arXiv subjects
Publications and source records attributed to Palle E. T. Jorgensen.
By way of concrete presentations, we construct two infinite-dimensional transforms at the crossroads of Gaussian fields and reproducing kernel Hilbert spaces (RKHS), thus leading to a new infinite-dimensional Fourier transform in a general setting of Gaussian processes. Our results serve to unify existing tools from infinite-dimensional analysis.
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.
The purpose of the paper is a general analysis of path space measures. Our focus is a certain path space analysis on generalized Bratteli diagrams. We use this in a systematic study of systems of self-similar measures (the term ``IFS measures'' is used in the paper) for both types of such diagrams, discrete and continuous. In special cases, such measures arise in the study of iterated function systems (IFS). In the literature, similarity may be defined by, e.g., systems of affine maps (Sierpinski), or systems of conformal maps (Julia). We study new classes of semi-branching function systems related to stationary Bratteli diagrams. The latter plays a big role in our understanding of new forms of harmonic analysis on fractals. The measures considered here arise in classes of discrete-time, multi-level dynamical systems where similarity is specified between levels. These structures are made precise by prescribed systems of functions which in turn serve to define self-similarity, i.e., the similarity of large scales, and small scales. For path space systems, in our main result, we give a necessary and sufficient condition for the existence of such generalized IFS measures. For the corresponding semi-branching function systems, we further identify the measures which are also shift-invariant.
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 transforms. Our results extend earlier work for probabilistic Karhunen-Loève transforms on compression of wavelet images. Our object is algorithms for optimization, selection of efficient bases, or components, which serve to minimize entropy and error; and hence to improve digital representation of images, and hence of optimal storage, and transmission. We prove several new theorems for data-dimension reduction. Moreover, with the use of frames in Hilbert space, and a new Hilbert-Schmidt analysis, we identify when a choice of Gaussian kernel is optimal.
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.
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.
We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral invariants. We focus on particular classes of infinite graphs, including such weighted graphs which arise in electrical network models, as well as new diagrammatic graph representations. We further stress some direct parallels between our present analysis on infinite graphs, on the one hand, and, on the other, specific areas of potential theory, Fourier duality, probability, harmonic functions, sampling/interpolation, and boundary theory. With the use of limit constructions, finite to infinite, and local to global, we outline how our results for infinite graphs may be viewed as extensions of Shannon's theory: Starting with a countable infinite graph $G$, and a suitable fixed positive weight function, we show that there are certain continua (certain ambient sets $X$) extending $G$, and associated notions of interpolation for (Hilbert spaces of) functions on $X$ from their restrictions to the discrete graph $G$.
The past decade has seen a flourishing of advances in harmonic analysis of graphs. They lie at the crossroads of graph theory and such analytical tools as graph Laplacians, Markov processes and associated boundaries, analysis of path-space, harmonic analysis, dynamics, and tail-invariant measures. Motivated by recent advances for the special case of Bratteli diagrams, our present focus will be on those graph systems $G$ with the property that the sets of vertices $V$ and edges $E$ admit discrete level structures. A choice of discrete levels in turn leads to new and intriguing discrete-time random-walk models. Our main extension (which greatly expands the earlier analysis of Bratteli diagrams) is the case when the levels in the graph system $G$ under consideration are now allowed to be standard measure spaces. Hence, in the measure framework, we must deal with systems of transition probabilities, as opposed to incidence matrices (for the traditional Bratteli diagrams). The paper is divided into two parts, (i) the special case when the levels are countable discrete systems, and (ii) the (non-atomic) measurable category, i.e., when each level is a prescribed measure space with standard Borel structure. The study of the two cases together is motivated in part by recent new results on graph-limits. Our results depend on a new analysis of certain duality systems for operators in Hilbert space; specifically, one dual system of operator for each level. We prove new results in both cases, (i) and (ii); and we further stress both similarities, and differences, between results and techniques involved in the two cases.
Recent applications of large network models to machine learning, and to neural network suggest a need for a systematic study of the general correspondence, (i) discrete vs (ii) continuous. Even if the starting point is (i), limit considerations lead to (ii), or, more precisely, to a measure theoretic framework which we make precise. Our motivation derives from graph analysis, e.g., studies of (infinite) electrical networks of resistors, but our focus will be (ii), i.e., the measure theoretic setting. In electrical networks of resistors, one considers pairs (of typically countably infinite), sets $V$ (vertices), $E$ (edges) a suitable subset of $V \times V$, and prescribed positive symmetric functions $c$ on $E$ . A conductance function $c$ is defined on $E$ (edges), or on $V \times V$, but with $E$ as its support. From an initial triple $(V, E, c)$ , one gets graph-Laplacians, generalized Dirichlet spaces (also called energy Hilbert spaces), dipoles, relative reproducing kernel-theory, dissipation spaces, reversible Markov chains, and more. Our main results include: spectral theory and Green's functions for measure theoretic graph-Laplace operators; the theory of reproducing kernel Hilbert spaces related to Laplace operators; a rigorous analysis of the Laplacian on Borel equivalence relations; a new decomposition theory; irreducibility criteria; dynamical systems governed by endomorphisms and measurable fields; orbit equivalence criteria; and path-space measures and induced dissipation Hilbert spaces. We consider several applications of our results to other fields such as machine learning problems, reproducing kernel Hilbert spaces, Gaussian and determinantal processes, and joinings.
With view to applications, we here give an explicit correspondence between the following two: (i) the set of symmetric and positive measures $ρ$ on one hand, and (ii) a certain family of generalized Markov transition measures $P$, with their associated Markov random walk models, on the other. By a generalized Markov transition measure we mean a measurable and measure-valued function $P$ on $(V, \mathcal B)$, such that for every $x \in V , P(x; \cdot)$ is a probability measure on $(V, \mathcal B$). Hence, with the use of our correspondence (i) - (ii), we study generalized Markov transitions $P$ and path-space dynamics. Given $P$, we introduce an associated operator, also denoted by $P$ , and we analyze its spectral theoretic properties with reference to a system of precise $L^2$ spaces. Our setting is more general than that of earlier treatments of reversible Markov processes. In a potential theoretic analysis of our processes, we introduce and study an associated energy Hilbert space $\mathcal H_E$, not directly linked to the initial $L^2$-spaces. Its properties are subtle, and our applications include a study of the $P$-harmonic functions. They may be in $\mathcal H_E$, called finite-energy harmonic functions. A second reason for $\mathcal H_E$ is that it plays a key role in our introduction of a generalized Greens function. (The latter stands in relation to our present measure theoretic Laplace operator in a way that parallels more traditional settings of Greens functions from classical potential theory.) A third reason for $\mathcal H_E$ is its use in our analysis of path-space dynamics for generalized Markov transition systems.
In this paper, we discuss a method of constructing separable representations of the $C^*$-algebras associated to strongly connected row-finite $k$-graphs $Λ$. We begin by giving an alternative characterization of the $Λ$-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to construct a faithful separable representation for any row-finite source-free $k$-graph.
Spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure. We consider in this paper the reverse: for a positive matrix in the Hardy space of the unit disc we consider which measures, if any, yield a boundary representation of the positive matrix. We prove a characterization of those representing measures via a matrix identity by introducing a new operator product called the Abel Product.
The main goal of this paper is to build a measurable analogue to the theory of weighted networks on infinite graphs. Our basic setting is an infinite $σ$-finite measure space $(V, \mathcal B, μ)$ and a symmetric measure $ρ$ on $(V\times V, \mathcal B\times \mathcal B)$ supported by a measurable symmetric subset $E\subset V\times V$. This applies to such diverse areas as optimization, graphons (limits of finite graphs), symbolic dynamics, measurable equivalence relations, to determinantal processes, to jump-processes; and it extends earlier studies of infinite graphs $G = (V, E)$ which are endowed with a symmetric weight function $c_{xy}$ defined on the set of edges $E$. As in the theory of weighted networks, we consider the Hilbert spaces $L^2(μ), L^2(cμ)$ and define two other Hilbert spaces, the dissipation space $Diss$ and finite energy space $\mathcal H_E$. Our main results include a number of explicit spectral theoretic and potential theoretic theorems that apply to two realizations of Laplace operators, and the associated jump-diffusion semigroups, one in $L^2(μ)$, and, the second, its counterpart in $\mathcal H_E$. We show in particular that it is the second setting (the energy-Hilbert space and the dissipation Hilbert space) which is needed in a detailed study of transient Markov processes.
Spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure. We consider in this paper the reverse: for a positive matrix in the Hardy space of the unit disc we consider which measures, if any, yield a boundary representation of the positive matrix. We introduce a potential characterization of those measures via a matrix identity and show that the characterization holds in several important special cases.
We develop a new duality between endomorphisms of measure spaces, on the one hand, and a certain family of positive operators, called transfer operators, acting in spaces of measurable functions on, on the other. A framework of standard Borel spaces is adopted; and this generality is wide enough to cover a host of applications. While the mathematical structures of positive operators, endomorphisms, transfer operators, measurable partitions, and Markov processes arise in a host of settings, both pure and applied, we propose here a unified study. This is the general setting of dynamics in Borel measure spaces. Hence the corresponding linear structures are infinite-dimensional. Nonetheless, we prove a number of analogues of the more familiar finite-dimensional settings, for example, the Perron-Frobenius theorem for positive matrices, and the corresponding Markov chains. Tools from the theory of operators in Hilbert space of special significance to us will be the use of a certain universal Hilbert space, as well as classes of operators in it, directly related to the central theme of duality for transfer operators. From ergodic theory, we address such questions as measurable cross sections, partitions, and Rohlin analysis of endomorphisms of measure spaces. While there are classical theorems dealing with analogous questions for automorphisms of measure spaces, a systematic study of endomorphisms is of more recent vintage;-- in its infancy. In order to make the exposition accessible to students and to researchers in neighboring areas, we have included a number of explicit examples and applications.
We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs infinite dimensions. For the case of infinite-dimensional Hilbert space $\mathcal{H}$, we study three cases of measures. We first show that, for $\mathcal{H}$ infinite dimensional, 1 one must resort to infinite dimensional measure spaces which properly contain $\mathcal{H}$. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures.
In a previous paper, the authors introduced the idea of a symmetric pair of operators as a way to compute self-adjoint extensions of symmetric operators. In brief, a symmetric pair consists of two densely defined linear operators $A$ and $B$, with $A \subseteq B^\star$ and $B \subseteq A^\star$. In this paper, we will show by example that symmetric pairs may be used to deduce closability of operators and sometimes even compute adjoints. In particular, we prove that the Malliavin derivative and Skorokhod integral of stochastic calculus are closable, and the closures are mutually adjoint. We also prove that the basic involutions of Tomita-Takesaki theory are closable and that their closures are mutually adjoint. Applications to functions of finite energy on infinite graphs are also discussed, wherein the Laplace operator and inclusion operator form a symmetric pair.
Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations.