Speaker: Yasu Hiraoka
Title: Topological data analysis on materials science: statistical characterization of glass transition
Abstract: In this talk, topological data analysis (TDA) on materials science will be presented. In particular, I will focus on amorphous structural analysis, and show how TDA is useful for revealing topological and geometric structure in disordered systems. Moreover, I will demonstrate that
several mathematical tools (persistence weighted Gaussian kernel and persistence modules on commutative ladder) can characterize new materials properties such as glass transitions and rigidity.
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Speaker: Gunnar Carlsson
Title: Topological modeling of complex data
Abstract: Topological data analysis has been making great strides as a tool for the analysis of numerous kinds of data. It turns out that it can be viewed as a complete modeling framework, which can perform very useful unsupervised analysis in additional to analogues of standard modeling operations. I will discuss this with examples.
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Speaker: David Spivak
Title: Pixel matrices for big, messy, real-world data
Abstract: Natural, real-world data does not come equipped with
mathematical equations that it satisfies. A data set is just a
collection of observed relationships, so it is naturally messy and
probabilistic. When combining and analyzing these observed
relationships, it is destructive, yet common practice, to first
"normalize" the data by choosing models, fitting curves, removing
outliers, or estimating parameters. It would often be preferable to
forgo the clean-up steap and simply combine the data sets directly.
Pixel matrices are an applied category-theoretic technique for doing
just that. We will explain how pixel matrices offer a massively
parallelizable method for approximating the solution set for systems
of nonlinear equations, and how the same idea can be applied when
working with messy data.
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Speaker: Michael Kerber
Title: Geometry helps to Compare Persistence Diagrams
Abstract: Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft--Karp algorithm for bottleneck matching (based on previous work by Efrat el al.) and of the auction algorithm
by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams.
This is joint work with Arnur Nigmetov and Dmitriy Morozov
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Speaker: Clement Maria
Title: Zigzag persistent homology: Algorithms, software and applications
Abstract: Zigzag persistence is a generalisation of persistent homology with a promising range of new applications. In this talk, I will present the recent algorithmic approaches to compute zigzag persistence diagrams, as well as their advantages and limits compare to algorithms for standard persistent homology. I will also present the integration of these methods in software, and motivate their use through topological data analysis applications.
" In this talk we will present the Gudhi library, a state of the art software package to compute topological and geometrical information of various complexes. We will start by discussing the vailable data structures and the ways of creating them based on various types of data. Then, we will show how to compute different types of persistence homology informations based on those complexes.
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Speaker: Pawel Dłotko
Title: Geometry Understanding in Higher Dimensions, the Gudhi library
Abstract: Gudhi is an open source library whose initial development has been undertaken by. Jean-Daniel Boissonnat, Pawel Dlotko, Clément Jamin, Siargey Kachanovich, Marc Glisse, Clément Maria, Vincent Rouvreau and David Salinas.
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Speaker: Tamal Dey
Title: SimBa : A Tool for Approximating the Persistence of Rips Filtrations Efficiently
Abstract: In topological data analysis, a point cloud data $P$ extracted from a metric space is often analyzed by computing the persistence diagram or barcodes of a sequence of Rips complexes built on $P$ indexed by a scale parameter. Unfortunately, even for moderate number of points, the size of the Rips complex may become prohibitively large as the scale parameter increases. Some existing methods such as Sparse Rips filtration aims to reduce the size of the complex so as to improve the time efficiency as well. However, experiments show that the existing approaches still fall short of scaling well, especially for high dimensional data. Based on insights gained from the experiments, we present an efficient new algorithm and a software, called \emph{SimBa}, for approximating the persistent homology of Rips filtrations with quality guarantees. Our approach leverages a batch collapse strategy as well as a new sparse Rips-like filtration. Experiments show that this strategy presents significant size and time reduction in practice.
Joint work with Dayu Shi and Yusu Wang
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Speaker: Matthew Wright
Title: Visualizing Multidimensional Persistent Homology
Abstract: Multidimensional persistent homology is highly relevant in various applications in which data is simultaneously filtered by two or more parameters. However, the algebraic complexity of multidimensional persistence modules makes it difficult to extract useful invariants in this setting. In this talk, I will describe recent work with Mike Lesnick to efficiently compute and visualize both the multigraded Betti numbers and the rank invariant of multidimensional persistence modules. In addition, I will demonstrate a new software program for computing and visualizing these invariants.
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Speaker: Jisu Kim
Title: R package TDA for Statistical inference on Topological Data Analysis
Abstract: This talk gives an introduction to the R package TDA and how it can be used to do a statistical inference on topological data analysis. Given data, the salient topological features of underlying space can be quantified with persistent homology. Between the resulting persistence diagrams, bottleneck distance is defined. The bottleneck distance between persistence diagrams is bounded by the distance between corresponding functions or datasets, which is the stability theorem. Based on the stability theorem, the confidence band can be computed to distinguish significant homological features from the noisy features in the persistence diagrams. I present how this statistical inference on persistence diagrams can be done using the R package TDA.
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Speaker: Yonghjin Lee
Title: Topological Data Analysis of Nanoporous Materials Genome Using Pore-Geometry Recognition Technique
Abstract: In most applications of nanoporous materials the pore structure is as important as the chemical composition as a determinant of performance. For example, one can alter performance in applications like carbon capture or methane storage by orders of magnitude by only modifying the pore structure. For these applications it is therefore important to identify the optimal pore geometry and use this information to find similar materials. However, the mathematical language and tools to identify materials with similar pore structures, but different composition, has been lacking. In this talk, we present development of a pore recognition approach to quantify similarity of pore structures and classify them using topological data analysis. Our approach allows us to identify materials with similar pore geometries, and to screen for materials that are similar to given top-performing structures. Using methane storage and carbon capture as a case study, we show that materials can be divided into topologically distinct classes — and that each class requires different optimisation strategies. In this work, we have focused on pore space, but our topological approach can be generalised to quantify similarity of any geometric object, which, given the many different Materials Genomics initiatives, opens many interesting avenues for big-data science.
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Speaker: Ellen Gasparovic
Title: Multi-Scale Modeling for Stratified Space Data
Abstract: Geometric and topological methods in data analysis are capable of exposing essential shape information that may be hidden in the original data. Our goal is to introduce an algorithm for producing multi-scale models for data using an adaptive cover tree methodology, techniques from persistent homology, and multi-scale local principal component analysis. Our method takes in a point cloud sampled from a stratified space, and produces a model for the data that is fundamentally based on its local geometric properties and that captures how the different pieces of the dataset fit together. We consider applications of our methods to musical audio data.
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Speaker: Pablo Camara
Title: Topological Methods for Molecular Phylogenetics
Abstract: The recent explosion of genomic data has underscored the need for interpretable and comprehensive analyses that can capture complex phylogenetic relations within and across species. Recombination, re-assortment and horizontal gene transfer constitute examples of pervasive biological phenomena that cannot be captured by tree-like representations. Starting from hundreds of genomes, we are interested in the reconstruction of potential evolutionary histories leading to the observed data. Recently, applied topology methods have been proposed as robust and scalable methods that can capture the genetic scale and frequency of recombination. In this talk I will discuss recent developments in the study of recombination using persistent homology, and I will present several biological applications, including the construction of high-resolution whole-genome human recombination maps.