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Karthik Mohan

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4 papers

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JMLR Journal 2014 Journal Article

Learning Graphical Models With Hubs

Kean Ming Tan
Palma London
Karthik Mohan
Su-In Lee
Maryam Fazel
Daniela Witten

We consider the problem of learning a high-dimensional graphical model in which there are a few hub nodes that are densely-connected to many other nodes. Many authors have studied the use of an $\ell_1$ penalty in order to learn a sparse graph in the high-dimensional setting. However, the $\ell_1$ penalty implicitly assumes that each edge is equally likely and independent of all other edges. We propose a general framework to accommodate more realistic networks with hub nodes, using a convex formulation that involves a row-column overlap norm penalty. We apply this general framework to three widely- used probabilistic graphical models: the Gaussian graphical model, the covariance graph model, and the binary Ising model. An alternating direction method of multipliers algorithm is used to solve the corresponding convex optimization problems. On synthetic data, we demonstrate that our proposed framework outperforms competitors that do not explicitly model hub nodes. We illustrate our proposal on a webpage data set and a gene expression data set. [abs] [ pdf ][ bib ] &copy JMLR 2014. ( edit, beta )

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JMLR Journal 2014 Journal Article

Node-Based Learning of Multiple Gaussian Graphical Models

Karthik Mohan
Palma London
Maryam Fazel
Daniela Witten
Su-In Lee

We consider the problem of estimating high-dimensional Gaussian graphical models corresponding to a single set of variables under several distinct conditions. This problem is motivated by the task of recovering transcriptional regulatory networks on the basis of gene expression data containing heterogeneous samples, such as different disease states, multiple species, or different developmental stages. We assume that most aspects of the conditional dependence networks are shared, but that there are some structured differences between them. Rather than assuming that similarities and differences between networks are driven by individual edges, we take a node-based approach, which in many cases provides a more intuitive interpretation of the network differences. We consider estimation under two distinct assumptions: (1) differences between the $K$ networks are due to individual nodes that are perturbed across conditions, or (2) similarities among the $K$ networks are due to the presence of common hub nodes that are shared across all $K$ networks. Using a row-column overlap norm penalty function, we formulate two convex optimization problems that correspond to these two assumptions. We solve these problems using an alternating direction method of multipliers algorithm, and we derive a set of necessary and sufficient conditions that allows us to decompose the problem into independent subproblems so that our algorithm can be scaled to high-dimensional settings. Our proposal is illustrated on synthetic data, a webpage data set, and a brain cancer gene expression data set. [abs] [ pdf ][ bib ] &copy JMLR 2014. ( edit, beta )

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JMLR Journal 2012 Journal Article

Iterative Reweighted Algorithms for Matrix Rank Minimization

Karthik Mohan
Maryam Fazel

The problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning, and is known to be NP-hard. A tractable relaxation for this problem is nuclear norm (or trace norm) minimization, which is guaranteed to find the minimum rank matrix under suitable assumptions. In this paper, we propose a family of Iterative Reweighted Least Squares algorithms IRLS- p (with 0 ≤ p ≤ 1 ), as a computationally efficient way to improve over the performance of nuclear norm minimization. The algorithms can be viewed as (locally) minimizing certain smooth approximations to the rank function. When p=1, we give theoretical guarantees similar to those for nuclear norm minimization, that is, recovery of low-rank matrices under certain assumptions on the operator defining the constraints. For p [abs] [ pdf ][ bib ] &copy JMLR 2012. ( edit, beta )

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NeurIPS Conference 2012 Conference Paper

Structured Learning of Gaussian Graphical Models

Karthik Mohan
Mike Chung
Seungyeop Han
Daniela Witten
Su-In Lee
Maryam Fazel

We consider estimation of multiple high-dimensional Gaussian graphical models corresponding to a single set of nodes under several distinct conditions. We assume that most aspects of the networks are shared, but that there are some structured differences between them. Specifically, the network differences are generated from node perturbations: a few nodes are perturbed across networks, and most or all edges stemming from such nodes differ between networks. This corresponds to a simple model for the mechanism underlying many cancers, in which the gene regulatory network is disrupted due to the aberrant activity of a few specific genes. We propose to solve this problem using the structured joint graphical lasso, a convex optimization problem that is based upon the use of a novel symmetric overlap norm penalty, which we solve using an alternating directions method of multipliers algorithm. Our proposal is illustrated on synthetic data and on an application to brain cancer gene expression data.

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