从推广黎曼概念到代数空间上的optimization技巧
Towards optimization techniques on diffeological spaces by generalizing Riemannian concepts
摘要：
拓扑空间是由J.M. Souriau在20世纪80年代是光滑流形的自然推广。然而，迄今为止，优化技术仅在流形上已知。由于几个原因，将这些技术推广到拓扑空间是非常具有挑战性的。其中一个主要原因是切空间有各种不同的定义，它们并不一致。此外，需要处理黎曼空间的推广，以便定义梯度，这对于最优化方法是必不可少的。本文研究了拓扑空间上的一种优化技术。因此，本文的一个主要目的是考虑到最优化方法的切空间的合适的定义。在此基础上，我们给出了一个拓扑黎曼空间和一个拓扑梯度，并利用它们来构造拓扑空间上的优化算法。此外，为了能够更新优化算法中的迭代在拓扑空间，我们提出了一个拓扑收缩和Levi-Civita连接拓扑空间。我们给出了新的对象的例子，并应用所提出的代数算法的最优化问题。
英文摘要：
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.
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