Persistence Images: A Stable Vector Representation of Persistent Homology
Persistence Images: A Stable Vector Representation of Persistent Homology
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2015-07
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通讯作者：
Henry Adams;T. Emerson;M. Kirby;R. Neville;C. Peterson;Patrick D. Shipman;Sofya Chepushtanova;Eric M. Hanson;Francis C. Motta;Lori Ziegelmeier
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作者：
Henry Adams;T. Emerson;M. Kirby;R. Neville;C. Peterson;Patrick D. Shipman;Sofya Chepushtanova;Eric M. Hanson;Francis C. Motta;Lori Ziegelmeier
许多数据集可以看作是对基础空间的嘈杂采样，而拓扑数据分析的工具可以为知识发现目的表征这种结构。一种这样的工具就是持续的同源性，它提供了数据集中同源特征的多尺度描述。此同源信息的有用表示是持久图（PD）。已经努力将PD映射到具有机器学习任务有价值的其他结构的空间中。我们将PD转换为有限维矢量表示，我们称之为持续图像（PI），并证明了这种转换在输入中的小扰动方面的稳定性。将PI的歧视力与现有方法进行了比较，显示出显着的性能提高。我们探索使用基于向量的机器学习工具的PI的使用，例如线性稀疏支持向量机，这些机器识别包含区分拓扑信息的功能。最后，从离散动力学系统（链接的扭曲图）和部分微分方程（各向异性kuramoto-sivashinsky方程）中对参数值的高精度推断，可提供PIS歧视性的新颖应用。
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.