Object-oriented Persistent Homology.
Object-oriented Persistent Homology.
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DOI：
10.1016/j.jcp.2015.10.036
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发表时间：
2016-01-15
影响因子：
4.1
通讯作者：
Wei GW
中科院分区：
文献类型：
--
作者：
Wang B;Wei GW
持久同调通过测量过滤过程中固有拓扑特征的生命时间，为大数据的拓扑简化提供了一种新的方法，并在科学和工程应用中取得了成功。然而，这样的成功基本上局限于定性的数据分类和分析。事实上，持续同源性很少用于定量建模和预测。此外，目前的持久同源性是一种被动的工具，而不是一种主动的技术，用于分类和分析。在这项工作中，我们概述了一个通用协议来构造面向对象的持久同源方法。利用曲面的微分几何理论，构造了一个目标泛函，即在目标数据上定义的曲面自由能。目标函数的最小化导致拉普拉斯-贝尔特拉米算子生成初始数据的多尺度表示，并提供面向目标的过滤过程。所得到的基于微分几何的面向对象的持久同构能够在进化过滤中保留理想的几何特征，并增强相应的拓扑持久性。为了与拉普拉斯-贝尔特拉米流的笛卡尔表示相容，本文采用了基于三次复的同调算法。本文提出的基于Laplace-Beltrami流的持久同调方法得到了广泛的验证。在Vietoris-Rips复合体上进行了大量的数值试验，证实了基于Laplace-Beltrami流的过滤与基于欧几里得距离的过滤的一致性。分析了基于拉普拉斯-贝尔特拉米流的立方复滤法在不同时空网格尺寸下的收敛性和可靠性。基于拉普拉斯-贝尔特拉米流的持续同源性方法被用于研究蛋白质和富勒烯分子的内在拓扑结构。基于富勒烯中心腔的拓扑持久性与富勒烯结构的总曲率能之间的定量模型，该方法用于预测富勒烯异构体的稳定性。通过500多个富勒烯分子的实验验证了该方法的有效性和鲁棒性。结果表明，所提出的基于持续同源性的定量模型能很好地预测十种富勒烯异构体的总曲率能。本文首次设计了面向对象的持久同源性，在过滤过程中增强或保留原始数据中的理想特征，然后从数据中自动检测或提取相应的拓扑特征。
Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct object-oriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the Laplace-Beltrami flow. The proposed Laplace-Beltrami flow based persistent homology method is extensively validated. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The Laplace-Beltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design object-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.