论文标题
深层神经网络的单一riemannian几何方法I.理论基础
A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations
论文作者
论文摘要
深度神经网络被广泛用于解决多个科学领域的复杂问题，例如语音识别，机器翻译，图像分析。用于研究其理论特性的策略主要依赖于欧几里得的几何形状，但是在过去的几年中，已经开发了基于Riemannian几何形状的新方法。在某些开放问题的动机中，我们研究了歧管之间的特定地图序列，配备有riemannian度量的序列的最后一个歧管。我们研究了序列的其他歧管以及某些相关的商的结构引起的凹回率。特别是，我们表明，最终的里曼式度量的回调是该序列的任何歧管，是一个退化的riemannian公制，诱发了伪时间空间的结构，我们表明，该伪仪器的Kolmogorov商的商品是平稳的纵向歧管，这是特定垂直bundle的基本空间。我们研究了此类序列图的理论属性，最终我们关注的是实现实际关注的神经网络之间的地图的情况，并介绍了本文第一部分中引入的几何框架的某些应用。
Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely on Euclidean geometry, but in the last years new approaches based on Riemannian geometry have been developed. Motivated by some open problems, we study a particular sequence of maps between manifolds, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the structures induced trough pullbacks on the other manifolds of the sequence and on some related quotients. In particular, we show that the pullbacks of the final Riemannian metric to any manifolds of the sequence is a degenerate Riemannian metric inducing a structure of pseudometric space, we show that the Kolmogorov quotient of this pseudometric space yields a smooth manifold, which is the base space of a particular vertical bundle. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between manifolds implementing neural networks of practical interest and we present some applications of the geometric framework we introduced in the first part of the paper.
