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.
Highlights
Geometric approaches to analyzing data have been extensively used for many years, the first topological methods for data analysis were developed only recently, e.g., [1,2,3,4,5,6].Topological Data Analysis (TDA) is a fairly new field at the intersection of data science and algebraic topology, the aim of which is to provide robust mathematical, statistical, and algorithmic methods to infer and analyze the topological and geometric structures underlying complex data
Motivated by the desire to objectively classify neuronal morphologies, in a previous publication (Kanari and Hess in [13]), we designed a topological signature for trees, the Topological Morphology Descriptor (TMD), that assigns a barcode to any geometric tree
The TMD (Topological Morphology Descriptor) is a many-to-one function from the set of geometric trees to the set of barcodes, TMD ∶ T → B, that encodes the overall shape of the tree, both the topology of the branching structure of a tree and its embedding in R3 [13]
Summary
Geometric approaches to analyzing data have been extensively used for many years, the first topological methods for data analysis were developed only recently, e.g., [1,2,3,4,5,6]. Topological Data Analysis (TDA) is a fairly new field at the intersection of data science and algebraic topology, the aim of which is to provide robust mathematical, statistical, and algorithmic methods to infer and analyze the topological and geometric structures underlying complex data. That the probability that two bars of a barcode B will be permuted by applying TMD ○ TNS decreases exponentially with the distance between the terminations of the two bars, which is another form of stability Together, these stability results imply that the TNS is an excellent approximation to a (right) inverse to the TMD. The trees T and T ′ can be quite different combinatorially, as seen on the right
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