[Ml-stat-talks] High-Dimensional Structure Learning of Graphical Models: Trees, Latent trees and Beyond
blei at CS.Princeton.EDU
Tue Feb 15 12:32:20 EST 2011
anima anandkumar from u.c. irvine is giving an interesting talk tomorrow
about graphical models. see details below.
February 16, 2011
EQuad Room B-205
Title : High-Dimensional Structure Learning of Graphical Models: Trees,
Latent trees and Beyond
Speaker: Anima Anandkumar (U.C. Irvine)
Graphical models or Markov random fields provide a graph-based framework for
capturing dependencies between
random variables of a large-scale multivariate distribution. This
interdisciplinary topic has found widespread application in a variety of
areas including image processing, bioinformatics, combinatorial optimization
and machine learning. Estimating the graph structure of
the model using samples drawn from it forms an important task, since the
structure reveals important relationships between the
variables. However, structure estimation has several challenges: in general
graphical models, it is NP-hard, the models are typically
in the high-dimensional regime where the number of variables is much larger
than the number of samples obtained, and there could
be many latent variables which are unobserved. I will address these
challenges in the talk and provide solutions for certain classes of
models I will focus on latent tree models in the first part of the talk.
These are tree graphical models where there are latent variables,
but there is no knowledge of the number or the location of the latent
variables. We have developed two novel algorithms which are
consistent, computationally efficient and have low sample complexity. These
algorithms are based on the presence of an additive
metric on the tree, due to the properties of correlations on a tree model.
The first algorithm uses these properties to check for sibling
relationships between node pairs and builds the tree in a recursive fashion.
The second algorithm initially builds a tree over the
observed variables, and then adds hidden nodes in a step-by-step fashion by
only operating on small subsets of variables. This
leads to considerable computational savings compared to the first algorithm.
We modify the second algorithm for experiments on real
data by trading off number of added latent variables with the accuracy of
resulting model fitting via the Bayesian Information Criterion
(BIC). Experiments on the S&P 100 monthly returns data and on the occurrence
of words in newsgroups reveal interesting
relationships. In the second part, I will talk about recent results on
learning graphical models on sparse Erdos-Renyi random graphs.
These random graphs are relevant in social networks. Since these graphs are
locally tree-like, it is a natural question if structure
learning is feasible in these models, given that learning tree models is
tractable. We provide a positive answer when the model is in
the so-called uniqueness regime, where there is a decay of long-range
correlations. The algorithm is based on a set of conditional
mutual information tests and is shown to be consistent for structure
estimation with almost order-optimal sample complexity. A
simpler algorithm based on correlation thresholding is also consistent, but
under more stringent conditions. To the best of our
knowledge, this is the first work to show that the regime of consistent
structure learning contains the regime of correlation decay.
Finally, depending on the time availability, I will briefly mention related
works on consistent estimation of high-dimensional forest
distributions and the characterization of extremal tree structures with
respect to error rates for structure learning.
Anima Anandkumar received her B.Tech in Electrical Engineering from the
Indian Institute of Technology (IIT) Madras in 2004 and her MS
and PhD degrees in Electrical Engineering from Cornell University, Ithaca,
NY in 2009. She was at the Stochastic Systems Group at MIT, Cambridge, MA as
a post-doctoral researcher. She has been an assistant professor at EECS
Dept. at U.C.Irvine since July 2010. She is the recipient of the 2009 Best
Thesis Award by the ACM Sigmetrics Society, 2008 IEEE Signal Processing
Society Young Author Best Paper Award, 2008 IBM Fran Allen PhD fellowship,
and student paper award at 2006 IEEE ICASSP. Her research interests are in
the area of statistical-signal processing, network theory and information
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