论文标题
Riemannian Admm
A Riemannian ADMM
论文作者
论文摘要
我们考虑一类Riemannian优化问题，其中目标是在环境空间中考虑的平滑函数和非平滑函数的总和。这类问题在机器学习和统计数据中找到了重要的应用，例如稀疏主成分分析，稀疏的光谱聚类和正交词典学习。我们提出了一种乘数的Riemannian交替方向方法（ADMM）来解决此类问题。我们的算法在每次迭代中都采用了易于计算的步骤。在轻度假设下分析了提出的用于获得$ε$平稳点的算法的迭代复杂性。现有用于解决非凸问题的ADMM要么不允许nonconvex约束集，要么不允许非平滑目标函数。我们的算法是第一种ADMM类型算法，可最大程度地限制多种歧管（特定的非convex集）的非平滑目标。进行数值实验以证明该方法的优势。
We consider a class of Riemannian optimization problems where the objective is the sum of a smooth function and a nonsmooth function, considered in the ambient space. This class of problems finds important applications in machine learning and statistics such as the sparse principal component analysis, sparse spectral clustering, and orthogonal dictionary learning. We propose a Riemannian alternating direction method of multipliers (ADMM) to solve this class of problems. Our algorithm adopts easily computable steps in each iteration. The iteration complexity of the proposed algorithm for obtaining an $ε$-stationary point is analyzed under mild assumptions. Existing ADMM for solving nonconvex problems either does not allow nonconvex constraint set, or does not allow nonsmooth objective function. Our algorithm is the first ADMM type algorithm that minimizes a nonsmooth objective over manifold -- a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed method.
