulting in the determination of »optimal« feature representations.
Access status: Open Access ,
Decomposition of Gaussian processes, and factorization of positive definite kernels
(Wydawnictwa AGH, 2019) Jørgensen, Palle E.T.; Tian, Feng
We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $K$, vs for Gaussian process $V$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $K$ is the exact same as that which yield factorizations for $V$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.
Access status: Open Access ,
Functional models for Nevanlinna families
(2008) Behrndt, Jussi; Hassi, Seppo; Snoo, Henk de
The class of Nevanlinna families consists of $\mathbb{R}$-symmetric holomorphic multivalued functions on $\mathbb{C} \setminus \mathbb{R}$ with maximal dissipative (maximal accumulative) values on $\mathbb{C}_{+}$ ($\mathbb{C}_{-}$, respectively) and is a generalization of the class of operator-valued Nevanlinna functions. In this note Nevanlinna families are realized as Weyl families of boundary relations induced by multiplication operators with the independent variable in reproducing kernel Hilbert spaces.