Linear Convergence of a Frank-Wolfe Type Algorithm over Trace-Norm Balls

We propose a rank-k variant of the classical Frank-Wolfe algorithm to solve convex minimization over a trace-norm ball. Our algorithm replaces the top singular-vector computation (1-SVD) of Frank-Wolfe with a top-k singular-vector computation (k-SVD), and this can be done by repeatedly applying 1-SVD k times. Our algorithm has a linear convergence rate when the objective function is smooth and strongly convex, and the optimal solution has rank at most k. This improves the convergence rate and the total complexity of the Frank-Wolfe method and its variants.