The Proxy Step-Size Technique for Regularized Optimization on the Sphere Manifold

We give an effective solution to the regularized optimization problem g (x) + h (x), where x is constrained on the unit sphere ||x ||₂ = 1. Here g (·) is a smooth cost with Lipschitz continuous gradient within the unit ball {x : ||x ||₂ ≤ 1 } whereas h (·) is typically non-smooth but convex and absolutely homogeneous, e.g., norm regularizers and their combinations. Our solution is based on the Riemannian proximal gradient, using an idea we call proxy step-size - a scalar variable which we prove is monotone with respect to the actual step-size within an interval. The proxy step-size exists ubiquitously for convex and absolutely homogeneous h(·), and decides the actual step-size and the tangent update in closed-form, thus the complete proximal gradient iteration. Based on these insights, we design a Riemannian proximal gradient method using the proxy step-size. We prove that our method converges to a critical point, guided by a line-search technique based on the g(·) cost only. The proposed method can be implemented in a couple of lines of code. We show its usefulness by applying nuclear norm, l₁ norm, and nuclear-spectral norm regularization to three classical computer vision problems. The improvements are consistent and backed by numerical experiments. available at https://bitbucket.org/FangBai/proxystepsize-pgs.