Wojciech Chachólski

KTH Royal Institute of Technology

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Jérôme Scherer

École Polytechnique Fédérale de Lausanne

Martina Scolamiero

KTH Royal Institute of Technology

Francesca Tombari

Azienda USL di Bologna

Wolfgang Pitsch

Universitat Autònoma de Barcelona

Claudia Landi

University of Modena and Reggio Emilia

Alvin Jin

Massachusetts Institute of Technology

Barbara Giunti

Graz University of Technology

Ryan Ramanujam

Karolinska Institutet

Emmanuel Dror Farjoun

Hebrew University of Jerusalem

Henri Riihimäki

KTH Royal Institute of Technology

Cooperative Institutions

KTH Royal Institute of Technology

École Polytechnique Fédérale de Lausanne

Polytechnic University of Turin

Uniwersytet Komisji Edukacji Narodowej w Krakowie

Max Planck Institute for Mathematics

Max Planck Society

Universitat Autònoma de Barcelona

Australian Mathematical Sciences Institute

Australian National University

Northwestern University

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A topological data analysis based classification method for multiple measurements

arXiv (Cornell University) (2019)

Henri Riihimäki Wojciech Chachólski Jakob Theorell Jan Hillert Ryan Ramanujam

Machine learning models for repeated measurements are limited. Using topological data analysis (TDA), we present a classifier for repeated measurements which samples from the data space and builds a network graph based on the data topology. When applying this to two case studies, accuracy exceeds alternative models with additional benefits such as reporting data subsets with high purity along with feature values. For 300 examples of 3 tree species, the accuracy reached 80% after 30 datapoints, which was improved to 90% after increased sampling to 400 datapoints. Using data from 100 examples of each of 6 point processes, the classifier achieved 96.8% accuracy. In both datasets, the TDA classifier outperformed an alternative model. This algorithm and software can be beneficial for repeated measurement data common in biological sciences, as both an accurate classifier and a feature selection tool.

Topological data analysis

10.48550/arxiv.1904.02971

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Decomposing filtered chain complexes: geometry behind barcoding algorithms

arXiv (Cornell University) (2020)

Wojciech Chachólski Barbara Giunti Alvin Jin Claudia Landi

In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called interval spheres. In this paper, we provide an algorithm to decompose filtered chain complexes into such interval spheres. This algorithm provides geometric insights into various aspects of the standard persistence algorithm and two of its run-time optimizations. Moreover, since it works for any filtered chain complexes, our algorithm can be applied in more general cases. As an application, we show how to decompose filtered kernels with it.

Chain (unit)

Persistence (discontinuity)

10.48550/arxiv.2012.01033

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Homotopical decompositions of simplicial and Vietoris Rips complexes

Journal of Applied and Computational Topology (2021)

Wojciech Chachólski Alvin Jin Martina Scolamiero Francesca Tombari

Abstract Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings.

Simplicial approximation theorem

Simplicial homology

Simplicial manifold

Abstract simplicial complex

Model category

h-vector

Clique

10.1007/s41468-021-00066-2

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Citations (3)

Cellular generators

Proceedings of the American Mathematical Society (2004)

Wojciech Chachólski Paul-Eugène Parent Donald Stanley

The aim of this paper is twofold. On the one hand, we show that the kernel $\overline {C(A)}$ of the Bousfield periodization functor $P_A$ is cellularly generated by a space $B$, i.e., we construct a space $B$ such that the smallest closed class $C(B)$ containing $B$ is exactly $\overline {C(A)}$. On the other hand, we show that the partial order $(Spaces,\gg )$ is a complete lattice, where $B\gg A$ if $B\in C(A)$. Finally, as a corollary we obtain Bousfieldâs theorem, which states that $(Spaces,>)$ is a complete lattice, where $B>A$ if $B\in \overline {C(A)}$.

Corollary

Lattice (music)

Kernel (algebra)

10.1090/s0002-9939-04-07346-0

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Stable Invariants for Multiparameter Persistence

arXiv (Cornell University) (2017)

Oliver Gäfvert Wojciech Chachólski

In this paper we explain how to convert discrete invariants into stable ones via what we call hierarchical stabilization. We illustrate this process by constructing stable invariants for multi-parameter persistence modules with respect to the interleaving distance and so called simple noise systems. For one parameter, we recover the standard barcode information. For more than one parameter we prove that the constructed invariants are in general NP-hard to calculate. A consequence is that computing the feature counting function, proposed by Scolamiero et. al. (2016), is NP-hard.

Barcode

Persistence (discontinuity)

Interleaving

10.48550/arxiv.1703.03632

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Citations (1)

Homotopy excision and cellularity

arXiv (Cornell University) (2014)

Wojciech Chachólski Jérôme Scherer Kay Werndli

Consider a push-out diagram of spaces C <-- A --> B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem.

10.48550/arxiv.1408.3252

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Idempotent deformations of finite groups

arXiv (Cornell University) (2012)

Martin Blomgren Wojciech Chachólski Emmanuel Dror Farjoun Yoav Segev

The action of the idempotent deformations on finite groups is discussed. This action is described in terms of the homological properties of groups. The orbits of finite simple groups are determined.

10.48550/arxiv.1204.6566

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Metrics and stabilization in one parameter persistence

arXiv (Cornell University) (2019)

Wojciech Chachólski Henri Riihimäki

We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector spaces leads to stabilization of discrete invariants. We develop theory behind this stabilization and stable rank invariant. We give evidence of the usefulness of this approach in concrete data analysis.

Persistence (discontinuity)

Rank (graph theory)

10.48550/arxiv.1904.02905

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Stable Invariants for Multidimensional Persistence

arXiv (Cornell University) (2017)

Oliver Gäfvert Wojciech Chachólski

In this paper we explain how to convert discrete invariants into stable ones via what we call hierarchical stabilization. We illustrate this process by constructing stable invariants for multi parameter persistence modules with respect to so called simple noise systems. For one parameter we recover the standard bar code information. For more than one parameter we prove that the constructed invariants are in general NP-hard to calculate.

Persistence (discontinuity)

Code (set theory)

Bar (unit)

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Citations (5)

A topological data analysis based classification method for multiple measurements

arXiv (Cornell University) (2019)

Henri Riihimäki Wojciech Chachólski Jakob Theorell Jan Hillert Ryan Ramanujam

Topological data analysis

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Citations (2)