Скачать или смотреть I-AIM seminar 4 (Kathryn Hess): From trees to barcode and back again

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 talk I will focus on 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. I will provide an overview of the TMD and the TNS and then describe the results of our theoretical and computational analysis of their behavior and properties, in which symmetric groups play a key role. In particular, I will specify the extent to which the TNS provides an inverse to the TMD. This is joint work with Adélie Garin and Lida Kanari, based on earlier collaborations led by Lida.