  We consider the problem of maximizing the $\ell_1$ norm of a linear map over
the sphere, which arises in various machine learning applications such as
orthogonal dictionary learning (ODL) and robust subspace recovery (RSR). The
problem is numerically challenging due to its nonsmooth objective and nonconvex
constraint, and its algorithmic aspects have not been well explored. In this
paper, we show how the manifold structure of the sphere can be exploited to
design fast algorithms for tackling this problem. Specifically, our
contribution is threefold. First, we present a manifold proximal point
algorithm (ManPPA) for the problem and show that it converges at a sublinear
rate. Furthermore, we show that ManPPA can achieve a quadratic convergence rate
when applied to the ODL and RSR problems. Second, we propose a stochastic
variant of ManPPA called StManPPA, which is well suited for large-scale
computation, and establish its sublinear convergence rate. Both ManPPA and
StManPPA have provably faster convergence rates than existing subgradient-type
methods. Third, using ManPPA as a building block, we propose a new approach to
solving a matrix analog of the problem, in which the sphere is replaced by the
Stiefel manifold. The results from our extensive numerical experiments on the
ODL and RSR problems demonstrate the efficiency and efficacy of our proposed
methods.
