1,073 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
Generating and Sampling Orbits for Lifted Probabilistic Inference
A key goal in the design of probabilistic inference algorithms is identifying
and exploiting properties of the distribution that make inference tractable.
Lifted inference algorithms identify symmetry as a property that enables
efficient inference and seek to scale with the degree of symmetry of a
probability model. A limitation of existing exact lifted inference techniques
is that they do not apply to non-relational representations like factor graphs.
In this work we provide the first example of an exact lifted inference
algorithm for arbitrary discrete factor graphs. In addition we describe a
lifted Markov-Chain Monte-Carlo algorithm that provably mixes rapidly in the
degree of symmetry of the distribution
Lie Markov models with purine/pyrimidine symmetry
Continuous-time Markov chains are a standard tool in phylogenetic inference.
If homogeneity is assumed, the chain is formulated by specifying
time-independent rates of substitutions between states in the chain. In
applications, there are usually extra constraints on the rates, depending on
the situation. If a model is formulated in this way, it is possible to
generalise it and allow for an inhomogeneous process, with time-dependent rates
satisfying the same constraints. It is then useful to require that there exists
a homogeneous average of this inhomogeneous process within the same model. This
leads to the definition of "Lie Markov models", which are precisely the class
of models where such an average exists. These models form Lie algebras and
hence concepts from Lie group theory are central to their derivation. In this
paper, we concentrate on applications to phylogenetics and nucleotide
evolution, and derive the complete hierarchy of Lie Markov models that respect
the grouping of nucleotides into purines and pyrimidines -- that is, models
with purine/pyrimidine symmetry. We also discuss how to handle the subtleties
of applying Lie group methods, most naturally defined over the complex field,
to the stochastic case of a Markov process, where parameter values are
restricted to be real and positive. In particular, we explore the geometric
embedding of the cone of stochastic rate matrices within the ambient space of
the associated complex Lie algebra.
The whole list of Lie Markov models with purine/pyrimidine symmetry is
available at http://www.pagines.ma1.upc.edu/~jfernandez/LMNR.pdf.Comment: 32 page
The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
K-theory for Cuntz-Krieger algebras arising from real quadratic maps
We compute the -groups for the Cuntz-Krieger algebras
, where is
the Markov transition matrix arising from the \textit{kneading sequence
} of the one-parameter family of real quadratic maps
.Comment: 8 page
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