126 research outputs found
Beyond the Cut-Set Bound: Uncertainty Computations in Network Coding with Correlated Sources
Cut-set bounds on achievable rates for network communication protocols are
not in general tight. In this paper we introduce a new technique for proving
converses for the problem of transmission of correlated sources in networks,
that results in bounds that are tighter than the corresponding cut-set bounds.
We also define the concept of "uncertainty region" which might be of
independent interest. We provide a full characterization of this region for the
case of two correlated random variables. The bounding technique works as
follows: on one hand we show that if the communication problem is solvable, the
uncertainty of certain random variables in the network with respect to
imaginary parties that have partial knowledge of the sources must satisfy some
constraints that depend on the network architecture. On the other hand, the
same uncertainties have to satisfy constraints that only depend on the joint
distribution of the sources. Matching these two leads to restrictions on the
statistical joint distribution of the sources in communication problems that
are solvable over a given network architecture.Comment: 12 pages, A short version appears in ISIT 201
On Optimal Finite-length Binary Codes of Four Codewords for Binary Symmetric Channels
Finite-length binary codes of four codewords are studied for memoryless
binary symmetric channels (BSCs) with the maximum likelihood decoding. For any
block-length, best linear codes of four codewords have been explicitly
characterized, but whether linear codes are better than nonlinear codes or not
is unknown in general. In this paper, we show that for any block-length, there
exists an optimal code of four codewords that is either linear or in a subset
of nonlinear codes, called Class-I codes. Based on the analysis of Class-I
codes, we derive sufficient conditions such that linear codes are optimal. For
block-length less than or equal to 8, our analytical results show that linear
codes are optimal. For block-length up to 300, numerical evaluations show that
linear codes are optimal.Comment: accepted by ISITA 202
Expander Chunked Codes
Chunked codes are efficient random linear network coding (RLNC) schemes with
low computational cost, where the input packets are encoded into small chunks
(i.e., subsets of the coded packets). During the network transmission, RLNC is
performed within each chunk. In this paper, we first introduce a simple
transfer matrix model to characterize the transmission of chunks, and derive
some basic properties of the model to facilitate the performance analysis. We
then focus on the design of overlapped chunked codes, a class of chunked codes
whose chunks are non-disjoint subsets of input packets, which are of special
interest since they can be encoded with negligible computational cost and in a
causal fashion. We propose expander chunked (EC) codes, the first class of
overlapped chunked codes that have an analyzable performance,where the
construction of the chunks makes use of regular graphs. Numerical and
simulation results show that in some practical settings, EC codes can achieve
rates within 91 to 97 percent of the optimum and outperform the
state-of-the-art overlapped chunked codes significantly.Comment: 26 pages, 3 figures, submitted for journal publicatio
Upper Bound Scalability on Achievable Rates of Batched Codes for Line Networks
The capacity of line networks with buffer size constraints is an open, but
practically important problem. In this paper, the upper bound on the achievable
rate of a class of codes, called batched codes, is studied for line networks.
Batched codes enable a range of buffer size constraints, and are general enough
to include special coding schemes studied in the literature for line networks.
Existing works have characterized the achievable rates of batched codes for
several classes of parameter sets, but leave the cut-set bound as the best
existing general upper bound. In this paper, we provide upper bounds on the
achievable rates of batched codes as functions of line network length for these
parameter sets. Our upper bounds are tight in order of the network length
compared with the existing achievability results.Comment: 6 pages, 1 tabl
Weighted Flow Diffusion for Local Graph Clustering with Node Attributes: an Algorithm and Statistical Guarantees
Local graph clustering methods aim to detect small clusters in very large
graphs without the need to process the whole graph. They are fundamental and
scalable tools for a wide range of tasks such as local community detection,
node ranking and node embedding. While prior work on local graph clustering
mainly focuses on graphs without node attributes, modern real-world graph
datasets typically come with node attributes that provide valuable additional
information. We present a simple local graph clustering algorithm for graphs
with node attributes, based on the idea of diffusing mass locally in the graph
while accounting for both structural and attribute proximities. Using
high-dimensional concentration results, we provide statistical guarantees on
the performance of the algorithm for the recovery of a target cluster with a
single seed node. We give conditions under which a target cluster generated
from a fairly general contextual random graph model, which includes both the
stochastic block model and the planted cluster model as special cases, can be
fully recovered with bounded false positives. Empirically, we validate all
theoretical claims using synthetic data, and we show that incorporating node
attributes leads to superior local clustering performances using real-world
graph datasets.Comment: 30 pages, 2 figures, 9 table
On Linear Operator Channels over Finite Fields
Motivated by linear network coding, communication channels perform linear
operation over finite fields, namely linear operator channels (LOCs), are
studied in this paper. For such a channel, its output vector is a linear
transform of its input vector, and the transformation matrix is randomly and
independently generated. The transformation matrix is assumed to remain
constant for every T input vectors and to be unknown to both the transmitter
and the receiver. There are NO constraints on the distribution of the
transformation matrix and the field size.
Specifically, the optimality of subspace coding over LOCs is investigated. A
lower bound on the maximum achievable rate of subspace coding is obtained and
it is shown to be tight for some cases. The maximum achievable rate of
constant-dimensional subspace coding is characterized and the loss of rate
incurred by using constant-dimensional subspace coding is insignificant.
The maximum achievable rate of channel training is close to the lower bound
on the maximum achievable rate of subspace coding. Two coding approaches based
on channel training are proposed and their performances are evaluated. Our
first approach makes use of rank-metric codes and its optimality depends on the
existence of maximum rank distance codes. Our second approach applies linear
coding and it can achieve the maximum achievable rate of channel training. Our
code designs require only the knowledge of the expectation of the rank of the
transformation matrix. The second scheme can also be realized ratelessly
without a priori knowledge of the channel statistics.Comment: 53 pages, 3 figures, submitted to IEEE Transaction on Information
Theor
Capacity Analysis of Linear Operator Channels over Finite Fields
Motivated by communication through a network employing linear network coding,
capacities of linear operator channels (LOCs) with arbitrarily distributed
transfer matrices over finite fields are studied. Both the Shannon capacity
and the subspace coding capacity are analyzed. By establishing
and comparing lower bounds on and upper bounds on , various
necessary conditions and sufficient conditions such that are
obtained. A new class of LOCs such that is identified, which
includes LOCs with uniform-given-rank transfer matrices as special cases. It is
also demonstrated that is strictly less than for a broad
class of LOCs. In general, an optimal subspace coding scheme is difficult to
find because it requires to solve the maximization of a non-concave function.
However, for a LOC with a unique subspace degradation, can be
obtained by solving a convex optimization problem over rank distribution.
Classes of LOCs with a unique subspace degradation are characterized. Since
LOCs with uniform-given-rank transfer matrices have unique subspace
degradations, some existing results on LOCs with uniform-given-rank transfer
matrices are explained from a more general way.Comment: To appear in IEEE Transactions on Information Theor
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