187 research outputs found
Upper bound on list-decoding radius of binary codes
Consider the problem of packing Hamming balls of a given relative radius
subject to the constraint that they cover any point of the ambient Hamming
space with multiplicity at most . For odd an asymptotic upper bound
on the rate of any such packing is proven. Resulting bound improves the best
known bound (due to Blinovsky'1986) for rates below a certain threshold. Method
is a superposition of the linear-programming idea of Ashikhmin, Barg and Litsyn
(that was used previously to improve the estimates of Blinovsky for ) and
a Ramsey-theoretic technique of Blinovsky. As an application it is shown that
for all odd the slope of the rate-radius tradeoff is zero at zero rate.Comment: IEEE Trans. Inform. Theory, accepte
Coherent multiple-antenna block-fading channels at finite blocklength
In this paper we consider a channel model that is often used to describe the
mobile wireless scenario: multiple-antenna additive white Gaussian noise
channels subject to random (fading) gain with full channel state information at
the receiver. Dynamics of the fading process are approximated by a
piecewise-constant process (frequency non-selective isotropic block fading).
This work addresses the finite blocklength fundamental limits of this channel
model. Specifically, we give a formula for the channel dispersion -- a quantity
governing the delay required to achieve capacity. Multiplicative nature of the
fading disturbance leads to a number of interesting technical difficulties that
required us to enhance traditional methods for finding channel dispersion.
Alas, one difficulty remains: the converse (impossibility) part of our result
holds under an extra constraint on the growth of the peak-power with
blocklength.
Our results demonstrate, for example, that while capacities of and antenna configurations coincide (under fixed received
power), the coding delay can be quite sensitive to this switch. For example, at
the received SNR of dB the system achieves capacity with
codes of length (delay) which is only of the length required for the
system. Another interesting implication is that for the MISO
channel, the dispersion-optimal coding schemes require employing orthogonal
designs such as Alamouti's scheme -- a surprising observation considering the
fact that Alamouti's scheme was designed for reducing demodulation errors, not
improving coding rate. Finding these dispersion-optimal coding schemes
naturally gives a criteria for producing orthogonal design-like inputs in
dimensions where orthogonal designs do not exist
Dissipation of information in channels with input constraints
One of the basic tenets in information theory, the data processing inequality
states that output divergence does not exceed the input divergence for any
channel. For channels without input constraints, various estimates on the
amount of such contraction are known, Dobrushin's coefficient for the total
variation being perhaps the most well-known. This work investigates channels
with average input cost constraint. It is found that while the contraction
coefficient typically equals one (no contraction), the information nevertheless
dissipates. A certain non-linear function, the \emph{Dobrushin curve} of the
channel, is proposed to quantify the amount of dissipation. Tools for
evaluating the Dobrushin curve of additive-noise channels are developed based
on coupling arguments. Some basic applications in stochastic control,
uniqueness of Gibbs measures and fundamental limits of noisy circuits are
discussed.
As an application, it shown that in the chain of power-constrained relays
and Gaussian channels the end-to-end mutual information and maximal squared
correlation decay as , which is in stark
contrast with the exponential decay in chains of discrete channels. Similarly,
the behavior of noisy circuits (composed of gates with bounded fan-in) and
broadcasting of information on trees (of bounded degree) does not experience
threshold behavior in the signal-to-noise ratio (SNR). Namely, unlike the case
of discrete channels, the probability of bit error stays bounded away from
regardless of the SNR.Comment: revised; include appendix B on contraction coefficient for mutual
information on general alphabet
Algebraic Methods of Classifying Directed Graphical Models
Directed acyclic graphical models (DAGs) are often used to describe common
structural properties in a family of probability distributions. This paper
addresses the question of classifying DAGs up to an isomorphism. By considering
Gaussian densities, the question reduces to verifying equality of certain
algebraic varieties. A question of computing equations for these varieties has
been previously raised in the literature. Here it is shown that the most
natural method adds spurious components with singular principal minors, proving
a conjecture of Sullivant. This characterization is used to establish an
algebraic criterion for isomorphism, and to provide a randomized algorithm for
checking that criterion. Results are applied to produce a list of the
isomorphism classes of tree models on 4,5, and 6 nodes. Finally, some evidence
is provided to show that projectivized DAG varieties contain useful information
in the sense that their relative embedding is closely related to efficient
inference
Attracting Random Walks
This paper introduces the Attracting Random Walks model, which describes the
dynamics of a system of particles on a graph with vertices. At each step, a
single particle moves to an adjacent vertex (or stays at the current one) with
probability proportional to the exponent of the number of other particles at a
vertex. From an applied standpoint, the model captures the rich get richer
phenomenon. We show that the Markov chain exhibits a phase transition in mixing
time, as the parameter governing the attraction is varied. Namely, mixing time
is when the temperature is sufficiently high and
when temperature is sufficiently low. When is the complete graph,
the model is a projection of the Potts model, whose mixing properties and the
critical temperature have been known previously. However, for any other graph
our model is non-reversible and does not seem to admit a simple Gibbsian
description of a stationary distribution. Notably, we demonstrate existence of
the dynamic phase transition without decomposing the stationary distribution
into phases.Comment: 32 pages, 7 figure
- β¦