论文标题
Riemannian梯度和Hessian的操作员价值公式以及可拖动指标的家属
Operator-valued formulas for Riemannian Gradient and Hessian and families of tractable metrics
论文作者
论文摘要
我们为Riemannian歧管提供了Levi-Civita连接和Riemannian Hessian的明确公式，该公式是嵌入具有非恒定度量功能的内部产品空间中的歧管的商。连同经典的投影公式，这使我们能够为几个经典歧管的指标家族评估Riemannian梯度和Hessian，包括在Stiefel歧管上的指标家族，将恒定环境和规范环境与封闭形式的测量学连接起来。使用这些公式，我们将riemannian优化框架推出了符号歧管的商，包括旗帜歧管（包括旗杆），以及在固定等级的正数矩阵的歧管上，由固定等级的歧管（被认为是stiefel和affinite matrix歧管均与Affine-Ininvariant Metrics offinite Metrics Metrics offinite Metrics offient-Nextimentient矩阵）的新家族。该方法是程序性的，在许多情况下，riemannian梯度和黑森公式可以通过符号积分得出。该方法扩展了可用于多种优化和机器学习的潜在指标列表。
We provide an explicit formula for the Levi-Civita connection and Riemannian Hessian for a Riemannian manifold that is a quotient of a manifold embedded in an inner product space with a non-constant metric function. Together with a classical formula for projection, this allows us to evaluate Riemannian gradient and Hessian for several families of metrics on classical manifolds, including a family of metrics on Stiefel manifolds connecting both the constant and canonical ambient metrics with closed-form geodesics. Using these formulas, we derive Riemannian optimization frameworks on quotients of Stiefel manifolds, including flag manifolds, and a new family of complete quotient metrics on the manifold of positive-semidefinite matrices of fixed rank, considered as a quotient of a product of Stiefel and positive-definite matrix manifold with affine-invariant metrics. The method is procedural, and in many instances, the Riemannian gradient and Hessian formulas could be derived by symbolic calculus. The method extends the list of potential metrics that could be used in manifold optimization and machine learning.
