Efficient generalized fused lasso and its applications

Generalized fused lasso (GFL) penalizes variables with l₁ norms based both on the variables and their pairwise differences. GFL is useful when applied to data where prior information is expressed using a graph over the variables. However, the existing GFL algorithms incur high computational costs and do not scale to high-dimensional problems. In this study, we propose a fast and scalable algorithm for GFL. Based on the fact that fusion penalty is the Lovász extension of a cut function, we show that the key building block of the optimization is equivalent to recursively solving graph-cut problems. Thus, we use a parametric flow algorithm to solve GFL in an efficient manner. Runtime comparisons demonstrate a significant speedup compared to existing GFL algorithms. Moreover, the proposed optimization framework is very general; by designing different cut functions, we also discuss the extension of GFL to directed graphs. Exploiting the scalability of the proposed algorithm, we demonstrate the applications of our algorithm to the diagnosis of Alzheimer's disease (AD) and video background subtraction (BS). In the AD problem, we formulated the diagnosis of AD as a GFL regularized classification. Our experimental evaluations demonstrated that the diagnosis performance was promising. We observed that the selected critical voxels were well structured, i.e., connected, consistent according to cross validation, and in agreement with prior pathological knowledge. In the BS problem, GFL naturally models arbitrary foregrounds without predefined grouping of the pixels. Even by applying simple background models, e.g., a sparse linear combination of former frames, we achieved state-of-the-art performance on several public datasets.