Provable Non-convex Projections for High-dimensional Learning Problems - Part1
5Typical high-dimensional learning problems such as sparse regression, low-rank matrix completion, robust PCA etc can be solved using projections onto non-convex sets. However, providing theoretical guarantees for such methods is difficult due to the non-convexity in projections. In this talk, we will discuss some of our recent results that show that non-convex projections based methods can be used to solve several important problems in this area such as: a) sparse regression, b) low-rank matrix completion, c) robust PCA. In this talk, we will give an overview of the state-of-the-art for these problems and also discuss how simple non-convex techniques can significantly outperform state-of-the-art convex relaxation based techniques and provide solid theoretical results as well. For example, for robust PCA, we provide first provable algorithm with time complexity O(n 2 r) which matches the time complexity of normal SVD and is faster than the usual nuclear+L 1 -regularization methods that incur O(n 3 ) time complexity. This talk is based on joint works with Ambuj Tewari, Purushottam Kar, Praneeth Netrapalli, Animashree Anandkumar, U N Niranjan, and Sujay Sanghavi.