Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?

Correlations in the signal observed via functional Magnetic Resonance Imaging (fMRI), are expected to reveal the interactions in the underlying neural populations through hemodynamic response. In particular, they highlight distributed set of mutually correlated regions that correspond to brain networks related to different cognitive functions. Yet graph-theoretical studies of neural connections give a different picture: that of a highly integrated system with small-world properties: local clustering but with short pathways across the complete structure. We examine the conditional independence properties of the fMRI signal, i.e. its Markov structure, to find realistic assumptions on the connectivity structure that are required to explain the observed functional connectivity. In particular we seek a decomposition of the Markov structure into segregated functional networks using decomposable graphs: a set of strongly-connected and partially overlapping cliques. We introduce a new method to efficiently extract such cliques on a large, strongly-connected graph. We compare methods learning different graph structures from functional connectivity by testing the goodness of fit of the model they learn on new data. We find that summarizing the structure as strongly-connected networks can give a good description only for very large and overlapping networks. These results highlight that Markov models are good tools to identify the structure of brain connectivity from fMRI signals, but for this purpose they must reflect the small-world properties of the underlying neural systems

Lagrangian Relaxation for MAP Estimation in Graphical Models

We develop a general framework for MAP estimation in discrete and Gaussian graphical models using Lagrangian relaxation techniques. The key idea is to reformulate an intractable estimation problem as one defined on a more tractable graph, but subject to additional constraints. Relaxing these constraints gives a tractable dual problem, one defined by a thin graph, which is then optimized by an iterative procedure. When this iterative optimization leads to a consistent estimate, one which also satisfies the constraints, then it corresponds to an optimal MAP estimate of the original model. Otherwise there is a ``duality gap'', and we obtain a bound on the optimal solution. Thus, our approach combines convex optimization with dynamic programming techniques applicable for thin graphs. The popular tree-reweighted max-product (TRMP) method may be seen as solving a particular class of such relaxations, where the intractable graph is relaxed to a set of spanning trees. We also consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g. loops), and a connected tree obtained by ``unwinding'' cycles. In addition, we propose a new class of multiscale relaxations that introduce ``summary'' variables. The potential benefits of such generalizations include: reducing or eliminating the ``duality gap'' in hard problems, reducing the number or Lagrange multipliers in the dual problem, and accelerating convergence of the iterative optimization procedure.Comment: 10 pages, presented at 45th Allerton conference on communication, control and computing, to appear in proceeding

Sure Screening for Gaussian Graphical Models

We propose {graphical sure screening}, or GRASS, a very simple and computationally-efficient screening procedure for recovering the structure of a Gaussian graphical model in the high-dimensional setting. The GRASS estimate of the conditional dependence graph is obtained by thresholding the elements of the sample covariance matrix. The proposed approach possesses the sure screening property: with very high probability, the GRASS estimated edge set contains the true edge set. Furthermore, with high probability, the size of the estimated edge set is controlled. We provide a choice of threshold for GRASS that can control the expected false positive rate. We illustrate the performance of GRASS in a simulation study and on a gene expression data set, and show that in practice it performs quite competitively with more complex and computationally-demanding techniques for graph estimation