Computing multidimensional persistence

ArXiv, 2015

An algorithm is presented that constructs an acyclic partial matching on the cells of a given simplicial complex from a vector-valued function defined on the vertices and extended to each simplex by taking the least common upper bound of the values on its vertices. The resulting acyclic partial matching may be used to construct a reduced filtered c omplex with the same multidimensional persistent homology as the original simplicial complex filtered by the sublevel sets of the function. Numeri cal tests show that in practical cases the rate of reduction in the number of cells achieved by the algorithm is substantial. This promises to be useful for the computation of multidimensional persistent homology of simplicial complexes filtered by sublevel sets of vector-valued functions.

2012

The extraction of significant structures in arbitrary high-dimensional data sets is a challenging task. Moreover, classifying data points as noise in order to reduce a data set bears special relevance for many application domains. Standard methods such as clustering serve to reduce problem complexity by providing the user with classes of similar entities. However, they usually do not highlight relations between different entities and require a stopping criterion, e.g. the number of clusters to be detected. In this paper, we present a visualization pipeline based on recent advancements in algebraic topology. More precisely, we employ methods from persistent homology that enable topological data analysis on high-dimensional data sets. Our pipeline inherently copes with noisy data and data sets of arbitrary dimensions. It extracts central structures of a data set in a hierarchical manner by using a persistence-based filtering algorithm that is theoretically well-founded. We furthermore introduce persistence rings, a novel visualization technique for a class of topological features-the persistence intervals-of large data sets. Persistence rings provide a unique topological signature of a data set, which helps in recognizing similarities. In addition, we provide interactive visualization techniques that assist the user in evaluating the parameter space of our method in order to extract relevant structures. We describe and evaluate our analysis pipeline by means of two very distinct classes of data sets: First, a class of synthetic data sets containing topological objects is employed to highlight the interaction capabilities of our method. Second, in order to affirm the utility of our technique, we analyse a class of high-dimensional real-world data sets arising from current research in cultural heritage.

ArXiv, 2021

In Topological Data Analysis, a common way of quantifying the shape of data is to use a persistence diagram (PD). PDs are multisets of points in R computed using tools of algebraic topology. However, this multi-set structure limits the utility of PDs in applications. Therefore, in recent years efforts have been directed towards extracting informative and efficient summaries from PDs to broaden the scope of their use for machine learning tasks. We propose a computationally efficient framework to convert a PD into a vector in R, called a vectorized persistence block (VPB). We show that our representation possesses many of the desired properties of vector-based summaries such as stability with respect to input noise, low computational cost and flexibility. Through simulation studies, we demonstrate the effectiveness of VPBs in terms of performance and computational cost within various learning tasks, namely clustering, classification and change point detection.

Journal of Mathematical Imaging and Vision, 2018

Given a simplicial complex and a vector-valued function on its vertices, we present an algorithmic construction of an acyclic partial matching on the cells of the complex compatible with the given function. This implies the construction can be used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. The correctness of the algorithm is proved, and its complexity is analyzed. A combinatorial interpretation of our algorithm based on the concept of a multidimensional discrete Morse function is introduced for the first time in this paper. Numerical experiments show a substantial rate of reduction in the number of cells achieved by the algorithm.

Foundations of Computational Mathematics, 2016

In this paper we study multidimensional persistence modules 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.

ArXiv, 2019

Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category, so that functors to that category induce persistence functions. We port the interleaving and bottleneck distances to this novel framework and generalize classical equalities and inequalities. Unlike sets and vector spaces, in many categories the rank of an object does not identify it up to isomorphism: to preserve information about the structure of persistence modules, we define colorable ranks, persistence diagrams and prove the equality between multicolored bottleneck distance and interleaving distance in semisimple Abelian...

2017

Given a simplicial complex and a vector-valued function on its vertices, we present an algorithmic construction of an acyclic partial matching on the cells of the complex. This construction is used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. A number of numerical experiments show a substantial rate of reduction in the number of cells achieved by the algorithm.

Arxiv preprint arXiv: …, 2009

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence.

2021

Persistence landscapes are functional summaries of persistence diagrams designed to enable analysis of the diagrams using tools from functional data analysis. They comprise a collection of scalar functions such that birth and death times of topological features in persistence diagrams map to extrema of functions and intervals where they are non-zero. As a consequence, topological information is encoded in both amplitude and phase components of persistence landscapes. Through functional data analysis of persistence landscapes under an elastic Riemannian metric, we show how meaningful statistical summaries of persistence landscapes (e.g., mean, dominant directions of variation) can be obtained by decoupling topological signal present in amplitude and phase variations. The estimated phase functions are tied to the resolution parameter that determines the filtration of simplicial complexes used to construct persistence diagrams. For a dataset obtained under scale and sampling variabilit...

2019

Nowadays, data generation, representation and analysis occupy central roles in human society. Therefore, it is necessary to develop frameworks of analysis able of adapting to diverse data structures with minimal effort, much as guaranteeing robustness and stability. While topological persistence allows to swiftly study simplicial complexes paired with continuous functions, we propose a new theory of persistence that is easily generalizable to categories other than topological spaces and functors other than homology. Thus, in this framework, it is possible to study complex objects such as networks and quivers without the need of auxiliary topological constructions. We define persistence functions by directly considering relevant features (even discrete) of the objects of the category of interest, while maintaining the properties of topological persistence and persistent homology that are essential for a robust, stable and agile data analysis.