arXiv subjects

Palle E. T. Jorgensen

Publications and source records attributed to Palle E. T. Jorgensen.

At least 19 recordsLinked to original sources

arXiv2026

The role of parameter Jacobians in the stability of network outputs

In the framework of network dynamics, learning models, and neural tangent kernels (NTK), we show that the corresponding linearized dynamics leads naturally to a semigroup formulation. More precisely, in our analysis of input/output models, the time-dynamics is presented via special semigroups of linear operators on Hilbert spaces, together with an associated class of semigroup perturbations. In this context, we then present new and explicit a priori perturbation-bound results: for the fixed-kernel linearization constructions arising in the NTK setting, we prove norm-bounds on the corresponding semigroup perturbations, in the form of explicit finite-time perturbation estimates. We further present refinements on prescribed task spaces, Cesàro-averaged (ergodic) comparisons estimates, and versions in which the lower spectral edge assumption is replaced by a spectral-distribution condition. We also extend the comparison to nonautonomous NTK evolutions through piecewise-frozen approximations, record a corresponding discrete Euler specialization, and offer worked examples in order to illustrate our perturbation-bound estimates.

math.FA↗

arXiv2026

Use of operator defect identities in multi-channel signal plus residual-analysis via iterated products and telescoping energy-residuals: Applications to kernels in machine learning

We present a new operator theoretic framework for analysis of complex systems with intrinsic subdivisions into components, taking the form of "residuals" in general, and "telescoping energy residuals" in particular. We prove new results which yield admissibility/effectiveness, and new a priori bounds on energy residuals. Applications include infinite-dimensional Kaczmarz theory for $λ_{n}$-relaxed variants, and $λ_{n}$-effectiveness. And we give applications of our framework to generalized machine learning algorithms, greedy Kernel Principal Component Analysis (KPCA), proving explicit convergence results, residual energy decomposition, and criteria for stability under noise.

math.FA↗

arXiv2025

Infinite-Dimensional Operator/Block Kaczmarz Algorithms: Regret Bounds and $λ$-Effectiveness

We present a variety of projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance. We study the role of the relaxation parameter in generalized Kaczmarz algorithms and establish a priori regret bounds with explicit $λ$-dependence to quantify how much an algorithm's performance deviates from its optimal performance. A detailed analysis of relaxation parameter is also provided. Applications include: explicit regret bounds for the framework of Kaczmarz algorithm models, non-orthogonal Fourier expansions, and the use of regret estimates in modern machine learning models, including for noisy data, i.e., regret bounds for the noisy Kaczmarz algorithms. Motivated by machine-learning practice, our wider framework treats bounded operators (on infinite-dimensional Hilbert spaces), with updates realized as (block) Kaczmarz algorithms, leading to new and versatile results.

stat.ML↗

arXiv2025

Factorization of positive definite kernels. Correspondences: $C^{*}$-algebraic and operator valued context vs scalar valued kernels

We introduce and study a class $\mathcal{M}$ of generalized positive definite kernels of the form $K\colon X\times X\to L(\mathfrak{A},L(H))$, where $\mathfrak{A}$ is a unital $C^{*}$-algebra and $H$ a Hilbert space. These kernels encode operator-valued correlations governed by the algebraic structure of $\mathfrak{A}$, and generalize classical scalar-valued positive definite kernels, completely positive (CP) maps, and states on $C^{*}$-algebras. Our approach is based on a scalar-valued kernel $\tilde{K}\colon(X\times\mathfrak{A}\times H)^{2}\to\mathbb{C}$ associated to $K$, which defines a reproducing kernel Hilbert space (RKHS) and enables a concrete, representation-theoretic analysis of the structure of such kernels. We show that every $K\in\mathcal{M}$ admits a Stinespring-type factorization $K(s,t)(a)=V(s)^{*}π(a)V(t)$. In analogy with the Radon--Nikodym theory for CP maps, we characterize kernel domination $K\leq L$ in terms of a positive operator $A\inπ_{L}(\mathfrak{A})'$ satisfying $K(s,t)(a)=V_{L}(s)^{*}π_{L}(a)AV_{L}(t)$. We further show that when $π_{L}$ is irreducible, domination implies scalar proportionality, thus recovering the classical correspondence between pure states and irreducible representations.

math.OA↗

arXiv2025

Fuglede's conjecture, differential operators and unitary groups of local translations

The purpose of the present paper is to address multiple aspects of the Fuglede question dealing (Fourier spectra vs geometry) with a variety of $L^2$ contexts where we make precise the interplay between the three sides of the question: (i) existence of orthogonal families of Fourier basis functions (and associated spectra) on the one hand, (ii) extensions of partial derivative operators, and (iii) geometry of the corresponding domains, stressing systems of translation-tiles. We emphasize an account of old and new developments since the original 1974-paper by Bent Fuglede where the co-authors and Steen Pedersen have contributed.

math.FA↗

arXiv2025

Hilbert space-valued Gaussian processes, and quantum states

We offer new results and new directions in the study of operator-valued kernels and their factorizations. Our approach provides both more explicit realizations and new results, as well as new applications. These include: (i) an explicit covariance analysis for Hilbert space-valued Gaussian processes, (ii) optimization results for quantum gates (from quantum information), (iii) new results for positive operator-valued measures (POVMs), and (iv) a new approach/result in inverse problems for quantum measurements.

quant-ph↗

arXiv2025

Horizontally stationary generalized Bratteli diagrams

Bratteli diagrams with countably infinite levels exhibit a new phenomenon: they can be horizontally stationary. The incidence matrices of these horizontally stationary Bratteli diagrams are infinite banded Toeplitz matrices. In this paper, we study the fundamental properties of horizontally stationary Bratteli diagrams. In these diagrams, we provide an explicit description of ergodic tail invariant probability measures. For a certain class of horizontally stationary Bratteli diagrams, we prove that all ergodic tail invariant probability measures are extensions of measures from odometers. Additionally, we establish conditions for the existence of a continuous Vershik map on the path space of a horizontally stationary Bratteli diagram.

math.DS↗

arXiv2025

From Classical to Quantum: Polymorphisms in Non-Commutative Probability

We present a parallel between commutative and non-commutative polymorphisms. Our emphasis is the applications to conditional distributions from stochastic processes. In the classical case, both the measures and the positive definite kernels are scalar valued. But the non-commutative framework (as motivated by quantum theory) dictates a setting where instead now both the measures (in the form of quantum states), and the positive definite kernels, are operator valued. The non-commutative theory entails a systematic study of positive operator valued measures, abbreviated POVMs. And quantum states (normal states) are indexed by normalized positive trace-class operators. In the non-commutative theory, the parallel to the commutative/scalar valued theory helps us understand entanglement in quantum information. A further implication of our study of the non-commutative framework will entail an interplay between the two cases, scalar valued, vs operator valued.

quant-ph↗

arXiv2025

New duality in choices of feature spaces via kernel analysis

We present a systematic study of the family of positive definite (p.d.) kernels with the use of their associated feature maps and feature spaces. For a fixed set $X$, generalizing Loewner, we make precise the corresponding partially ordered set $Pos\left(X\right)$ of all p.d. kernels on $X$, as well as a study of its global properties. This new analysis includes both results dealing with applications and concrete examples, including such general notions for $Pos\left(X\right)$ as the structure of its partial order, its products, sums, and limits; as well as their Hilbert space-theoretic counterparts. For this purpose, we introduce a new duality for feature spaces, feature selections, and feature mappings. For our analysis, we further introduce a general notion of dual pairs of p.d. kernels. Three special classes of kernels are studied in detail: (a) the case when the reproducing kernel Hilbert spaces (RKHSs) may be chosen as Hilbert spaces of analytic functions, (b) when they are realized in spaces of Schwartz-distributions, and (c) arise as fractal limits. We further prove inverse theorems in which we derive results for the analysis of $Pos\left(X\right)$ from the operator theory of specified counterpart-feature spaces. We present constructions of new p.d. kernels in two ways: (i) as limits of monotone families in $Pos\left(X\right)$, and (ii) as p.d. kernels which model fractal limits, i.e., are invariant with respect to certain iterated function systems (IFS)-transformations.

math.FA↗

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↗

arXiv2024

Substitution-dynamics and invariant measures for infinite alphabet-path space

We study substitutions on countably infinite alphabet (without compactification) as Borel dynamical systems. We construct stationary and non-stationary generalized Bratteli-Vershik models for a class of such substitutions, known as left determined. In this setting of Borel dynamics, using a stationary generalized Bratteli-Vershik model, we provide a new and canonical construction of shift-invariant measures (both finite and infinite) for the associated class of subshifts.

math.DS↗

arXiv2024

Fourier series for singular measures in higher dimensions

For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal $L^2$ Fourier expansions. Our results hold for probability measures $μ$ with finite support in $\mathbb{R}^d$ that satisfy a certain disintegration condition that we refer to as ``slice-singular''. In this general framework, we present explicit $L^{2}(μ)$-Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every $f \in L^2(μ)$, are based on an extended Kaczmarz algorithm, and use a new recursive $μ$ Rokhlin disintegration representation. In detail, our Fourier series expansion for $f$ is in terms of the multivariate Fourier exponentials $\{e_n\}$, but the associated Fourier coefficients for $f$ are now computed from a Kaczmarz system $\{g_n\}$ in $L^{2}(μ)$ which is dual to the Fourier exponentials. The $\{g_n\}$ system is shown to be a Parseval frame for $L^{2}(μ)$. Explicit computations for our new Fourier expansions entail a detailed analysis of subspaces of the Hardy space on the polydisk, dual to $L^{2}(μ)$, and an associated d-variable Normalized Cauchy Transform. Our results extend earlier work for measures $μ$ in one and two dimensions, i.e., $d=1 (μ$ singular), and $d=2 (μ$ assumed slice-singular). Here our focus is the extension to the cases of measures $μ$ in dimensions $d >2$. Our results are illustrated with the use of explicit iterated function systems (IFSs), including the IFS generated Menger sponge for $d=3$.

math.FA↗

arXiv2023

The momentum operator on a union of intervals and the Fuglede conjecture

The purpose of the present paper is to place a number of geometric (and hands-on) configurations relating to spectrum and geometry inside a general framework for the {\it Fuglede conjecture}. Note that in its general form, the Fuglede conjecture concerns general Borel sets $Ω$ in a fixed number of dimensions $d$ such that $Ω$ has finite positive Lebesgue measure. The conjecture proposes a correspondence between two properties for $Ω$, one takes the form of spectrum, while the other refers to a translation-tiling property. We focus here on the case of dimension one, and the connections between the Fuglede conjecture and properties of the self-adjoint extensions of the momentum operator $\frac{1}{2πi}\frac{d}{dx}$, realized in $L^2$ of a union of intervals.

math.SP↗

arXiv2023

Bratteli diagrams in Borel dynamics

Bratteli-Vershik models have been very successfully applied to the study of various dynamical systems, in particular, in Cantor dynamics. In this paper, we study dynamics on the path spaces of generalized Bratteli diagrams that form models for non-compact Borel dynamical systems. Generalized Bratteli diagrams have countably infinite many vertices at each level, thus the corresponding incidence matrices are also countably infinite. We emphasize differences (and similarities) between generalized and classical Bratteli diagrams. Our main results: $(i)$ We utilize Perron-Frobenius theory for countably infinite matrices to establish criteria for the existence and uniqueness of tail-invariant path space measures (both probability and $σ$-finite). $(ii)$ We provide criteria for the topological transitivity of the tail equivalence relation. $(iii)$ We describe classes of stationary generalized Bratteli diagrams (hence Borel dynamical systems) that: $(a)$ do not support a probability tail-invariant measure, $(b)$ are not uniquely ergodic with respect to the tail equivalence relation. $(iv)$ We describe classes of generalized Bratteli diagrams which can or cannot admit a continuous Vershik map and construct a Vershik map which is a minimal homeomorphism of a (non locally compact) Polish space. $(v)$ We provide an application of the theory of stochastic matrices to analyze diagrams with positive recurrent incidence matrices.

math.DS↗

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

Measurable multiresolution systems, endomorphisms, and representations of Cuntz relations

The purpose of this paper is to present new classes of function systems as part of multiresolution analyses. Our approach is representation theoretic, and it makes use of generalized multiresolution function systems (MRSs). It further entails new ideas from measurable endomorphisms-dynamics. Our results yield applications that are not amenable to more traditional techniques used on metric spaces. As the main tool in our approach, we make precise new classes of generalized MRSs which arise directly from a dynamical theory approach to the study of surjective endomorphisms on measure spaces. In particular, we give the necessary and sufficient conditions for a family of functions to define generators of Cuntz relations. We find an explicit description of the set of generalized wavelet filters. Our results are motivated in part by analyses of sub-band filters in signal/image processing. But our paper goes further, and it applies to such wider contexts as measurable dynamical systems, and complex dynamics. A unifying theme in our results is a new analysis of endomorphisms in general measure space, and its connection to multi-resolutions, to representation theory, and generalized wavelet systems.

math.DS↗