Nonconvex Recovery of Low-complexity Models

Today we are living in the era of big data, there is a pressing need for efficient, scalable and robust optimization methods to analyze the data we create and collect. Although Convex methods offer tractable solutions with global optimality, heuristic nonconvex methods are often more attractive in practice due to their superior efficiency and scalability. Moreover, for better representations of the data, the mathematical model we are building today are much more complicated, which often results in highly nonlinear and nonconvex optimizations problems. Both of these challenges require us to go beyond convex optimization. While nonconvex optimization is extraordinarily successful in practice, unlike convex optimization, guaranteeing the correctness of nonconvex methods is notoriously difficult. In theory, even finding a local minimum of a general nonconvex function is NP-hard – nevermind the global minimum. This thesis aims to bridge the gap between practice and theory of nonconvex optimization, by developing global optimality guarantees for nonconvex problems arising in real-world engineering applications, and provable, efficient nonconvex optimization algorithms. First, this thesis reveals that for certain nonconvex problems we can construct a model specialized initialization that is close to the optimal solution, so that simple and efficient methods provably converge to the global solution with linear rate. These problem include sparse basis learning and convolutional phase retrieval. In addition, the work has led to the discovery of a broader class of nonconvex problems – the so-called ridable saddle functions. Those problems possess characteristic structures, in which (i) all local minima are global, (ii) the energy landscape does not have any ''flat'' saddle points. More interestingly, when data are large and random, this thesis reveals that many problems in the real world are indeed ridable saddle, those problems include complete dictionary learning and generalized phase retrieval. For each of the aforementioned problems, the benign geometric structure allows us to obtain global recovery guarantees by using efficient optimization methods with arbitrary initialization

Proximal Methods for Hierarchical Sparse Coding

Sparse coding consists in representing signals as sparse linear combinations of atoms selected from a dictionary. We consider an extension of this framework where the atoms are further assumed to be embedded in a tree. This is achieved using a recently introduced tree-structured sparse regularization norm, which has proven useful in several applications. This norm leads to regularized problems that are difficult to optimize, and we propose in this paper efficient algorithms for solving them. More precisely, we show that the proximal operator associated with this norm is computable exactly via a dual approach that can be viewed as the composition of elementary proximal operators. Our procedure has a complexity linear, or close to linear, in the number of atoms, and allows the use of accelerated gradient techniques to solve the tree-structured sparse approximation problem at the same computational cost as traditional ones using the L1-norm. Our method is efficient and scales gracefully to millions of variables, which we illustrate in two types of applications: first, we consider fixed hierarchical dictionaries of wavelets to denoise natural images. Then, we apply our optimization tools in the context of dictionary learning, where learned dictionary elements naturally organize in a prespecified arborescent structure, leading to a better performance in reconstruction of natural image patches. When applied to text documents, our method learns hierarchies of topics, thus providing a competitive alternative to probabilistic topic models