From Trees to Barcodes and Back Again: Theoretical and Statistical Perspectives

Abstract

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.

Stratifying the space of barcodes using Coxeter complexes

“…We call the coordinates that we obtain from this description Coxeter coordinates. [27,14]. The stratification one obtains is induced by the simplicial structure of Σ(Sym n ).…”

“…Therefore, they offer a new way to study statistics of barcodes by using both the average and standard deviation of births and deaths, which are commonly used summaries in Topological Data Analysis (TDA), and permutation statistics tools. The latter include the number of descents for instance, or the inversion numbers, which have proven useful for the study of the inverse problem for trees and barcodes [27,14].…”

“…by the permutation type of a barcode as defined in [27,14]. By associating to a barcode the coordinates of its region, we define a new invariant of barcodes.…”

“…We call a barcode strict if there are no two pairs in it that have the same birth or death. It was observed in [27] that to a strict barcode B = {(b i , d i )} i∈{1,...,n} with n bars, one can associate a permutation σ B ∈ Sym n . It is the permutation such that the bar with the i-th smallest death has the σ B (i)-th smallest birth.…”

“…A first approach to this, taken in [27,14], is to consider the Cayley graph of the symmetric group with respect to the generating set given by adjacent transpositions (i, i + 1). This yields a combinatorial representation of the elements of Sym n .…”