17,064 research outputs found
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
Virtually all questions that one can ask about the behavioral and structural
complexity of a stochastic process reduce to a linear algebraic framing of a
time evolution governed by an appropriate hidden-Markov process generator. Each
type of question---correlation, predictability, predictive cost, observer
synchronization, and the like---induces a distinct generator class. Answers are
then functions of the class-appropriate transition dynamic. Unfortunately,
these dynamics are generically nonnormal, nondiagonalizable, singular, and so
on. Tractably analyzing these dynamics relies on adapting the recently
introduced meromorphic functional calculus, which specifies the spectral
decomposition of functions of nondiagonalizable linear operators, even when the
function poles and zeros coincide with the operator's spectrum. Along the way,
we establish special properties of the projection operators that demonstrate
how they capture the organization of subprocesses within a complex system.
Circumventing the spurious infinities of alternative calculi, this leads in the
sequel, Part II, to the first closed-form expressions for complexity measures,
couched either in terms of the Drazin inverse (negative-one power of a singular
operator) or the eigenvalues and projection operators of the appropriate
transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht
Algorithmic and Statistical Perspectives on Large-Scale Data Analysis
In recent years, ideas from statistics and scientific computing have begun to
interact in increasingly sophisticated and fruitful ways with ideas from
computer science and the theory of algorithms to aid in the development of
improved worst-case algorithms that are useful for large-scale scientific and
Internet data analysis problems. In this chapter, I will describe two recent
examples---one having to do with selecting good columns or features from a (DNA
Single Nucleotide Polymorphism) data matrix, and the other having to do with
selecting good clusters or communities from a data graph (representing a social
or information network)---that drew on ideas from both areas and that may serve
as a model for exploiting complementary algorithmic and statistical
perspectives in order to solve applied large-scale data analysis problems.Comment: 33 pages. To appear in Uwe Naumann and Olaf Schenk, editors,
"Combinatorial Scientific Computing," Chapman and Hall/CRC Press, 201
Quark-Hadron Duality
I review the notion of the quark-hadron duality from the modern perspective.
Both, the theoretical foundation and practical applications are discussed. The
proper theoretical framework in which the problem can be formulated and treated
is Wilson's operator product expansion (OPE). Two models developed for the
description of duality violations are considered in some detail: one is
instanton-based, another resonance-based. The mechanisms they represent are
complementary. Although both models are rather primitive (their largest virtue
is their simplicity) they hopefully capture important features of the
phenomenon. Being open for improvements, they can be used "as is" for
orientation in the studies of duality violations in the processes of practical
interest.Comment: Based on the talks delivered at the VIII-th International Symposium
on Heavy Flavor Physics, Southampton, UK, 25-29 July 1999, and the
International Workshop "Gribov-70", Orsay, France, 27-29 March 2000. To be
published in the Boris Ioffe Festschrift "At the Frontier of Particle
Physics/Handbook of QCD", Ed. M. Shifman (World Scientific, Singapore, 2001);
41 pages, 14 eps figures, Late
On the noncommutative geometry of tilings
This is a chapter in an incoming book on aperiodic order. We review results
about the topology, the dynamics, and the combinatorics of aperiodically
ordered tilings obtained with the tools of noncommutative geometry
The Einstein Relation on Metric Measure Spaces
This note is based on F. Burghart's master thesis at Stuttgart university
from July 2018, supervised by Prof. Freiberg.
We review the Einstein relation, which connects the Hausdorff, local walk and
spectral dimensions on a space, in the abstract setting of a metric measure
space equipped with a suitable operator. This requires some twists compared to
the usual definitions from fractal geometry. The main result establishes the
invariance of the three involved notions of fractal dimension under
bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more
generally, how the transport of the analytic and stochastic structure behind
the Einstein relation works. While any homeomorphism suffices for this
transport of structure, non-Lipschitz maps distort the Hausdorff and the local
walk dimension in different ways. To illustrate this, we take a look at
H\"older regular transformations and how they influence the local walk
dimension and prove some partial results concerning the Einstein relation on
graphs of fractional Brownian motions. We conclude by giving a short list of
further questions that may help building a general theory of the Einstein
relation.Comment: 28 pages, 3 figure
A Closed-Form Shave from Occam's Quantum Razor: Exact Results for Quantum Compression
The causal structure of a stochastic process can be more efficiently
transmitted via a quantum channel than a classical one, an advantage that
increases with codeword length. While previously difficult to compute, we
express the quantum advantage in closed form using spectral decomposition,
leading to direct computation of the quantum communication cost at all encoding
lengths, including infinite. This makes clear how finite-codeword compression
is controlled by the classical process' cryptic order and allows us to analyze
structure within the length-asymptotic regime of infinite-cryptic order (and
infinite Markov order) processes.Comment: 21 pages, 13 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/eqc.ht
Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."
This is a technical report on the proceedings of the workshop held July 21 to
July 25, 2008 at the American Institute of Mathematics, Palo Alto, California,
organized by Joseph Landsberg, Lek-Heng Lim, Jason Morton, and Jerzy Weyman. We
include a list of open problems coming from applications in 4 different areas:
signal processing, the Mulmuley-Sohoni approach to P vs. NP, matchgates and
holographic algorithms, and entanglement and quantum information theory. We
emphasize the interactions between geometry and representation theory and these
applied areas
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