182,499 research outputs found
Multi-Stage Robust Chinese Remainder Theorem
It is well-known that the traditional Chinese remainder theorem (CRT) is not
robust in the sense that a small error in a remainder may cause a large error
in the reconstruction solution. A robust CRT was recently proposed for a
special case when the greatest common divisor (gcd) of all the moduli is more
than 1 and the remaining integers factorized by the gcd of all the moduli are
co-prime. In this special case, a closed-form reconstruction from erroneous
remainders was proposed and a necessary and sufficient condition on the
remainder errors was obtained. It basically says that the reconstruction error
is upper bounded by the remainder error level if is smaller than
a quarter of the gcd of all the moduli. In this paper, we consider the robust
reconstruction problem for a general set of moduli. We first present a
necessary and sufficient condition for the remainder errors for a robust
reconstruction from erroneous remainders with a general set of muduli and also
a corresponding robust reconstruction method. This can be thought of as a
single stage robust CRT. We then propose a two-stage robust CRT by grouping the
moduli into several groups as follows. First, the single stage robust CRT is
applied to each group. Then, with these robust reconstructions from all the
groups, the single stage robust CRT is applied again across the groups. This is
then easily generalized to multi-stage robust CRT. Interestingly, with this
two-stage robust CRT, the robust reconstruction holds even when the remainder
error level is above the quarter of the gcd of all the moduli. In this
paper, we also propose an algorithm on how to group a set of moduli for a
better reconstruction robustness of the two-stage robust CRT in some special
cases.Comment: 26 pages, 2 figure
An Information Theoretic Study for Noisy Compressed Sensing With Joint Sparsity Model-2
In this paper, we study a support set reconstruction problem in which the
signals of interest are jointly sparse with a common support set, and sampled
by joint sparsity model-2 (JSM-2) in the presence of noise. Using mathematical
tools, we develop upper and lower bounds on the failure probability of support
set reconstruction in terms of the sparsity, the ambient dimension, the minimum
signal to noise ratio, the number of measurement vectors and the number of
measurements. These bounds can be used to provide a guideline to determine the
system parameters in various applications of compressed sensing with noisy
JSM-2. Based on the bounds, we develop necessary and sufficient conditions for
reliable support set reconstruction. We interpret these conditions to give
theoretical explanations about the benefits enabled by joint sparsity structure
in noisy JSM-2. We compare our sufficient condition with the existing result of
noisy multiple measurement vectors model (MMV). As a result, we show that noisy
JSM-2 may require less number of measurements than noisy MMV for reliable
support set reconstruction
Coding With Action-dependent Side Information and Additional Reconstruction Requirements
Constrained lossy source coding and channel coding with side information
problems which extend the classic Wyner-Ziv and Gel'fand-Pinsker problems are
considered. Inspired by applications in sensor networking and control, we first
consider lossy source coding with two-sided partial side information where the
quality/availability of the side information can be influenced by a
cost-constrained action sequence. A decoder reconstructs a source sequence
subject to the distortion constraint, and at the same time, an encoder is
additionally required to be able to estimate the decoder's reconstruction.
Next, we consider the channel coding "dual" where the channel state is assumed
to depend on the action sequence, and the decoder is required to decode both
the transmitted message and channel input reliably. Implications on the
fundamental limits of communication in discrete memoryless systems due to the
additional reconstruction constraints are investigated. Single-letter
expressions for the rate-distortion-cost function and channel capacity for the
respective source and channel coding problems are derived. The dual relation
between the two problems is discussed. Additionally, based on the two-stage
coding structure and the additional reconstruction constraint of the channel
coding problem, we discuss and give an interpretation of the two-stage coding
condition which appears in the channel capacity expression. Besides the rate
constraint on the message, this condition is a necessary and sufficient
condition for reliable transmission of the channel input sequence over the
channel in our "two-stage" communication problem. It is also shown in one
example that there exists a case where the two-stage coding condition can be
active in computing the capacity, and it thus can actively restrict the set of
capacity achieving input distributions.Comment: submitted to IEEE Trans. Information Theor
Reconstruction of convex bodies from moments
We investigate how much information about a convex body can be retrieved from
a finite number of its geometric moments. We give a sufficient condition for a
convex body to be uniquely determined by a finite number of its geometric
moments, and we show that among all convex bodies, those which are uniquely
determined by a finite number of moments form a dense set. Further, we derive a
stability result for convex bodies based on geometric moments. It turns out
that the stability result is improved considerably by using another set of
moments, namely Legendre moments. We present a reconstruction algorithm that
approximates a convex body using a finite number of its Legendre moments. The
consistency of the algorithm is established using the stability result for
Legendre moments. When only noisy measurements of Legendre moments are
available, the consistency of the algorithm is established under certain
assumptions on the variance of the noise variables
Random and universal metric spaces
We introduce a model of the set of all Polish (=separable complete metric)
spaces: the cone of distance matrices, and consider geometric and
probabilistic problems connected with this object. The notion of the universal
distance matrix is defined and we proved that the set of such matrices is
everywhere dense set in weak topology in the cone .
Universality of distance matrix is the necessary and sufficient condition on
the distance matrix of the countable everywhere dense set of so called
universal Urysohn space which he had defined in 1924 in his last paper. This
means that Urysohn space is generic in the set of all Polish spaces. Then we
consider metric spaces with measures (metric triples) and define a complete
invariant: its - matrix distribution. We give an intrinsic characterization of
the set of matrix distributions, and using the ergodic theorem, give a new
proof of Gromov's ``reconstruction theorem'. A natural construction of a wide
class of measures on the cone is given and for these we show that {\it
with probability one a random Polish space is again the Urysohn space}. There
is a close connection between these questions, metric classification of
measurable functions of several arguments, and classification of the actions of
the infinite symmetric group [V1,V2].Comment: 30 pages, substantially revised versio
Direct measurement of the biphoton Wigner function through two-photon interference
The Hong-Ou-Mandel (HOM) experiment was a benchmark in quantum optics,
evidencing the quantum nature of the photon. In order to go deeper, and obtain
the complete information about the quantum state of a system, for instance,
composed by photons, the direct measurement or reconstruction of the Wigner
function or other quasi--probability distribution in phase space is necessary.
In the present paper, we show that a simple modification in the well-known HOM
experiment provides the direct measurement of the Wigner function. We apply our
results to a widely used quantum optics system, consisting of the biphoton
generated in the parametric down conversion process. In this approach, a
negative value of the Wigner function is a sufficient condition for
non-gaussian entanglement between two photons. In the general case, the Wigner
function provides all the required information to infer entanglement using well
known necessary and sufficient criteria. We analyze our results using two
examples of parametric down conversion processes taken from recent experiments.
The present work offers a new vision of the HOM experiment that further
develops its possibilities to realize fundamental tests of quantum mechanics
involving decoherence and entanglement using simple optical set-ups.Comment: 5 pages, 2 figures plus supplementary information; submitted;
comments welcom
Robust Compressed Sensing Under Matrix Uncertainties
Compressed sensing (CS) shows that a signal having a sparse or compressible
representation can be recovered from a small set of linear measurements. In
classical CS theory, the sampling matrix and representation matrix are assumed
to be known exactly in advance. However, uncertainties exist due to sampling
distortion, finite grids of the parameter space of dictionary, etc. In this
paper, we take a generalized sparse signal model, which simultaneously
considers the sampling and representation matrix uncertainties. Based on the
new signal model, a new optimization model for robust sparse signal
reconstruction is proposed. This optimization model can be deduced with
stochastic robust approximation analysis. Both convex relaxation and greedy
algorithms are used to solve the optimization problem. For the convex
relaxation method, a sufficient condition for recovery by convex relaxation is
given; For the greedy algorithm, it is realized by the introduction of a
pre-processing of the sensing matrix and the measurements. In numerical
experiments, both simulated data and real-life ECG data based results show that
the proposed method has a better performance than the current methods.Comment: 17 pages, 8 figure
Randomized transmit and receive ultrasound tomography
A tomographic method is considered that forms images from sets of spatially
randomized source signals and receiver sensitivities. The method is designed to
allow image reconstruction for an extended number of transmitters and receivers
in the presence noise and without plane wave approximation or otherwise
approximation on the size or regularity of source and receiver functions. An
overdetermined set of functions are formed from the Hadamard product between a
Gaussian function and a uniformly distributed random number set. It is shown
that this particular type of randomization tends to produce well-conditioned
matrices whose pseudoinverses may be determined without implementing relaxation
methods. When the inverted sets are applied to simulated first-order scattering
from a Shepp-Logan phantom, successful image reconstructions are achieved for
signal-to-noise ratios (SNR) as low as 1. Evaluation of the randomization
approach is conducted by comparing condition numbers with other forms of signal
randomization. Image quality resulting from tomographic reconstructions is then
compared with an idealized synthetic aperture approach, which is subjected to a
comparable SNR. By root-mean-square-difference comparisons it is concluded that
- provided a sufficient level of oversampling - the dynamic transmit and
dynamic receive approach produces superior images, particularly in the presence
of low SNR.Comment: 16 pages, 7 figure
Spectral neighbor joining for reconstruction of latent tree models
A common assumption in multiple scientific applications is that the
distribution of observed data can be modeled by a latent tree graphical model.
An important example is phylogenetics, where the tree models the evolutionary
lineages of a set of observed organisms. Given a set of independent
realizations of the random variables at the leaves of the tree, a key challenge
is to infer the underlying tree topology. In this work we develop Spectral
Neighbor Joining (SNJ), a novel method to recover the structure of latent tree
graphical models. Given a matrix that contains a measure of similarity between
all pairs of observed variables, SNJ computes a spectral measure of cohesion
between groups of observed variables. We prove that SNJ is consistent, and
derive a sufficient condition for correct tree recovery from an estimated
similarity matrix. Combining this condition with a concentration of measure
result on the similarity matrix, we bound the number of samples required to
recover the tree with high probability. We illustrate via extensive simulations
that in comparison to several other reconstruction methods, SNJ requires fewer
samples to accurately recover trees with a large number of leaves or long
edges
Theoretical Bounds on Mate-Pair Information for Accurate Genome Assembly
Over the past two decades, a series of works have aimed at studying the
problem of genome assembly: the process of reconstructing a genome from
sequence reads. An early formulation of the genome assembly problem showed that
genome reconstruction is NP-hard when framed as finding the shortest sequence
that contains all observed reads. Although this original formulation is very
simplistic and does not allow for mate-pair information, subsequent
formulations have also proven to be NP-hard, and/or may not be guaranteed to
return a correct assembly.
In this paper, we provide an alternate perspective on the genome assembly
problem by showing genome assembly is easy when provided with sufficient
mate-pair information. Moreover, we quantify the number of mate-pair libraries
necessary and sufficient for accurate genome assembly, in terms of the length
of the longest repetitive region within a genome. In our analysis, we consider
an idealized sequencing model where each mate-pair library generates pairs of
error free reads with a fixed and known insert size at each position in the
genome.
Even in this idealized model, we show that accurate genome reconstruction
cannot be guaranteed in the worst case unless at least roughly R/2L mate-pair
libraries are produced, where R is the length of the longest repetitive region
in the genome and L is the length of each read. On the other hand, if (R/L)+1
mate-pair libraries are provided, then a simple algorithm can be used to find a
correct genome assembly easily in polynomial time. Although (R/L)+1 mate-pair
libraries can be too much to produce in practice, the previous bounds only hold
in the worst case. In our last result, we show that if additional conditions
hold on a genome, a correct assembly can be guaranteed with only O(log (R/L))
mate-pair libraries
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