8,457 research outputs found
Discrete torsion, symmetric products and the Hilbert scheme
We combine our results on symmetric products and second quantization with our
description of discrete torsion in order to explain the ring structure of the
cohomology of the Hilbert scheme of points on a K3 surface. This is achieved in
terms of an essentially unique symmetric group Frobenius algebra twisted by a
specific discrete torsion. This twist is realized in form of a tensor product
with a twisted group algebra that is defined by a discrete torsion cocycle. We
furthermore show that the form of this cocycle is dictated by the geometry of
the Hilbert scheme as a resolution of singularities of the symmetric product.Comment: This is an older paper (2004), which we would like to make more
widely accessibl
Second quantized Frobenius algebras
We show that given a Frobenius algebra there is a unique notion of its second
quantization, which is the sum over all symmetric group quotients of n--th
tensor powers, where the quotients are given by symmetric group twisted
Frobenius algebras. To this end, we consider the setting of Frobenius algebras
given by functors from geometric categories whose objects are endowed with
geometric group actions and prove structural results, which in turn yield a
constructive realization in the case of n--th tensor powers and the natural
permutation action. We also show that naturally graded symmetric group twisted
Frobenius algebras have a unique algebra structure already determined by their
underlying additive data together with a choice of super--grading. Furthermore
we discuss several notions of discrete torsion andshow that indeed a
non--trivial discrete torsion leads to a non--trivial super structure on the
second quantization.Comment: 39p. Latex. New version fixes sign mistake and includes the full
description of discrete torsio
On the Automatic Parameter Selection for Permutation Entropy
Permutation Entropy (PE) has been shown to be a useful tool for time series
analysis due to its low computational cost and noise robustness. This has drawn
for its successful application in many fields. Some of these include damage
detection, disease forecasting, and financial volatility analysis. However, to
successfully use PE, an accurate selection of two parameters is needed: the
permutation dimension and embedding delay . These parameters are
often suggested by experts based on a heuristic or by a trial and error
approach. unfortunately, both of these methods can be time-consuming and lead
to inaccurate results. To help combat this issue, in this paper we investigate
multiple schemes for automatically selecting these parameters with only the
corresponding time series as the input. Specifically, we develop a
frequency-domain approach based on the least median of squares and the Fourier
spectrum, as well as extend two existing methods: Permutation Auto-Mutual
Information (PAMI) and Multi-scale Permutation Entropy (MPE) for determining
. We then compare our methods as well as current methods in the
literature for obtaining both and against expert-suggested values in
published works. We show that the success of any method in automatically
generating the correct PE parameters depends on the category of the studied
system. Specifically, for the delay parameter , we show that our
frequency approach provides accurate suggestions for periodic systems,
nonlinear difference equations, and ECG/EEG data, while the mutual information
function computed using adaptive partitions provides the most accurate results
for chaotic differential equations. For the permutation dimension , both
False Nearest Neighbors and MPE provide accurate values for for most of the
systems with being suitable in most cases.Comment: Abstract with all example systems provided in appendi
Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
Compressed Sensing aims to capture attributes of -sparse signals using
very few measurements. In the standard Compressed Sensing paradigm, the
\m\times \n measurement matrix \A is required to act as a near isometry on
the set of all -sparse signals (Restricted Isometry Property or RIP).
Although it is known that certain probabilistic processes generate \m \times
\n matrices that satisfy RIP with high probability, there is no practical
algorithm for verifying whether a given sensing matrix \A has this property,
crucial for the feasibility of the standard recovery algorithms. In contrast
this paper provides simple criteria that guarantee that a deterministic sensing
matrix satisfying these criteria acts as a near isometry on an overwhelming
majority of -sparse signals; in particular, most such signals have a unique
representation in the measurement domain. Probability still plays a critical
role, but it enters the signal model rather than the construction of the
sensing matrix. We require the columns of the sensing matrix to form a group
under pointwise multiplication. The construction allows recovery methods for
which the expected performance is sub-linear in \n, and only quadratic in
\m; the focus on expected performance is more typical of mainstream signal
processing than the worst-case analysis that prevails in standard Compressed
Sensing. Our framework encompasses many families of deterministic sensing
matrices, including those formed from discrete chirps, Delsarte-Goethals codes,
and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in
Signal Processing, the special issue on Compressed Sensin
Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis
Permutation Entropy (PE) is a powerful tool for quantifying the
predictability of a sequence which includes measuring the regularity of a time
series. Despite its successful application in a variety of scientific domains,
PE requires a judicious choice of the delay parameter . While another
parameter of interest in PE is the motif dimension , Typically is
selected between and with or giving optimal results for the
majority of systems. Therefore, in this work we focus solely on choosing the
delay parameter. Selecting is often accomplished using trial and error
guided by the expertise of domain scientists. However, in this paper, we show
that persistent homology, the flag ship tool from Topological Data Analysis
(TDA) toolset, provides an approach for the automatic selection of . We
evaluate the successful identification of a suitable from our TDA-based
approach by comparing our results to a variety of examples in published
literature
WARP: Wavelets with adaptive recursive partitioning for multi-dimensional data
Effective identification of asymmetric and local features in images and other
data observed on multi-dimensional grids plays a critical role in a wide range
of applications including biomedical and natural image processing. Moreover,
the ever increasing amount of image data, in terms of both the resolution per
image and the number of images processed per application, requires algorithms
and methods for such applications to be computationally efficient. We develop a
new probabilistic framework for multi-dimensional data to overcome these
challenges through incorporating data adaptivity into discrete wavelet
transforms, thereby allowing them to adapt to the geometric structure of the
data while maintaining the linear computational scalability. By exploiting a
connection between the local directionality of wavelet transforms and recursive
dyadic partitioning on the grid points of the observation, we obtain the
desired adaptivity through adding to the traditional Bayesian wavelet
regression framework an additional layer of Bayesian modeling on the space of
recursive partitions over the grid points. We derive the corresponding
inference recipe in the form of a recursive representation of the exact
posterior, and develop a class of efficient recursive message passing
algorithms for achieving exact Bayesian inference with a computational
complexity linear in the resolution and sample size of the images. While our
framework is applicable to a range of problems including multi-dimensional
signal processing, compression, and structural learning, we illustrate its work
and evaluate its performance in the context of 2D and 3D image reconstruction
using real images from the ImageNet database. We also apply the framework to
analyze a data set from retinal optical coherence tomography
Reconstruction Codes for DNA Sequences with Uniform Tandem-Duplication Errors
DNA as a data storage medium has several advantages, including far greater
data density compared to electronic media. We propose that schemes for data
storage in the DNA of living organisms may benefit from studying the
reconstruction problem, which is applicable whenever multiple reads of noisy
data are available. This strategy is uniquely suited to the medium, which
inherently replicates stored data in multiple distinct ways, caused by
mutations. We consider noise introduced solely by uniform tandem-duplication,
and utilize the relation to constant-weight integer codes in the Manhattan
metric. By bounding the intersection of the cross-polytope with hyperplanes, we
prove the existence of reconstruction codes with greater capacity than known
error-correcting codes, which we can determine analytically for any set of
parameters.Comment: 11 pages, 2 figures, Latex; version accepted for publicatio
Query Complexity of Mastermind Variants
We study variants of Mastermind, a popular board game in which the objective
is sequence reconstruction. In this two-player game, the so-called
\textit{codemaker} constructs a hidden sequence
of colors selected from an alphabet
(\textit{i.e.,} for all ). The game
then proceeds in turns, each of which consists of two parts: in turn , the
second player (the \textit{codebreaker}) first submits a query sequence with for all , and second
receives feedback , where is some agreed-upon function
of distance between two sequences with components. The game terminates when
, and the codebreaker seeks to end the game in as few turns as
possible. Throughout we let denote the smallest integer such that the
codebreaker can determine any in turns. We prove three main
results: First, when is known to be a permutation of ,
we prove that for all sufficiently large .
Second, we show that Knuth's Minimax algorithm identifies any in at most
queries. Third, when feedback is not received until all queries have been
submitted, we show that .Comment: Revised and trimmed- 17 page
Geometrical Ambiguity of Pair Statistics. I. Point Configurations
Point configurations have been widely used as model systems in condensed
matter physics, materials science and biology. Statistical descriptors such as
the -body distribution function is usually employed to characterize
the point configurations, among which the most extensively used is the pair
distribution function . An intriguing inverse problem of practical
importance that has been receiving considerable attention is the degree to
which a point configuration can be reconstructed from the pair distribution
function of a target configuration. Although it is known that the pair-distance
information contained in is in general insufficient to uniquely determine
a point configuration, this concept does not seem to be widely appreciated and
general claims of uniqueness of the reconstructions using pair information have
been made based on numerical studies. In this paper, we introduce the idea of
the distance space, called the space. The pair distances of a
specific point configuration are then represented by a single point in the
space. We derive the conditions on the pair distances that can be
associated with a point configuration, which are equivalent to the
realizability conditions of the pair distribution function . Moreover, we
derive the conditions on the pair distances that can be assembled into distinct
configurations. These conditions define a bounded region in the
space. By explicitly constructing a variety of degenerate point configurations
using the space, we show that pair information is indeed
insufficient to uniquely determine the configuration in general. We also
discuss several important problems in statistical physics based on the
space.Comment: 28 pages, 8 figure
Limits of discrete distributions and Gibbs measures on random graphs
Building upon the theory of graph limits and the Aldous-Hoover representation
and inspired by Panchenko's work on asymptotic Gibbs measures (Annals of
Probability 2013), we construct continuous embeddings of discrete probability
distributions. We show that the theory of graph limits induces a meaningful
notion of convergence and derive a corresponding version of the Szemer\'edi
regularity lemma. Moreover, complementing recent work (Bapst et. al. 2015), we
apply these results to Gibbs measures induced by sparse random factor graphs
and verify the "replica symmetric solution" predicted in the physics literature
under the assumption of non-reconstruction
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