From trees to barcodes and back again: theoretical and statistical perspectives
From trees to barcodes and back again: theoretical and statistical perspectives
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从树木到条形码再返回：理论和统计视角
作者：
Lida Kanari;A. Garin;K. Hess
拓扑数据分析方法已成功地应用于广泛的领域，提供有用的拓扑描述符，如持久性图方面的复杂数据集的结构的摘要。虽然存在许多用于计算拓扑描述符的强大技术，但是逆问题，即，从拓扑描述符恢复输入数据已被证明是具有挑战性的。在这篇文章中，我们详细研究了拓扑形态描述符（TMD），它分配一个持久性图嵌入在欧氏空间中的任何树，和一种随机逆TMD，拓扑神经元合成（TNS）算法，获得理论和计算的见解两者之间的关系。我们提出了一种新的方法来分类条码使用对称组，这提供了一个具体的语言来制定我们的结果。我们调查到什么程度的TNS恢复几何树从TMD和描述的影响，不同类型的噪声的过程中的树生成持久性图。此外，我们证明了TNS算法是稳定的特定类型的噪声。
Methods of topological data analysis have been successfully applied in a wide range of fields to provide useful summaries of the structure of complex data sets in terms of topological descriptors, such as persistence diagrams. While there are many powerful techniques for computing topological descriptors, the inverse problem, i.e., recovering the input data from topological descriptors, has proved to be challenging. In this article, we study in detail the Topological Morphology Descriptor (TMD), which assigns a persistence diagram to any tree embedded in Euclidean space, and a sort of stochastic inverse to the TMD, the Topological Neuron Synthesis (TNS) algorithm, gaining both theoretical and computational insights into the relation between the two. We propose a new approach to classify barcodes using symmetric groups, which provides a concrete language to formulate our results. We investigate to what extent the TNS recovers a geometric tree from its TMD and describe the effect of different types of noise on the process of tree generation from persistence diagrams. We prove moreover that the TNS algorithm is stable with respect to specific types of noise.