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Structured sparsityinducing norms through submodular functions
 IN ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS
, 2010
"... Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turnedinto a convex optimization problem byreplacing the cardinality function by its convex en ..."
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Cited by 60 (10 self)
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Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turnedinto a convex optimization problem byreplacing the cardinality function by its convex envelope (tightest convex lower bound), in this case the ℓ1norm. In this paper, we investigate more general setfunctions than the cardinality, that may incorporate prior knowledge or structural constraints which are common in many applications: namely, we show that for nonincreasing submodular setfunctions, the corresponding convex envelope can be obtained from its Lovász extension, a common tool in submodular analysis. This defines a family of polyhedral norms, for which we provide generic algorithmic tools (subgradients and proximal operators) and theoretical results (conditions for support recovery or highdimensional inference). By selecting specific submodular functions, we can give a new interpretation to known norms, such as those based on rankstatistics or grouped norms with potentially overlapping groups; we also define new norms, in particular ones that can be used as nonfactorial priors for supervised learning.
Structured Sparsity through Convex Optimization
, 2012
"... Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the ℓ1norm. In this paper, we consider sit ..."
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Cited by 47 (6 self)
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Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the ℓ1norm. In this paper, we consider situations where we are not only interested in sparsity, but where some structural prior knowledge is available as well. We show that the ℓ1norm can then be extended to structured norms built on either disjoint or overlapping groups of variables, leading to a flexible framework that can deal with various structures. We present applications to unsupervised learning, for structured sparse principal component analysis and hierarchical dictionary learning, and to supervised learning in the context of nonlinear variable selection.
Machine learning and convex optimization with Submodular Functions
 WORKSHOP ON COMBINATORIAL OPTIMIZATION CARGESE, 2013
, 2013
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Higher Order Fused Regularization for Supervised Learning with Grouped Parameters
"... Abstract. We often encounter situations in supervised learning where there exist possibly groups that consist of more than two parameters. For example, we might work on parameters that correspond to words expressing the same meaning, music pieces in the same genre, and books released in the same ye ..."
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Abstract. We often encounter situations in supervised learning where there exist possibly groups that consist of more than two parameters. For example, we might work on parameters that correspond to words expressing the same meaning, music pieces in the same genre, and books released in the same year. Based on such auxiliary information, we could suppose that parameters in a group have similar roles in a problem and similar values. In this paper, we propose the Higher Order Fused (HOF) regularization that can incorporate smoothness among parameters with group structures as prior knowledge in supervised learning. We define the HOF penalty as the Lovász extension of a submodular higherorder potential function, which encourages parameters in a group to take similar estimated values when used as a regularizer. Moreover, we develop an efficient network flow algorithm for calculating the proximity operator for the regularized problem. We investigate the empirical performance of the proposed algorithm by using synthetic and realworld data. 1
An Algorithmic Theory of Dependent Regularizers Part 1: Submodular Structure
, 2013
"... We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems under a common theory, leading to novel methods for working with several impor ..."
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We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems under a common theory, leading to novel methods for working with several important models of interest in statistics, machine learning and computer vision. In Part 1, we review the concepts of network flows and submodular function optimization theory foundational to our results. We then examine the connections between network flows and the minimumnorm algorithm from submodular optimization, extending and improving several current results. This leads to a concise representation of the structure of a large class of pairwise regularized models important in machine learning, statistics and computer vision. In Part 2, we describe the full regularization path of a class of penalized regression problems with dependent variables that includes the graphguided LASSO and total variation constrained models. This description also motivates a practical algorithm. This allows us to efficiently find the regularization path of the discretized version of TV penalized models. Ultimately, our new algorithms scale up to highdimensional problems with millions of variables. 1 ar