arXiv subjects

Wojciech Chachólski

Publications and source records attributed to Wojciech Chachólski.

10 recordsLinked to original sources

arXiv2026

Choosing the Geometry: valuations, norms, and contours

We develop a general framework for constructing pseudometrics on objects of Abelian categories using valuations, which assign non-negative costs to morphisms. Specializing to finitely presented vector space representations of the poset $[0,\infty)^\ell$, a natural setting for multiparameter persistence modules, we show that a large class of these distances is governed by contours, right-continuous lax actions of $[0,\infty)$ on the parameter poset. Properties of contours translate into geometric properties of the induced distances, including conditions under which the resulting pseudometrics are metrics and allowing us to identify a rich family of compact subsets of finitely presented representations.

math.AT↗

arXiv2024

Abelian and model structures on tame functors

In this paper, we discuss certain circumstances in which the category of tame functors inherits an abelian category structure with minimal resolutions and a model category structure with minimal cofibrant replacements. We also present a structure theorem for cofibrant objects in the category of tame functors indexed by realizations of posets of dimension $1$ with values in the category of chain complexes in an abelian category whose all objects are projectives. Moreover, we introduce a general technique to generate indecomposable objects in the abelian category of functors indexed by finite posets.

math.AT↗

arXiv2024

The Shift-Dimension of Multipersistence Modules

We present the shift-dimension of multipersistence modules and investigate its algebraic properties. This gives rise to a new invariant of multigraded modules over the multivariate polynomial ring arising from the hierarchical stabilization of the zeroth total multigraded Betti number. We give a fast algorithm for the computation of the shift-dimension of interval modules in the bivariate case. We construct multipersistence contours that are parameterized by multivariate functions and hence provide a large class of feature maps for machine learning tasks.

math.AC↗

arXiv2022

Decomposing filtered chain complexes: geometry behind barcoding algorithms

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.

math.AT↗

arXiv2021

Stable Invariants for Multiparameter Persistence

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.

math.AT↗

arXiv2020

Invariants for tame parametrised chain complexes

We set the foundations for a new approach to Topological Data Analysis (TDA) based on homotopical methods at chain complexes level. We present the category of tame parametrised chain complexes as a comprehensive environment that includes several cases that usually TDA handles separately, such as persistence modules, zigzag modules, and commutative ladders. We extract new invariants in this category using a model structure and various minimal cofibrant approximations. Such approximations and their invariants retain some of the topological, and not just homological, aspects of the objects they approximate.

math.AT↗

arXiv2020

Metrics and stabilization in one parameter persistence

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.

math.AT↗

arXiv2019

A topological data analysis based classification method for multiple measurements

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.

cs.LG↗

arXiv2016

Multidimensional Persistence and Noise

In this paper we study multidimensional persistence modules [5,13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

math.AT↗

arXiv2012

Idempotent deformations of finite groups

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.

math.GR↗