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We give an effective solution to the regularized optimization problem   $g (\boldsymbol{x}) + h (\boldsymbol{x})$g  (  x  )  +  h  (  x  ), where   $\boldsymbol{x}$    xis constrained on the unit sphere   $\Vert \boldsymbol{x} \Vert _{2} = 1$&#x2225;  x  &#x2225;   2   =  1. Here   $g (\cdot)$g  (  &#x00B7;  )is a smooth cost with Lipschitz continuous gradient within the unit ball   $\lbrace \boldsymbol{x} : \Vert \boldsymbol{x} \Vert _{2} \leq 1 \rbrace$&#x007B;  x  :  &#x2225;  x   &#x2225;  2   &#x2264;  1  &#x007D;whereas   $h (\cdot)$h  (  &#x00B7;  )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  &#x2013; 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   $h(\cdot)$h  (  &#x00B7;  ), 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   $g(\cdot)$g  (  &#x00B7;  )cost only. The proposed method can be implemented in a couple of lines of code. We show its usefulness by applying nuclear norm,   $\ell _{1}$&#x2113;  1norm, 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 .

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