6,690 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
Chromatic PAC-Bayes Bounds for Non-IID Data: Applications to Ranking and Stationary -Mixing Processes
Pac-Bayes bounds are among the most accurate generalization bounds for
classifiers learned from independently and identically distributed (IID) data,
and it is particularly so for margin classifiers: there have been recent
contributions showing how practical these bounds can be either to perform model
selection (Ambroladze et al., 2007) or even to directly guide the learning of
linear classifiers (Germain et al., 2009). However, there are many practical
situations where the training data show some dependencies and where the
traditional IID assumption does not hold. Stating generalization bounds for
such frameworks is therefore of the utmost interest, both from theoretical and
practical standpoints. In this work, we propose the first - to the best of our
knowledge - Pac-Bayes generalization bounds for classifiers trained on data
exhibiting interdependencies. The approach undertaken to establish our results
is based on the decomposition of a so-called dependency graph that encodes the
dependencies within the data, in sets of independent data, thanks to graph
fractional covers. Our bounds are very general, since being able to find an
upper bound on the fractional chromatic number of the dependency graph is
sufficient to get new Pac-Bayes bounds for specific settings. We show how our
results can be used to derive bounds for ranking statistics (such as Auc) and
classifiers trained on data distributed according to a stationary {\ss}-mixing
process. In the way, we show how our approach seemlessly allows us to deal with
U-processes. As a side note, we also provide a Pac-Bayes generalization bound
for classifiers learned on data from stationary -mixing distributions.Comment: Long version of the AISTATS 09 paper:
http://jmlr.csail.mit.edu/proceedings/papers/v5/ralaivola09a/ralaivola09a.pd
Local Graph Coloring and Index Coding
We present a novel upper bound for the optimal index coding rate. Our bound
uses a graph theoretic quantity called the local chromatic number. We show how
a good local coloring can be used to create a good index code. The local
coloring is used as an alignment guide to assign index coding vectors from a
general position MDS code. We further show that a natural LP relaxation yields
an even stronger index code. Our bounds provably outperform the state of the
art on index coding but at most by a constant factor.Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013;
typos correcte
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