论文标题
计算欧几里得距离和最大似然回缩图以进行约束优化
Computing Euclidean distance and maximum likelihood retraction maps for constrained optimization
论文作者
论文摘要
Riemannian优化使用本地方法来解决约束集的优化问题。沿着某个下降方向的线性步骤通常会留下约束，因此使用回缩图来近似指数图并返回歧管。对于许多常见的矩阵歧管，可以使用缩回图，并具有或多或少的显式公式。对于隐式定义的歧管，很难计算合适的缩回图。因此，我们开发了一种算法，该算法使用同型延续来计算任何隐式定义的r^n子曼物的欧几里得距离回收，并证明了收敛结果。 我们还将统计模型视为带有Fisher度量的概率单纯性的Riemannian Submanifolds。用最大似然取代欧几里得距离会导致我们证明是回缩的地图。实际上，我们证明了撤回是二阶的。与Fisher指标相关的Levi-Civita连接，它近似于二阶精度。
Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to approximate the exponential map and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For implicitly-defined manifolds, suitable retraction maps are difficult to compute. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of R^n, and prove convergence results. We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. Replacing Euclidean distance with maximum likelihood results in a map which we prove is a retraction. In fact, we prove the retraction is second-order; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.
