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Atomic Decomposition by Basis Pursuit
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Information Theory, Inference & Learning Algorithms
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Convex and Network Flow Optimization for Structured Sparsity
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Structured Variable Selection with Sparsity-Inducing Norms
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Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries
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Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering
IEEE Transactions on Image Processing
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We consider supervised learning problems where the features are embedded in a graph, such as gene expressions in a gene network. In this context, it is of much interest to automatically select a subgraph with few connected components; by exploiting prior knowledge, one can indeed improve the prediction performance or obtain results that are easier to interpret. Regularization or penalty functions for selecting features in graphs have recently been proposed, but they raise new algorithmic challenges. For example, they typically require solving a combinatorially hard selection problem among all connected subgraphs. In this paper, we propose computationally feasible strategies to select a sparse and well-connected subset of features sitting on a directed acyclic graph (DAG). We introduce structured sparsity penalties over paths on a DAG called "path coding" penalties. Unlike existing regularization functions that model long-range interactions between features in a graph, path coding penalties are tractable. The penalties and their proximal operators involve path selection problems, which we efficiently solve by leveraging network flow optimization. We experimentally show on synthetic, image, and genomic data that our approach is scalable and leads to more connected subgraphs than other regularization functions for graphs.