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We consider optimization problems in which the goal is find a $k$-dimensionalsubspace of $\mathbb{R}^n$, $k<<n$, which minimizes a convex and smooth loss.Such problems generalize the fundamental task of principal component analysis(PCA) to include robust and sparse counterparts, and logistic PCA for binarydata, among others. This problem could be approached either via nonconvexgradient methods with highly-efficient iterations, but for which arguing aboutfast convergence to a global minimizer is difficult or, via a convex relaxationfor which arguing about convergence to a global minimizer is straightforward,but the corresponding methods are often inefficient in high dimensions. In thiswork we bridge these two approaches under a strict complementarity assumption,which in particular implies that the optimal solution to the convex relaxationis unique and is also the optimal solution to the original nonconvex problem.Our main result is a proof that a natural nonconvex gradient method which is\textit{SVD-free} and requires only a single QR-factorization of an $n\times k$matrix per iteration, converges locally with a linear rate. We also establishlinear convergence results for the nonconvex projected gradient method, and theFrank-Wolfe method when applied to the convex relaxation.

Local Linear Convergence of Gradient Methods for Subspace Optimization via Strict Complementarity