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

We give an effective solution to the regularized optimization problem
$g (\boldsymbol{x}) + h (\boldsymbol{x})$
, where
$\boldsymbol{x}$
is constrained on the unit sphere
$\Vert \boldsymbol{x} \Vert _{2} = 1$
. Here
$g (\cdot)$
is a smooth cost with Lipschitz continuous gradient within the unit ball
$\lbrace \boldsymbol{x} : \Vert \boldsymbol{x} \Vert _{2} \leq 1 \rbrace$
whereas
$h (\cdot)$
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
$h(\cdot)$
, 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
$g(\cdot)$
cost only. The proposed method can be implemented in a couple of lines of code. We show its usefulness by applying nuclear norm,
$\ell _{1}$
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
.