4,707 research outputs found
On Index Coding and Graph Homomorphism
In this work, we study the problem of index coding from graph homomorphism
perspective. We show that the minimum broadcast rate of an index coding problem
for different variations of the problem such as non-linear, scalar, and vector
index code, can be upper bounded by the minimum broadcast rate of another index
coding problem when there exists a homomorphism from the complement of the side
information graph of the first problem to that of the second problem. As a
result, we show that several upper bounds on scalar and vector index code
problem are special cases of one of our main theorems.
For the linear scalar index coding problem, it has been shown in [1] that the
binary linear index of a graph is equal to a graph theoretical parameter called
minrank of the graph. For undirected graphs, in [2] it is shown that
if and only if there exists a homomorphism from
to a predefined graph . Combining these two results, it
follows that for undirected graphs, all the digraphs with linear index of at
most k coincide with the graphs for which there exists a homomorphism from
to . In this paper, we give a direct proof to this result
that works for digraphs as well.
We show how to use this classification result to generate lower bounds on
scalar and vector index. In particular, we provide a lower bound for the scalar
index of a digraph in terms of the chromatic number of its complement.
Using our framework, we show that by changing the field size, linear index of
a digraph can be at most increased by a factor that is independent from the
number of the nodes.Comment: 5 pages, to appear in "IEEE Information Theory Workshop", 201
Subdeterminant Maximization via Nonconvex Relaxations and Anti-concentration
Several fundamental problems that arise in optimization and computer science
can be cast as follows: Given vectors and a
constraint family , find a set that
maximizes the squared volume of the simplex spanned by the vectors in . A
motivating example is the data-summarization problem in machine learning where
one is given a collection of vectors that represent data such as documents or
images. The volume of a set of vectors is used as a measure of their diversity,
and partition or matroid constraints over are imposed in order to ensure
resource or fairness constraints. Recently, Nikolov and Singh presented a
convex program and showed how it can be used to estimate the value of the most
diverse set when corresponds to a partition matroid. This result was
recently extended to regular matroids in works of Straszak and Vishnoi, and
Anari and Oveis Gharan. The question of whether these estimation algorithms can
be converted into the more useful approximation algorithms -- that also output
a set -- remained open.
The main contribution of this paper is to give the first approximation
algorithms for both partition and regular matroids. We present novel
formulations for the subdeterminant maximization problem for these matroids;
this reduces them to the problem of finding a point that maximizes the absolute
value of a nonconvex function over a Cartesian product of probability
simplices. The technical core of our results is a new anti-concentration
inequality for dependent random variables that allows us to relate the optimal
value of these nonconvex functions to their value at a random point. Unlike
prior work on the constrained subdeterminant maximization problem, our proofs
do not rely on real-stability or convexity and could be of independent interest
both in algorithms and complexity.Comment: in FOCS 201
On the Sample Information About Parameter and Prediction
The Bayesian measure of sample information about the parameter, known as
Lindley's measure, is widely used in various problems such as developing prior
distributions, models for the likelihood functions and optimal designs. The
predictive information is defined similarly and used for model selection and
optimal designs, though to a lesser extent. The parameter and predictive
information measures are proper utility functions and have been also used in
combination. Yet the relationship between the two measures and the effects of
conditional dependence between the observable quantities on the Bayesian
information measures remain unexplored. We address both issues. The
relationship between the two information measures is explored through the
information provided by the sample about the parameter and prediction jointly.
The role of dependence is explored along with the interplay between the
information measures, prior and sampling design. For the conditionally
independent sequence of observable quantities, decompositions of the joint
information characterize Lindley's measure as the sample information about the
parameter and prediction jointly and the predictive information as part of it.
For the conditionally dependent case, the joint information about parameter and
prediction exceeds Lindley's measure by an amount due to the dependence. More
specific results are shown for the normal linear models and a broad subfamily
of the exponential family. Conditionally independent samples provide relatively
little information for prediction, and the gap between the parameter and
predictive information measures grows rapidly with the sample size.Comment: Published in at http://dx.doi.org/10.1214/10-STS329 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Time-ordering and a generalized Magnus expansion
Both the classical time-ordering and the Magnus expansion are well-known in
the context of linear initial value problems. Motivated by the noncommutativity
between time-ordering and time derivation, and related problems raised recently
in statistical physics, we introduce a generalization of the Magnus expansion.
Whereas the classical expansion computes the logarithm of the evolution
operator of a linear differential equation, our generalization addresses the
same problem, including however directly a non-trivial initial condition. As a
by-product we recover a variant of the time ordering operation, known as
T*-ordering. Eventually, placing our results in the general context of
Rota-Baxter algebras permits us to present them in a more natural algebraic
setting. It encompasses, for example, the case where one considers linear
difference equations instead of linear differential equations
Fractional forcing number of graphs
The notion of forcing sets for perfect matchings was introduced by Harary,
Klein, and \v{Z}ivkovi\'{c}. The application of this problem in chemistry, as
well as its interesting theoretical aspects, made this subject very active. In
this work, we introduce the notion of the forcing function of fractional
perfect matchings which is continuous analogous to forcing sets defined over
the perfect matching polytope of graphs. We show that our defined object is a
continuous and concave function extension of the integral forcing set. Then, we
use our results about this extension to conclude new bounds and results about
the integral case of forcing sets for the family of edge and vertex-transitive
graphs and in particular, hypercube graphs
Exponential renormalization
Moving beyond the classical additive and multiplicative approaches, we
present an "exponential" method for perturbative renormalization. Using Dyson's
identity for Green's functions as well as the link between the Faa di Bruno
Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the
composition of formal power series is analyzed. Eventually, we argue that the
new method has several attractive features and encompasses the BPHZ method. The
latter can be seen as a special case of the new procedure for renormalization
scheme maps with the Rota-Baxter property. To our best knowledge, although very
natural from group-theoretical and physical points of view, several ideas
introduced in the present paper seem to be new (besides the exponential method,
let us mention the notions of counterfactors and of order n bare coupling
constants).Comment: revised version; accepted for publication in Annales Henri Poincar
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