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We present an exploration of the rich theoretical connections between severalclasses of regularized models, network flows, and recent results in submodularfunction theory. This work unifies key aspects of these problems under a commontheory, leading to novel methods for working with several important models ofinterest in statistics, machine learning and computer vision.  In Part 1, we review the concepts of network flows and submodular functionoptimization theory foundational to our results. We then examine theconnections between network flows and the minimum-norm algorithm fromsubmodular optimization, extending and improving several current results. Thisleads to a concise representation of the structure of a large class of pairwiseregularized models important in machine learning, statistics and computervision.  In Part 2, we describe the full regularization path of a class of penalizedregression problems with dependent variables that includes the graph-guidedLASSO and total variation constrained models. This description also motivates apractical algorithm. This allows us to efficiently find the regularization pathof the discretized version of TV penalized models. Ultimately, our newalgorithms scale up to high-dimensional problems with millions of variables.

An Algorithmic Theory of Dependent Regularizers, Part 1: Submodular Structure