494 research outputs found
Fastest mixing Markov chain on graphs with symmetries
We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue modulus of
the transition probability matrix). Exploiting symmetry can lead to significant
reduction in both the number of variables and the size of matrices in the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible. We obtain
analytic or semi-analytic results for particular classes of graphs, such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection between
these two approaches.Comment: 39 pages, 15 figure
Improved mixing rates of directed cycles by added connection
We investigate the mixing rate of a Markov chain where a combination of long
distance edges and non-reversibility is introduced: as a first step, we focus
here on the following graphs: starting from the cycle graph, we select random
nodes and add all edges connecting them. We prove a square factor improvement
of the mixing rate compared to the reversible version of the Markov chain
Accelerating Consensus by Spectral Clustering and Polynomial Filters
It is known that polynomial filtering can accelerate the convergence towards
average consensus on an undirected network. In this paper the gain of a
second-order filtering is investigated. A set of graphs is determined for which
consensus can be attained in finite time, and a preconditioner is proposed to
adapt the undirected weights of any given graph to achieve fastest convergence
with the polynomial filter. The corresponding cost function differs from the
traditional spectral gap, as it favors grouping the eigenvalues in two
clusters. A possible loss of robustness of the polynomial filter is also
highlighted
Lifted Probabilistic Inference: An MCMC Perspective
The general consensus seems to be that lifted
inference is concerned with exploiting model
symmetries and grouping indistinguishable
objects at inference time. Since first-order
probabilistic formalisms are essentially tem-
plate languages providing a more compact
representation of a corresponding ground
model, lifted inference tends to work especially well in these models. We show that the
notion of indistinguishability manifests itself
on several dferent levels {the level of constants, the level of ground atoms (variables),
the level of formulas (features), and the level
of assignments (possible worlds). We discuss
existing work in the MCMC literature on ex-
ploiting symmetries on the level of variable
assignments and relate it to novel results in
lifted MCMC
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