Biconvex Relaxation for Semidefinite Programming in Computer Vision
  Semidefinite programming is an indispensable tool in computer vision, but general-purpose solvers for semidefinite programs are often too slow and memory intensive for large-scale problems. We propose a general framework to approximately solve large-scale semidefinite problems (SDPs) at low complexity. Our approach, referred to as biconvex relaxation (BCR), transforms a general SDP into a specific biconvex optimization problem, which can then be solved in the original, low-dimensional variable space at low complexity. The resulting biconvex problem is solved using an efficient alternating minimization (AM) procedure. Since AM has the potential to get stuck in local minima, we propose a general initialization scheme that enables BCR to start close to a global optimum - this is key for our algorithm to quickly converge to optimal or near-optimal solutions. We showcase the efficacy of our approach on three applications in computer vision, namely segmentation, co-segmentation, and manifold metric learning. BCR achieves solution quality comparable to state-of-the-art SDP methods with speedups between 4X and 35X. At the same time, BCR handles a more general set of SDPs than previous approaches, which are more specialized. 
Sohil Shah, Abhay Kumar, Carlos Castillo, David Jacobs, Christoph   Studer, Tom Goldstein
cs.CV cs.NA math.NA math.OC