On the geometrical properties of the coherent matching distance in 2D persistent homology

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including its stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a normal filtering function as well as of transport of a matching. As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.Comment: 39 pages, 15 figures. Corrected the definition of multiplicity of points in the extended Pareto grid and the definition of normal function. Removed Rem. 3.3. Added Ex. 3.9, Fig. 11, Fig. 12, Rem. 5.3 and Fig. 15. Changed Rem. 4.9 into regular text. Reformulated statements of Theorems 5.1, 5.2, 5.4. Some changes in their proofs. Added references. Some small changes in the text and in the figure

Deformation of surfaces in 2D persistent homology

In the context of 2D persistent homology a new metric has been recently introduced, the coherent matching distance. In order to study this metric, the filtering function is required to present particular “regularity” properties, based on a geometrical construction of the real plane, called extended Pareto grid. This dissertation shows a new result for modifying the extended Pareto grid associated to a filtering function defined on a smooth closed surface, with values in the real plane. In future, the technical result presented here could be used to prove the genericity of the regularity conditions assumed for the filtering function

A Brief Introduction to Multidimensional Persistent Betti Numbers

In this paper, we propose a brief overview about multidimensional persistent Betti numbers (PBNs) and the metric that is usually used to compare them, i.e., the multidimensional matching distance. We recall the main definitions and results, mainly focusing on the 2-dimensional case. An algorithm to approximate n-dimensional PBNs with arbitrary precision is described

The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for topological data analysis in scientific visualization. TTK provides a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependence-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website

Unified Topological Inference for Brain Networks in Temporal Lobe Epilepsy Using the Wasserstein Distance

Persistent homology can extract hidden topological signals present in brain networks. Persistent homology summarizes the changes of topological structures over multiple different scales called filtrations. Doing so detect hidden topological signals that persist over multiple scales. However, a key obstacle of applying persistent homology to brain network studies has always been the lack of coherent statistical inference framework. To address this problem, we present a unified topological inference framework based on the Wasserstein distance. Our approach has no explicit models and distributional assumptions. The inference is performed in a completely data driven fashion. The method is applied to the resting-state functional magnetic resonance images (rs-fMRI) of the temporal lobe epilepsy patients collected at two different sites: University of Wisconsin-Madison and the Medical College of Wisconsin. However, the topological method is robust to variations due to sex and acquisition, and thus there is no need to account for sex and site as categorical nuisance covariates. We are able to localize brain regions that contribute the most to topological differences. We made MATLAB package available at https://github.com/laplcebeltrami/dynamicTDA that was used to perform all the analysis in this study

AVIDA: Alternating method for Visualizing and Integrating Data

High-dimensional multimodal data arises in many scientific fields. The integration of multimodal data becomes challenging when there is no known correspondence between the samples and the features of different datasets. To tackle this challenge, we introduce AVIDA, a framework for simultaneously performing data alignment and dimension reduction. In the numerical experiments, Gromov-Wasserstein optimal transport and t-distributed stochastic neighbor embedding are used as the alignment and dimension reduction modules respectively. We show that AVIDA correctly aligns high-dimensional datasets without common features with four synthesized datasets and two real multimodal single-cell datasets. Compared to several existing methods, we demonstrate that AVIDA better preserves structures of individual datasets, especially distinct local structures in the joint low-dimensional visualization, while achieving comparable alignment performance. Such a property is important in multimodal single-cell data analysis as some biological processes are uniquely captured by one of the datasets. In general applications, other methods can be used for the alignment and dimension reduction modules.Comment: To appear in Journal of Computational Science (Accepted, 2023

Computing multiparameter persistent homology through a discrete Morse-based approach

Persistent homology allows for tracking topological features, like loops, holes and their higher-dimensional analogues, along a single-parameter family of nested shapes. Computing descriptors for complex data characterized by multiple parameters is becoming a major challenging task in several applications, including physics, chemistry, medicine, and geography. Multiparameter persistent homology generalizes persistent homology to allow for the exploration and analysis of shapes endowed with multiple filtering functions. Still, computational constraints prevent multiparameter persistent homology to be a feasible tool for analyzing large size data sets. We consider discrete Morse theory as a strategy to reduce the computation of multiparameter persistent homology by working on a reduced dataset. We propose a new preprocessing algorithm, well suited for parallel and distributed implementations, and we provide the first evaluation of the impact of multiparameter persistent homology on computations

Topological Learning for Brain Networks

This paper proposes a novel topological learning framework that can integrate networks of different sizes and topology through persistent homology. This is possible through the introduction of a new topological loss function that enables such challenging task. The use of the proposed loss function bypasses the intrinsic computational bottleneck associated with matching networks. We validate the method in extensive statistical simulations with ground truth to assess the effectiveness of the topological loss in discriminating networks with different topology. The method is further applied to a twin brain imaging study in determining if the brain network is genetically heritable. The challenge is in overlaying the topologically different functional brain networks obtained from the resting-state functional MRI (fMRI) onto the template structural brain network obtained through the diffusion MRI (dMRI)