论文标题
在一类通过欧几里得MM方法解决的地理凸优化问题上
On a class of geodesically convex optimization problems solved via Euclidean MM methods
论文作者
论文摘要
我们研究了可以写入欧几里得凸函数的差异的地质凸（G-Convex）问题。这种结构出现在统计和机器学习中的几个优化问题中，例如，用于基质缩放，协方差的M估计器和Brascamp-Lieb不平等。我们的工作提供有效的算法，一方面利用G-Convexity来确保全球最优性以及保证迭代复杂性。另一方面，拆分结构使我们能够开发欧几里得 - 最小化算法，这些算法可以帮助我们绕开计算昂贵的Riemannian操作（例如指数式地图和并行运输）的需求。我们通过将其专门针对机器学习文献中以前研究过的一些具体优化问题来说明我们的结果。最终，我们希望我们的工作有助于激励人们更广泛地寻找混合的欧几罗南优化算法
We study geodesically convex (g-convex) problems that can be written as a difference of Euclidean convex functions. This structure arises in several optimization problems in statistics and machine learning, e.g., for matrix scaling, M-estimators for covariances, and Brascamp-Lieb inequalities. Our work offers efficient algorithms that on the one hand exploit g-convexity to ensure global optimality along with guarantees on iteration complexity. On the other hand, the split structure permits us to develop Euclidean Majorization-Minorization algorithms that help us bypass the need to compute expensive Riemannian operations such as exponential maps and parallel transport. We illustrate our results by specializing them to a few concrete optimization problems that have been previously studied in the machine learning literature. Ultimately, we hope our work helps motivate the broader search for mixed Euclidean-Riemannian optimization algorithms
