This paper considers a decentralized projection free algorithm for non-convex optimization in high dimension. More specifically, we propose a Decentralized Frank-Wolfe (DeFW)
algorithm which is suitable when high dimensional optimization constraints are difficult to handle by conventional projection/proximal-based gradient descent methods. We present conditions under which the DeFW algorithm converges to a stationary point and prove that the rate of convergence is as fast as ${\cal O}( 1/\sqrt{T} )$, where
$T$ is the iteration number. This paper provides the first convergence guarantee for Frank-Wolfe methods applied to non-convex decentralized optimization. Utilizing our theoretical findings, we formulate a novel robust matrix completion problem and apply DeFW to give an efficient decentralized solution. Numerical experiments are performed on realistic and synthetic data to support our findings.