Papers › The Proxy Step-size Technique for Regularized Optimization on the Sphere Manifold
The Proxy Step-size Technique for Regularized Optimization on the Sphere Manifold
Fang Bai, Adrien Bartoli
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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, \textit{e.g.,}~norm regularizers and their combinations. Our solution is based on the Riemannian proximal gradient, using an idea we call \textit{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.
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