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arXiv 2009.01786cs.CVcs.LGcs.NAmath.NA
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变形流形的计算分析:从几何建模到深度学习

Computational Analysis of Deformable Manifolds: from Geometric Modelling to Deep Learning

Rensselaer Polytechnic Institute(伦斯勒理工学院)

Stefan C Schonsheck

摘要:

列夫·托尔斯泰(Leo Tolstoy)在他的不朽小说《安娜·卡列尼娜》(Anna Karenina)中以现在著名的话开头:幸福的家庭都是相似的;每个不幸的家庭都有自己的不幸类似的概念也适用于数学空间:每个平坦的空间都是相似的;每个不平坦的空间都有自己的不平坦。然而,我们将展示非平坦空间的多样性提供了一个丰富的研究领域,而不是成为不快乐的来源。所谓的大数据时代的起源以及不断增长的社会和科学数据库的激增导致了对能够有效处理、分析甚至生成高维数据的算法的需求。然而,维数灾难导致的事实是,许多经典的方法不缩放以及这些问题的大小。避免这些不良影响的一种技术是利用相干数据的几何结构。在这篇论文中,我们将探讨几何方法的形状处理和数据分析。更具体地说,我们将通过各种数学工具,包括但不限于计算微分几何,变分PDE建模和深度学习,研究表示流形和信号的技术。首先,我们将探索通过变分建模的非等距形状匹配。接下来,我们将使用流形上的并行传输的思想来将卷积和卷积神经网络推广到可变形流形。最后,我们提出了一种新的自回归模型来捕获数据的内在几何和拓扑结构。在整个工作中,我们将使用计算对应的想法作为思路,以激励我们的工作和分析我们的结果。

英文摘要:

Leo Tolstoy opened his monumental novel Anna Karenina with the now famous words: Happy families are all alike; every unhappy family is unhappy in its own way A similar notion also applies to mathematical spaces: Every flat space is alike; every unflat space is unflat in its own way. However, rather than being a source of unhappiness, we will show that the diversity of non-flat spaces provides a rich area of study. The genesis of the so-called big data era and the proliferation of social and scientific databases of increasing size has led to a need for algorithms that can efficiently process, analyze and, even generate high dimensional data. However, the curse of dimensionality leads to the fact that many classical approaches do not scale well with respect to the size of these problems. One technique to avoid some of these ill-effects is to exploit the geometric structure of coherent data. In this thesis, we will explore geometric methods for shape processing and data analysis. More specifically, we will study techniques for representing manifolds and signals supported on them through a variety of mathematical tools including, but not limited to, computational differential geometry, variational PDE modeling, and deep learning. First, we will explore non-isometric shape matching through variational modeling. Next, we will use ideas from parallel transport on manifolds to generalize convolution and convolutional neural networks to deformable manifolds. Finally, we conclude by proposing a novel auto-regressive model for capturing the intrinsic geometry and topology of data. Throughout this work, we will use the idea of computing correspondences as a though-line to both motivate our work and analyze our results.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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