1,083 research outputs found
Multivariable generalizations of the Schur class: positive kernel characterization and transfer function realization
The operator-valued Schur-class is defined to be the set of holomorphic
functions mapping the unit disk into the space of contraction operators
between two Hilbert spaces. There are a number of alternate characterizations:
the operator of multiplication by defines a contraction operator between
two Hardy Hilbert spaces, satisfies a von Neumann inequality, a certain
operator-valued kernel associated with is positive-definite, and can be
realized as the transfer function of a dissipative (or even conservative)
discrete-time linear input/state/output linear system. Various multivariable
generalizations of this class have appeared recently,one of the most
encompassing being that of Muhly and Solel where the unit disk is replaced by
the strict unit ball of the elements of a dual correspondence
associated with a -correspondence over a -algebra \cA
together with a -representation of \cA. The main new point which
we add here is the introduction of the notion of reproducing kernel Hilbert
correspondence and identification of the Muhly-Solel Hardy spaces as
reproducing kernel Hilbert correspondences associated with a completely
positive analogue of the classical Szeg\"o kernel. In this way we are able to
make the analogy between the Muhly-Solel Schur class and the classical Schur
class more complete. We also illustrate the theory by specializing it to some
well-studied special cases; in some instances there result new kinds of
realization theorems.Comment: adjusted the definition of completely positve kernel on page 12 and
did minor modifications corresponding to this adjustmen
Weighted Bergman spaces: shift-invariant subspaces and input/state/output linear systems
It is well known that subspaces of the Hardy space over the unit disk which
are invariant under the backward shift occur as the image of an observability
operator associated with a discrete-time linear system with stable
state-dynamics, as well as the functional-model space for a Hilbert space
contraction operator, while forward shift-invariant subspaces have a
representation in terms of an inner function. We discuss several variants of
these statements in the context of weighted Bergman spaces on the unit disk
A Novel Fourier Theory on Non-linear Phases and Applications
Positive time varying frequency representation for transient signals has been
a hearty desire of signal analysts due to its theoretical and practical
importance. During approximately the last two decades there has formulated a
signal decomposition and reconstruction method rooted in harmonic and complex
analysis giving rise to the desired signal representation. The method
decomposes any signal into a few basic signals that possess positive
instantaneous frequencies. The theory has profound relations with classical
mathematics and can be generalized to signals defined in higher dimensional
manifolds with vector and matrix values, and in particular, promotes rational
approximation in higher dimensions. This article mainly serves as a survey. It
also gives a new proof for a general convergence result, as well as a proof for
the necessity of multiple selection of the parameters.
Mono-components are crucial to understand the concept instantaneous
frequency. We will present several most important mono-component function
classes. Decompositions of signals into mono-components are called adaptive
Fourier decompositions (AFDs). We note that some scopes of the studies on the
1D mono-components and AFDs can be extended to vector-valued or even
matrix-valued signals defined on higher dimensional manifolds. We finally
provide an account of related studies in pure and applied mathematics, and in
signal analysis, as well as applications of the theory found in the literature.Comment: 33 page
Function spaces obeying a time-varying bandlimit
Motivated by applications to signal processing and mathematical physics,
recent work on the concept of time-varying bandwidth has produced a class of
function spaces which generalize the Paley-Wiener spaces of bandlimited
functions: any regular simple symmetric linear transformation with deficiency
indices is naturally represented as multiplication by the independent
variable in one of these spaces. We explicitly demonstrate the equivalence of
this model for such linear transformations to several other functional models
based on the theories of meromorphic model spaces of Hardy space and purely
atomic Herglotz measures on the real line, respectively. This theory provides a
precise notion of a time-varying or local bandwidth, and we describe how it may
be applied to construct signal processing techniques that are adapted to
signals obeying a time-varying bandlimit
Adaptive Estimation for Nonlinear Systems using Reproducing Kernel Hilbert Spaces
This paper extends a conventional, general framework for online adaptive
estimation problems for systems governed by unknown nonlinear ordinary
differential equations. The central feature of the theory introduced in this
paper represents the unknown function as a member of a reproducing kernel
Hilbert space (RKHS) and defines a distributed parameter system (DPS) that
governs state estimates and estimates of the unknown function. This paper 1)
derives sufficient conditions for the existence and stability of the infinite
dimensional online estimation problem, 2) derives existence and stability of
finite dimensional approximations of the infinite dimensional approximations,
and 3) determines sufficient conditions for the convergence of finite
dimensional approximations to the infinite dimensional online estimates. A new
condition for persistency of excitation in a RKHS in terms of its evaluation
functionals is introduced in the paper that enables proof of convergence of the
finite dimensional approximations of the unknown function in the RKHS. This
paper studies two particular choices of the RKHS, those that are generated by
exponential functions and those that are generated by multiscale kernels
defined from a multiresolution analysis.Comment: 24 pages, Submitted to CMAM
Online dictionary learning for kernel LMS. Analysis and forward-backward splitting algorithm
Adaptive filtering algorithms operating in reproducing kernel Hilbert spaces
have demonstrated superiority over their linear counterpart for nonlinear
system identification. Unfortunately, an undesirable characteristic of these
methods is that the order of the filters grows linearly with the number of
input data. This dramatically increases the computational burden and memory
requirement. A variety of strategies based on dictionary learning have been
proposed to overcome this severe drawback. Few, if any, of these works analyze
the problem of updating the dictionary in a time-varying environment. In this
paper, we present an analytical study of the convergence behavior of the
Gaussian least-mean-square algorithm in the case where the statistics of the
dictionary elements only partially match the statistics of the input data. This
allows us to emphasize the need for updating the dictionary in an online way,
by discarding the obsolete elements and adding appropriate ones. We introduce a
kernel least-mean-square algorithm with L1-norm regularization to automatically
perform this task. The stability in the mean of this method is analyzed, and
its performance is tested with experiments
Random Euler Complex-Valued Nonlinear Filters
Over the last decade, both the neural network and kernel adaptive filter have
successfully been used for nonlinear signal processing. However, they suffer
from high computational cost caused by their complex/growing network
structures. In this paper, we propose two random Euler filters for
complex-valued nonlinear filtering problem, i.e., linear random Euler
complex-valued filter (LRECF) and its widely-linear version (WLRECF), which
possess a simple and fixed network structure. The transient and steady-state
performances are studied in a non-stationary environment. The analytical
minimum mean square error (MSE) and optimum step-size are derived. Finally,
numerical simulations on complex-valued nonlinear system identification and
nonlinear channel equalization are presented to show the effectiveness of the
proposed methods
Two-Manifold Problems with Applications to Nonlinear System Identification
Recently, there has been much interest in spectral approaches to learning
manifolds---so-called kernel eigenmap methods. These methods have had some
successes, but their applicability is limited because they are not robust to
noise. To address this limitation, we look at two-manifold problems, in which
we simultaneously reconstruct two related manifolds, each representing a
different view of the same data. By solving these interconnected learning
problems together, two-manifold algorithms are able to succeed where a
non-integrated approach would fail: each view allows us to suppress noise in
the other, reducing bias. We propose a class of algorithms for two-manifold
problems, based on spectral decomposition of cross-covariance operators in
Hilbert space, and discuss when two-manifold problems are useful. Finally, we
demonstrate that solving a two-manifold problem can aid in learning a nonlinear
dynamical system from limited data.Comment: ICML2012. arXiv admin note: text overlap with arXiv:1112.639
Sufficient Conditions for Parameter Convergence over Embedded Manifolds using Kernel Techniques
The persistence of excitation (PE) condition is sufficient to ensure
parameter convergence in adaptive estimation problems. Recent results on
adaptive estimation in reproducing kernel Hilbert spaces (RKHS) introduce PE
conditions for RKHS. This paper presents sufficient conditions for PE for the
particular class of uniformly embedded reproducing kernel Hilbert spaces (RKHS)
defined over smooth Riemannian manifolds. This paper also studies the
implications of the sufficient condition in the case when the RKHS is finite or
infinite-dimensional. When the RKHS is finite-dimensional, the sufficient
condition implies parameter convergence as in the conventional analysis. On the
other hand, when the RKHS is infinite-dimensional, the same condition implies
that the function estimate error is ultimately bounded by a constant that
depends on the approximation error in the infinite-dimensional RKHS. We
illustrate the effectiveness of the sufficient condition in a practical
example.Comment: 11 pages, 2 figure
Short-term time series prediction using Hilbert space embeddings of autoregressive processes
Linear autoregressive models serve as basic representations of discrete time
stochastic processes. Different attempts have been made to provide non-linear
versions of the basic autoregressive process, including different versions
based on kernel methods. Motivated by the powerful framework of Hilbert space
embeddings of distributions, in this paper we apply this methodology for the
kernel embedding of an autoregressive process of order . By doing so, we
provide a non-linear version of an autoregressive process, that shows increased
performance over the linear model in highly complex time series. We use the
method proposed for one-step ahead forecasting of different time-series, and
compare its performance against other non-linear methods
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