论文标题
通过概括Riemannian概念，朝着差异空间进行优化技术
Towards optimization techniques on diffeological spaces by generalizing Riemannian concepts
论文作者
论文摘要
J.M. Souriau在1980年代首先引入的差异空间是平滑歧管的自然概括。但是，到目前为止，优化技术仅在流形上才知道。由于几个原因，将这些技术概括为差异空间非常具有挑战性。主要原因之一是，切线空间的各种定义与不一致。此外，需要处理Riemannian空间的概括，以定义对于优化方法必不可少的梯度。本文专门研究了关于差异空间的优化技术。因此，本文的一个主要目的是对切线空间进行适当的定义，以考虑到优化方法。基于此定义，我们提出了一个差异的riemannian空间和一个差异梯度，我们需要在差异空间上制定优化算法。此外，为了能够在差异空间的优化算法中更新迭代元素，我们在差异空间上提出了差异缩回和LEVI-CIVITA连接。我们提供了新物体的例子，并将提出的差异算法应用于优化问题。
Diffeological spaces firstly introduced by J.M. Souriau in the 1980s are a natural generalization of smooth manifolds. However, optimization techniques are only known on manifolds so far. Generalizing these techniques to diffeological spaces is very challenging because of several reasons. One of the main reasons is that there are various definitions of tangent spaces which do not coincide. Additionally, one needs to deal with a generalization of a Riemannian space in order to define gradients which are indispensable for optimization methods. This paper is devoted to an optimization technique on diffeological spaces. Thus, one main aim of this paper is a suitable definition of a tangent space in view to optimization methods. Based on this definition, we present a diffeological Riemannian space and a diffeological gradient, which we need to formulate an optimization algorithm on diffeological spaces. Moreover, in order to be able to update the iterates in an optimization algorithm on diffeological spaces, we present a diffeological retraction and the Levi-Civita connection on diffeological spaces. We give examples for the novel objects and apply the presented diffeological algorithm to an optimization problem.
