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.
翻译:深神经网络被广泛用于解决若干科学领域的复杂问题,如语音识别、机器翻译、图像分析等。调查其理论特性的战略主要依靠欧洲大陆的几何学,但在过去几年中,根据里曼的几何学发展了新办法。受一些开放问题的驱使,我们研究了不同方位之间的地图序列,其中最后一个方位配有里曼度度量仪。我们调查了这些序列的其他方位和一些相关的商数上引出回的结构。我们特别表明,该序列中任何方位的最后里曼度量仪的拉回是一个退化的里曼度量仪,引出了一个假体空间的结构。我们表明,这个假体空间的科尔莫戈夫特方位产生一个滑动的方位,这是一个特定的垂直捆绑的基点空间。我们研究了这种序列的地图的理论属性,最终我们侧重于执行有实际兴趣的神经网络的多个方位和我们在文件第一部分提出的几何框架的一些应用。