  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.
