Arian Maleki, Columbia

In many application areas ranging from bioinformatics to imaging we are faced with the following question: Can we recover a sparse vector x of dimension p from its undersampled set of n noisy observations y= Ax + z. The last decade has witnessed a surge of algorithms to address this question. One of the most popular algorithms is the l_q-regularized least squares, where 0 < q <= 2. Despite the non-convexity of these optimization problems for 0 < q < 1 , they are still appealing for their closer proximity to the “ideal” l_0 -regularized least squares. In this talk, we adopt the asymptotic framework where p tends to infinity and n/p tends to a constant and analyze the properties of the global minimizer of (1) under the optimal tuning of the regularization parameter. Our goal is to answer the following questions: (i) Do non-convex regularizers outperform convex regularizers? (ii) Does q = 1 outperform other convex optimization problems when the vector x is sparse?

We discuss both the predictive power and variable selection accuracy of these algorithms. If time permits, we also discuss algorithms that can provably reach to the global minima of the non-convex problems in certain regimes.

This talk is based on a joint work with Haolei Weng, Shuaiwen Wang, and Le Zheng.