Optimization Algorithms for Structured Machine Learning and Image Processing Problems

de Zhiwei Qin

Optimization algorithms are often the solution engine for machine learning and image processing techniques, but they can also become the bottleneck in applying these techniques if they are unable to cope with the size of the data. With the rapid advancement of modern technology, data of unprecedented size has become more and more available, and there is an increasing demand to process and interpret the data. Traditional optimization methods, such as the interior-point method, can solve a wide array of problems arising from the machine learning domain, but it is also this generality that often prevents them from dealing with large data efficiently. Hence, specialized algorithms that can readily take advantage of the problem structure are highly desirable and of immediate practical interest. This thesis focuses on developing efficient optimization algorithms for machine learning and image processing problems of diverse types, including supervised learning (e.g., the group lasso), unsupervised learning (e.g., robust tensor decompositions), and total-variation image denoising. These algorithms are of wide interest to the optimization, machine learning, and image processing communities. Specifically, (i) we present two algorithms to solve the Group Lasso problem. First, we propose a general version of the Block Coordinate Descent (BCD) algorithm for the Group Lasso that employs an efficient approach for optimizing each subproblem exactly. We show that it exhibits excellent performance when the groups are of moderate size. For groups of large size, we propose an extension of the proximal gradient algorithm based on variable step-lengths that can be viewed as a simplified version of BCD. By combining the two approaches we obtain an implementation that is very competitive and often outperforms other state-of-the-art approaches for this problem. We show how these methods fit into the globally convergent general block coordinate gradient descent framework in (Tseng and Yun, 2009). We also show that the proposed approach is more efficient in practice than the one implemented in (Tseng and Yun, 2009). In addition, we apply our algorithms to the Multiple Measurement Vector (MMV) recovery problem, which can be viewed as a special case of the Group Lasso problem, and compare their performance to other methods in this particular instance; (ii) we further investigate sparse linear models with two commonly adopted general sparsity-inducing regularization terms, the overlapping Group Lasso penalty l1/l2-norm and the l1/l_infty-norm. We propose a unified framework based on the augmented Lagrangian method, under which problems with both types of regularization and their variants can be efficiently solved. As one of the core building-blocks of this framework, we develop new algorithms using a partial-linearization/splitting technique and prove that the accelerated versions of these algorithms require $O(1 sqrt(epsilon) ) iterations to obtain an epsilon-optimal solution. We compare the performance of these algorithms against that of the alternating direction augmented Lagrangian and FISTA methods on a collection of data sets and apply them to two real-world problems to compare the relative merits of the two norms; (iii) we study the problem of robust low-rank tensor recovery in a convex optimization framework, drawing upon recent advances in robust Principal Component Analysis and tensor completion. We propose tailored optimization algorithms with global convergence guarantees for solving both the constrained and the Lagrangian formulations of the problem. These algorithms are based on the highly efficient alternating direction augmented Lagrangian and accelerated proximal gradient methods. We also propose a nonconvex model that can often improve the recovery results from the convex models. We investigate the empirical recoverability properties of the convex and nonconvex formulations and compare the computational performance of the algorithms on simulated data. We demonstrate through a number of real applications the practical effectiveness of this convex optimization framework for robust low-rank tensor recovery; (iv) we consider the image denoising problem using total variation regularization. This problem is computationally challenging to solve due to the non-differentiability and non-linearity of the regularization term. We propose a new alternating direction augmented Lagrangian method, involving subproblems that can be solved efficiently and exactly. The global convergence of the new algorithm is established for the anisotropic total variation model. We compare our method with the split Bregman method and demonstrate the superiority of our method in computational performance on a set of standard test images.