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Principles of Riemannian Geometry in...

Hauser, Michael.

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Principles of Riemannian Geometry in Neural Networks.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Principles of Riemannian Geometry in Neural Networks./
作者:	Hauser, Michael.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2018,
面頁冊數:	156 p.
附註:	Source: Dissertation Abstracts International, Volume: 79-12(E), Section: B.
Contained By:	Dissertation Abstracts International79-12B(E).
標題:	Mechanical engineering. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10903668
ISBN:	9780438134935

Principles of Riemannian Geometry in Neural Networks.
Hauser, Michael.

Principles of Riemannian Geometry in Neural Networks. - Ann Arbor : ProQuest Dissertations & Theses, 2018 - 156 p.

Source: Dissertation Abstracts International, Volume: 79-12(E), Section: B.

Thesis (Ph.D.)--The Pennsylvania State University, 2018.

The first part of this dissertation deals with neural networks in the sense of geometric transformations acting on the coordinate representation of the underlying data manifold from which the data is sampled. It forms part of an attempt to construct a formalized general theory of neural networks in the setting of algebraic and Riemannian geometry. Geometry allows for a rigorous formulation, and therefore understanding, of neural networks as they act on data manifolds, which is certainly of great importance as these tools become ubiquitous in engineering applications. From this perspective, the following theoretical results are developed and proven for feedforward networks. First it is shown that residual neural networks are finite difference approximations to dynamical systems of first order differential equations, as opposed to ordinary networks that are static. This implies that the network is learning systems of differential equations governing the coordinate transformations that represent the data. Second it is shown that a closed form solution of the metric tensor on the underlying data manifold can be found by backpropagating the coordinate representation through the neural network. This is formulated in a formal abstract sense as a sequence of pullback / Lie group actions on the metric fibre space in the principal and associated bundles on the data manifold, where backpropagation is shown to be the pullback / Lie group actions on tensor bundles. A model based on perturbation theory is developed and used to understand how neural networks treat testing data differently than training data. The second part of this dissertation makes use of neural networks for forecasting probability distributions of time series in terms of discrete symbols that are quantized from real-valued data. The developed framework formulates the forecasting problem into a probabilistic paradigm as htheta : X x Y → [0, 1] such that Sigmay∈Y htheta(x,y) = 1, where X is the finite-dimensional state space, Y is the symbol set, and theta is the set of model parameters. The main advantage of formulating probabilistic forecasting in the symbolic setting is that density predictions are obtained without any significantly restrictive assumptions, such as second order statistics. The efficacy of the proposed method has been demonstrated by forecasting probability distributions on chaotic time series data collected from a laboratory-scale experimental apparatus.

ISBN: 9780438134935Subjects--Topical Terms:

649730
Mechanical engineering.

Principles of Riemannian Geometry in Neural Networks.
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