Palle E. T. Jørgensen

Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz

We here use notions from the theory linear shift-invariant dynamical systems to provide an easy-to-compute characterization of all rational wavelet filters. For a given N bigger or equql to 2, the number of inputs, the construction is based on a factorization to an elementary wavelet filter along with of m elementary unitary matrices. We shall call this m the index of the filter. It turns out that the resulting wavelet filter is of McMillan degree $N((N-1)/2+m). Rational wavelet filters bounded at infinity, admit state space realization. The above input-output parameterization is exploited for a step-by-step construction (where in each the index m is increased by one) of state space model of wavelet filters.

Dorin Ervin Dutkay, Palle E. T. Jorgensen

We establish a spectral duality for certain unbounded operators in Hilbert space. The class of operators includes discrete graph Laplacians arising from infinite weighted graphs. The problem in this context is to establish a practical approximation of infinite models with suitable sequences of finite models which in turn allow (relatively) easy computations. Let $X$ be an infinite set and let $\H$ be a Hilbert space of functions on $X$ with inner product $\ip{\cdot}{\cdot}=\ip{\cdot}{\cdot}_{\H}$. We will be assuming that the Dirac masses $δ_x$, for $x\in X$, are contained in $\H$. And we then define an associated operator $Δ$ in $\H$ given by $$(Δv)(x):=\ip{δ_x}{v}_{\H}.$$ Similarly, for every finite subset $F\subset X$, we get an operator $Δ_F$. If $F_1\subset F_2\subset...$ is an ascending sequence of finite subsets such that $\cup_{k\in\bn}F_k=X$, we are interested in the following two problems: (a) obtaining an approximation formula $$\lim_{k\to\infty}Δ_{F_k}=Δ;$$ and (b) establish a computational spectral analysis for the truncated operators $Δ_F$ in (a).

Palle E. T. Jorgensen, Myung-Sin Song, James Tian

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.

Palle E. T. Jorgensen, Myung-Sin Song, James Tian

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 $C$.

Palle Jorgensen, Myung-Sin Song, Feng Tian

We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are central to spectral theory of operators in Hilbert space, to optimization, to large sparse systems, to iterated function systems (IFS), and to fractal harmonic analysis. We present a new recursive iteration scheme involving as input a prescribed sequence of selfadjoint projections. Applications include random Kaczmarz recursions, their limits, and their error-estimates.