629 research outputs found
A Simple Derivation of the Refined Sphere Packing Bound Under Certain Symmetry Hypotheses
A judicious application of the Berry-Esseen theorem via suitable Augustin
information measures is demonstrated to be sufficient for deriving the sphere
packing bound with a prefactor that is
for all codes on certain
families of channels -- including the Gaussian channels and the non-stationary
Renyi symmetric channels -- and for the constant composition codes on
stationary memoryless channels. The resulting non-asymptotic bounds have
definite approximation error terms. As a preliminary result that might be of
interest on its own, the trade-off between type I and type II error
probabilities in the hypothesis testing problem with (possibly non-stationary)
independent samples is determined up to some multiplicative constants, assuming
that the probabilities of both types of error are decaying exponentially with
the number of samples, using the Berry-Esseen theorem.Comment: 20 page
The Third-Order Term in the Normal Approximation for the AWGN Channel
This paper shows that, under the average error probability formalism, the
third-order term in the normal approximation for the additive white Gaussian
noise channel with a maximal or equal power constraint is at least . This matches the upper bound derived by
Polyanskiy-Poor-Verd\'{u} (2010).Comment: 13 pages, 1 figur
Model for Spreading of Liquid Monolayers
Manipulating fluids at the nanoscale within networks of channels or chemical
lanes is a crucial challenge in developing small scale devices to be used in
microreactors or chemical sensors. In this context, ultra-thin (i.e.,
monolayer) films, experimentally observed in spreading of nano-droplets or upon
extraction from reservoirs in capillary rise geometries, represent an extreme
limit which is of physical and technological relevance since the dynamics is
governed solely by capillary forces. In this work we use kinetic Monte Carlo
(KMC) simulations to analyze in detail a simple, but realistic model proposed
by Burlatsky \textit{et al.} \cite{Burlatsky_prl96,Oshanin_jml} for the
two-dimensional spreading on homogeneous substrates of a fluid monolayer which
is extracted from a reservoir. Our simulations confirm the previously predicted
time-dependence of the spreading, , with as
the average position of the advancing edge at time , and they reveal a
non-trivial dependence of the prefactor on the strength of
inter-particle attraction and on the fluid density at the reservoir as
well as an -dependent spatial structure of the density profile of the
monolayer. The asymptotic density profile at long time and large spatial scale
is carefully analyzed within the continuum limit. We show that including the
effect of correlations in an effective manner into the standard mean-field
description leads to predictions both for the value of the threshold
interaction above which phase segregation occurs and for the density profiles
in excellent agreement with KMC simulations results.Comment: 21 pages, 9 figures, submitted to Phys. Rev.
The Saddlepoint Approximation: Unified Random Coding Asymptotics for Fixed and Varying Rates
This paper presents a saddlepoint approximation of the random-coding union
bound of Polyanskiy et al. for i.i.d. random coding over discrete memoryless
channels. The approximation is single-letter, and can thus be computed
efficiently. Moreover, it is shown to be asymptotically tight for both fixed
and varying rates, unifying existing achievability results in the regimes of
error exponents, second-order coding rates, and moderate deviations. For fixed
rates, novel exact-asymptotics expressions are specified to within a
multiplicative 1+o(1) term. A numerical example is provided for which the
approximation is remarkably accurate even at short block lengths.Comment: Accepted to ISIT 2014, presented without publication at ITA 201
The Sphere Packing Bound For Memoryless Channels
Sphere packing bounds (SPBs) ---with prefactors that are polynomial in the
block length--- are derived for codes on two families of memoryless channels
using Augustin's method: (possibly non-stationary) memoryless channels with
(possibly multiple) additive cost constraints and stationary memoryless
channels with convex constraints on the composition (i.e. empirical
distribution, type) of the input codewords. A variant of Gallager's bound is
derived in order to show that these sphere packing bounds are tight in terms of
the exponential decay rate of the error probability with the block length under
mild hypotheses.Comment: 29 page
Mean-field methods in evolutionary duplication-innovation-loss models for the genome-level repertoire of protein domains
We present a combined mean-field and simulation approach to different models
describing the dynamics of classes formed by elements that can appear,
disappear or copy themselves. These models, related to a paradigm
duplication-innovation model known as Chinese Restaurant Process, are devised
to reproduce the scaling behavior observed in the genome-wide repertoire of
protein domains of all known species. In view of these data, we discuss the
qualitative and quantitative differences of the alternative model formulations,
focusing in particular on the roles of element loss and of the specificity of
empirical domain classes.Comment: 10 Figures, 2 Table
The Sphere Packing Bound via Augustin's Method
A sphere packing bound (SPB) with a prefactor that is polynomial in the block
length is established for codes on a length product channel
assuming that the maximum order Renyi capacity among the component
channels, i.e. , is . The
reliability function of the discrete stationary product channels with feedback
is bounded from above by the sphere packing exponent. Both results are proved
by first establishing a non-asymptotic SPB. The latter result continues to hold
under a milder stationarity hypothesis.Comment: 30 pages. An error in the statement of Lemma 2 is corrected. The
change is inconsequential for the rest of the pape
Group testing with Random Pools: Phase Transitions and Optimal Strategy
The problem of Group Testing is to identify defective items out of a set of
objects by means of pool queries of the form "Does the pool contain at least a
defective?". The aim is of course to perform detection with the fewest possible
queries, a problem which has relevant practical applications in different
fields including molecular biology and computer science. Here we study GT in
the probabilistic setting focusing on the regime of small defective probability
and large number of objects, and . We construct and
analyze one-stage algorithms for which we establish the occurrence of a
non-detection/detection phase transition resulting in a sharp threshold, , for the number of tests. By optimizing the pool design we construct
algorithms whose detection threshold follows the optimal scaling . Then we consider two-stages algorithms and analyze their
performance for different choices of the first stage pools. In particular, via
a proper random choice of the pools, we construct algorithms which attain the
optimal value (previously determined in Ref. [16]) for the mean number of tests
required for complete detection. We finally discuss the optimal pool design in
the case of finite
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