28,838 research outputs found
Labeled Directed Acyclic Graphs: a generalization of context-specific independence in directed graphical models
We introduce a novel class of labeled directed acyclic graph (LDAG) models
for finite sets of discrete variables. LDAGs generalize earlier proposals for
allowing local structures in the conditional probability distribution of a
node, such that unrestricted label sets determine which edges can be deleted
from the underlying directed acyclic graph (DAG) for a given context. Several
properties of these models are derived, including a generalization of the
concept of Markov equivalence classes. Efficient Bayesian learning of LDAGs is
enabled by introducing an LDAG-based factorization of the Dirichlet prior for
the model parameters, such that the marginal likelihood can be calculated
analytically. In addition, we develop a novel prior distribution for the model
structures that can appropriately penalize a model for its labeling complexity.
A non-reversible Markov chain Monte Carlo algorithm combined with a greedy hill
climbing approach is used for illustrating the useful properties of LDAG models
for both real and synthetic data sets.Comment: 26 pages, 17 figure
Locally Testable Codes and Cayley Graphs
We give two new characterizations of (\F_2-linear) locally testable
error-correcting codes in terms of Cayley graphs over \F_2^h:
\begin{enumerate} \item A locally testable code is equivalent to a Cayley
graph over \F_2^h whose set of generators is significantly larger than
and has no short linear dependencies, but yields a shortest-path metric that
embeds into with constant distortion. This extends and gives a
converse to a result of Khot and Naor (2006), which showed that codes with
large dual distance imply Cayley graphs that have no low-distortion embeddings
into .
\item A locally testable code is equivalent to a Cayley graph over \F_2^h
that has significantly more than eigenvalues near 1, which have no short
linear dependencies among them and which "explain" all of the large
eigenvalues. This extends and gives a converse to a recent construction of
Barak et al. (2012), which showed that locally testable codes imply Cayley
graphs that are small-set expanders but have many large eigenvalues.
\end{enumerate}Comment: 22 page
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