Graph-based sparse learning: models, algorithms, and applications

Sparse learning is a powerful tool to generate models of high-dimensional data with high interpretability, and it has many important applications in areas such as bioinformatics, medical image processing, and computer vision. Recently, the a priori structural information has been shown to be powerful for improving the performance of sparse learning models. A graph is a fundamental way to represent structural information of features. This dissertation focuses on graph-based sparse learning. The first part of this dissertation aims to integrate a graph into sparse learning to improve the performance. Specifically, the problem of feature grouping and selection over a given undirected graph is considered. Three models are proposed along with efficient solvers to achieve simultaneous feature grouping and selection, enhancing estimation accuracy. One major challenge is that it is still computationally challenging to solve large scale graph-based sparse learning problems. An efficient, scalable, and parallel algorithm for one widely used graph-based sparse learning approach, called anisotropic total variation regularization is therefore proposed, by explicitly exploring the structure of a graph. The second part of this dissertation focuses on uncovering the graph structure from the data. Two issues in graphical modeling are considered. One is the joint estimation of multiple graphical models using a fused lasso penalty and the other is the estimation of hierarchical graphical models. The key technical contribution is to establish the necessary and sufficient condition for the graphs to be decomposable. Based on this key property, a simple screening rule is presented, which reduces the size of the optimization problem, dramatically reducing the computational cost.