In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode.Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order.We show how to lift this combinatorial characterization of barcodes to an analogous combinatorialization of merge trees.As result of this study, we provide the first clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees: generic phylogenetic trees on n + 1 leaf nodes fall into (2n -1)!! distinct equivalence classes, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., (n + 1)!n!2 -n .The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group.We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation of our distribution in terms of Dirichlet convolution.This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data, which opens the door to doing more precise statistics and science.

Persistent Homology (PH) has been successfully used to train networks to detect curvilinear structures and to improve the topological quality of their results. However, existing methods are very global and ignore the location of topological features. In this paper, we remedy this by introducing a new filtration function that fuses two earlier approaches: thresholding-based filtration, previously used to train deep networks to segment medical images, and filtration with height functions, typically used to compare 2D and 3D shapes. We experimentally demonstrate that deep networks trained using our PH-based loss function yield reconstructions of road networks and neuronal processes that reflect ground-truth connectivity better than networks trained with existing loss functions based on PH.

We introduce giotto-tda, a Python library that integrates high-performance topological data analysis with machine learning via a scikit-learn-compatible API and state-of-the-art C++ implementations. The library's ability to handle various types of data is rooted in a wide range of preprocessing techniques, and its strong focus on data exploration and interpretability is aided by an intuitive plotting API. Source code, binaries, examples, and documentation can be found at https://github.com/giotto-ai/giotto-tda.

MOTIVATION: Unbiased clustering methods are needed to analyze growing numbers of complex datasets. Currently available clustering methods often depend on parameters that are set by the user, they lack stability, and are not applicable to small datasets. To overcome these shortcomings we used topological data analysis, an emerging field of mathematics that discerns additional feature and discovers hidden insights on datasets and has a wide application range. RESULTS: We have developed a topology-based clustering method called Two-Tier Mapper (TTMap) for enhanced analysis of global gene expression datasets. First, TTMap discerns divergent features in the control group, adjusts for them, and identifies outliers. Second, the deviation of each test sample from the control group in a high-dimensional space is computed, and the test samples are clustered using a new Mapper-based topological algorithm at two levels: a global tier and local tiers. All parameters are either carefully chosen or data-driven, avoiding any user-induced bias. The method is stable, different datasets can be combined for analysis, and significant subgroups can be identified. It outperforms current clustering methods in sensitivity and stability on synthetic and biological datasets, in particular when sample sizes are small; outcome is not affected by removal of control samples, by choice of normalization, or by subselection of data. TTMap is readily applicable to complex, highly variable biological samples and holds promise for personalized medicine. AVAILABILITY AND IMPLEMENTATION: TTMap is supplied as an R package in Bioconductor. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.