Riemannian submanifold optimization with momentum is computationally challenging because ensuring iterates remain on the submanifold often requires solving difficult differential equations. We simplify such optimization algorithms for the submanifold of symmetric positive-definite matrices with the affine invariant metric. We propose a generalized version of the Riemannian normal coordinates which dynamically trivializes the problem into a Euclidean unconstrained problem. We use our approach to explain and simplify existing approaches for structured covariances and develop efficient second-order optimizers for deep learning without explicit matrix inverses.

翻译：使用动力优化里曼尼亚次元在计算上具有挑战性,因为确保迭代留在亚元上往往需要解决困难的差别方程式。我们简化了用于对称正定矩阵的次元端的优化算法,并采用偏差指数。我们提出了一个通用的里曼尼正常坐标,该坐标动态地将问题简单化,形成一个欧盟不受约束的问题。我们用我们的方法来解释和简化结构化共变法的现有方法,并开发高效的二阶优化器,用于没有明确矩阵反向的深层学习。