52 research outputs found
Spectral Theory of Discrete Processes
We offer a spectral analysis for a class of transfer operators. These
transfer operators arise for a wide range of stochastic processes, ranging from
random walks on infinite graphs to the processes that govern signals and
recursive wavelet algorithms; even spectral theory for fractal measures. In
each case, there is an associated class of harmonic functions which we study.
And in addition, we study three questions in depth:
In specific applications, and for a specific stochastic process, how do we
realize the transfer operator as an operator in a suitable Hilbert space?
And how to spectral analyze once the right Hilbert space has
been selected? Finally we characterize the stochastic processes that are
governed by a single transfer operator.
In our applications, the particular stochastic process will live on an
infinite path-space which is realized in turn on a state space . In the case
of random walk on graphs , will be the set of vertices of . The
Hilbert space on which the transfer operator acts will then
be an space on , or a Hilbert space defined from an energy-quadratic
form.
This circle of problems is both interesting and non-trivial as it turns out
that may often be an unbounded linear operator in ; but even
if it is bounded, it is a non-normal operator, so its spectral theory is not
amenable to an analysis with the use of von Neumann's spectral theorem. While
we offer a number of applications, we believe that our spectral analysis will
have intrinsic interest for the theory of operators in Hilbert space.Comment: 34 pages with figures removed, for the full version with all the
figures please go to http://www.siue.edu/~msong/Research/spectrum.pd
Markov chains and generalized wavelet multiresolutions
We develop some new results for a general class of transfer operators, as they are used in a construction of multi-resolutions. We then proceed to give explicit and concrete applications. We further discuss the need for such a constructive harmonic analysis/dynamical systems approach to fractals
Infinite-Dimensional Measure Spaces and Frame Analysis
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 â„‹, we study three cases of measures. We first show that, for â„‹ infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain â„‹. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures
Reproducing Kernel Hilbert Space vs. Frame Estimates
We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn Hinto a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes
Filters and Matrix Factorization
We give a number of explicit matrix-algorithms for analysis/synthesis
in multi-phase filtering; i.e., the operation on discrete-time signals which
allow a separation into frequency-band components, one for each of the
ranges of bands, say N , starting with low-pass, and then corresponding
filtering in the other band-ranges. If there are N bands, the individual
filters will be combined into a single matrix action; so a representation of
the combined operation on all N bands by an N x N matrix, where the
corresponding matrix-entries are periodic functions; or their extensions to
functions of a complex variable. Hence our setting entails a fixed N x N
matrix over a prescribed algebra of functions of a complex variable. In the
case of polynomial filters, the factorizations will always be finite. A novelty
here is that we allow for a wide family of non-polynomial filter-banks.
Working modulo N in the time domain, our approach also allows for
a natural matrix-representation of both down-sampling and up-sampling.
The implementation encompasses the combined operation on input, filtering,
down-sampling, transmission, up-sampling, an action by dual filters,
and synthesis, merges into a single matrix operation. Hence our matrixfactorizations
break down the global filtering-process into elementary steps.
To accomplish this, we offer a number of adapted matrix factorizationalgorithms,
such that each factor in our product representation implements
in a succession of steps the filtering across pairs of frequency-bands; and so
it is of practical significance in implementing signal processing, including
filtering of digitized images. Our matrix-factorizations are especially useful
in the case of the processing a fixed, but large, number of bands
Dimension Reduction and Kernel Principal Component Analysis
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-Lo\`eve 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.Comment: 38 page
Entropy encoding, hilbert space, and karhunen-loève transforms
By introducing Hilbert space and operators, we show how probabilities, approximations, and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which optimize distinct measures of data encoding
- …