论文标题
在球体歧管上进行正则优化的代理级尺寸技术
The Proxy Step-size Technique for Regularized Optimization on the Sphere Manifold
论文作者
论文摘要
我们为正规化优化问题$ g（\ boldsymbol {x}） + h（\ boldsymbol {x}）$提供了有效的解决方案，其中$ \ boldsymbol {x} $在单位sphere $ \ vert \ vert \ boldsymbol {x}}} \ vert_2 = 1 $上受约束。这里的$ g（\ cdot）$是Lipschitz在单位球$ \ {\ boldsymbol {x}中连续梯度的平稳成本：\ vert \ boldsymbol {x} \ vert_2 \ le 1 \ le 1 \} $，而$ h（\ cdot）$ h（\ cdot）$ nord noce noce noce note note nord norde norde and norde and quorte and coneene and coneene and and and and and and〜正规机构及其组合。我们的解决方案基于Riemannian近端梯度，使用我们称为\ textIt {代理步进式}的想法 - 一个标量变量，我们证明，在一个间隔内，我们证明它是单调的。对于凸面和绝对均匀的$ h（\ cdot）$，替代步骤尺寸存在，并确定封闭形式中的实际阶梯尺寸和切线更新，因此是完整的近端梯度迭代。基于这些见解，我们使用代理步骤设计了Riemannian近端梯度方法。我们证明，我们的方法仅基于$ g（\ cdot）$成本的线条搜索技术而收敛到关键点。提出的方法可以用几行代码实现。我们通过应用核规范，$ \ ell_1 $规范和核谱规范化来显示其有用性，以解决三个经典的计算机视觉问题。这些改进是一致的，并得到数值实验的支持。
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 $\{\boldsymbol{x} : \Vert \boldsymbol{x} \Vert_2 \le 1 \}$ whereas $h (\cdot)$ 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 $h(\cdot)$, 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.
