Network Flow Algorithms for Structured Sparsity (1008.5209v1)

Abstract: We consider a class of learning problems that involve a structured sparsity-inducing norm defined as the sum of $\ell_\infty$-norms over groups of variables. Whereas a lot of effort has been put in developing fast optimization methods when the groups are disjoint or embedded in a specific hierarchical structure, we address here the case of general overlapping groups. To this end, we show that the corresponding optimization problem is related to network flow optimization. More precisely, the proximal problem associated with the norm we consider is dual to a quadratic min-cost flow problem. We propose an efficient procedure which computes its solution exactly in polynomial time. Our algorithm scales up to millions of variables, and opens up a whole new range of applications for structured sparse models. We present several experiments on image and video data, demonstrating the applicability and scalability of our approach for various problems.

• The paper presents a novel method that computes the proximal operator for structured sparsity by reducing it to a quadratic min-cost flow problem.
• The proposed ProxFlow algorithm efficiently handles overlapping group models, enhancing regularization performance in sparse optimization.
• Numerical experiments demonstrate that the approach accelerates large-scale optimization, effectively scaling to problems with millions of variables.

Network Flow Algorithms for Structured Sparsity

In their research paper, Mairal et al. explore an advanced optimization framework for solving machine learning problems that involve structured sparsity. Structured sparsity models are designed to efficiently encode high-order information from datasets, wherein variables sharing structural relationships such as spatial or temporal dependencies can be regularized collectively. This paper specifically investigates the optimization problems related to the sparse models involving overlapping groups, and introduces an efficient solution via network flow algorithms.

The authors begin by highlighting the limitations of the traditional $\ell_1$-norm in sparse models, which neglects existing relationships between variables. They address this by proposing a sparsity-inducing norm defined as a sum of $\ell_\infty$-norms over groups of variables. The optimization problem, relevant to this structured norm, is shown to be reducible to network flow optimization — particularly quadratic min-cost flow problems.

Key contributions of this paper include:

1. Proximal Operator Calculation: The proximal operator associated with the structured sparsity norm is computed by solving a quadratic min-cost flow problem. This connects sparse optimization intricacies with network flow optimization literature.
2. Efficient Algorithm: An algorithm termed ProxFlow is proposed, designed to compute solutions in polynomial time. Remarkably, this algorithm scales to handle problems composed of millions of variables, opening prospects for broader applications in structured sparse modeling.
3. Dual Norm Evaluation: The paper efficiently evaluates the dual norm of the proposed sparsity-inducing regularization, enabling computation of duality gaps which are useful for assessing optimization problem convergence.

Numerical experiments conducted by the authors demonstrate the efficacy of their approach. In particular, their optimization procedure accelerates computations on large-scale datasets compared to previous methods, like the subgradient descent and interior point methods, without compromising the sparsity of solutions.

The practical applications advised in this research include video background subtraction and hierarchical structure learning for dictionary generation in natural image patch processing. For background subtraction tasks, the proposed method delivers improved results by removing scattered artifacts often present with traditional norms. In multi-task learning, hierarchical structures of dictionary elements are fine-tuned to improve the performance over standard sparse coding practices.

In summary, the paper elucidates significant theoretical and practical advancements in structured sparsity. It provides a robust framework utilizing network flow algorithms to address optimization challenges inherent in structured sparsity models with overlapping variable groups. Looking ahead, this research could drive advancements in the field of artificial intelligence, especially in applications such as computer vision, bioinformatics, and beyond, where data complexity mandates structured sparsity solutions. Future research could explore enhancing the scalability of the algorithm to encompass even more extensive datasets or exploring additional real-world applications.