2,237 research outputs found
Exact Inference on Gaussian Graphical Models of Arbitrary Topology using Path-Sums
We present the path-sum formulation for exact statistical inference of
marginals on Gaussian graphical models of arbitrary topology. The path-sum
formulation gives the covariance between each pair of variables as a branched
continued fraction of finite depth and breadth. Our method originates from the
closed-form resummation of infinite families of terms of the walk-sum
representation of the covariance matrix. We prove that the path-sum formulation
always exists for models whose covariance matrix is positive definite: i.e.~it
is valid for both walk-summable and non-walk-summable graphical models of
arbitrary topology. We show that for graphical models on trees the path-sum
formulation is equivalent to Gaussian belief propagation. We also recover, as a
corollary, an existing result that uses determinants to calculate the
covariance matrix. We show that the path-sum formulation formulation is valid
for arbitrary partitions of the inverse covariance matrix. We give detailed
examples demonstrating our results
Arriving on time: estimating travel time distributions on large-scale road networks
Most optimal routing problems focus on minimizing travel time or distance
traveled. Oftentimes, a more useful objective is to maximize the probability of
on-time arrival, which requires statistical distributions of travel times,
rather than just mean values. We propose a method to estimate travel time
distributions on large-scale road networks, using probe vehicle data collected
from GPS. We present a framework that works with large input of data, and
scales linearly with the size of the network. Leveraging the planar topology of
the graph, the method computes efficiently the time correlations between
neighboring streets. First, raw probe vehicle traces are compressed into pairs
of travel times and number of stops for each traversed road segment using a
`stop-and-go' algorithm developed for this work. The compressed data is then
used as input for training a path travel time model, which couples a Markov
model along with a Gaussian Markov random field. Finally, scalable inference
algorithms are developed for obtaining path travel time distributions from the
composite MM-GMRF model. We illustrate the accuracy and scalability of our
model on a 505,000 road link network spanning the San Francisco Bay Area
Polynomial Linear Programming with Gaussian Belief Propagation
Interior-point methods are state-of-the-art algorithms for solving linear
programming (LP) problems with polynomial complexity. Specifically, the
Karmarkar algorithm typically solves LP problems in time O(n^{3.5}), where
is the number of unknown variables. Karmarkar's celebrated algorithm is known
to be an instance of the log-barrier method using the Newton iteration. The
main computational overhead of this method is in inverting the Hessian matrix
of the Newton iteration. In this contribution, we propose the application of
the Gaussian belief propagation (GaBP) algorithm as part of an efficient and
distributed LP solver that exploits the sparse and symmetric structure of the
Hessian matrix and avoids the need for direct matrix inversion. This approach
shifts the computation from realm of linear algebra to that of probabilistic
inference on graphical models, thus applying GaBP as an efficient inference
engine. Our construction is general and can be used for any interior-point
algorithm which uses the Newton method, including non-linear program solvers.Comment: 7 pages, 1 figure, appeared in the 46th Annual Allerton Conference on
Communication, Control and Computing, Allerton House, Illinois, Sept. 200
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