Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS165, part 1: Multiparameter persistence: algebra, algorithms, and applications
Time:
Tuesday, 09/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F006
30 seats, 57m^2

Presentations
3:00pm - 5:00pm

Multiparameter persistence: algebra, algorithms, and applications

Chair(s): Matthew Wright (St. Olaf College, United States of America)

Multiparameter persistent homology is an area of applied algebraic topology that studies topological spaces, often arising from complex data, simultaneously indexed by multiple parameters. In the usual setting, persistent homology studies a single-parameter filtration associated with a topological space. The homology of such a filtration is a persistence module, which can be conveniently described by its barcode decomposition. In many applications, however, a single-parameter filtration is not adequate to encode the structures of interest in complex data; two or more filtrations may be required. Multiparameter persistence studies the homology of spaces equipped with multiple filtrations. The homological invariants of these spaces are far more complicated than in the single-parameter setting, requiring new algebraic, computational, and statistical techniques. This work has deep connections to representation theory and commutative algebra, with compelling applications to data analysis.

Recent years have seen considerable advances in multiparameter persistent homology, including algorithms for working with large multiparameter persistence modules, software for computing and visualizing invariants, statistical techniques, and applications. This minisymposium will highlight recent work in multiparameter persistence. Talks will include including theoretical results, algorithmic advances, and applications to data analysis. As many important questions remain to be answered in order to advance the theory and to increase the applicability of multiparameter persistence, this minisymposium seeks to cultivate discussion and collaboration that will lead to new results in the practical use of multiparameter persistent homology.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Multiparameter persistence: brief background and current challenges

Matthew Wright
St. Olaf College

Persistent homology is a popular tool in topological data analysis, providing a method for discerning the shape of complex data. Applied in areas including computer graphics, biology, neuroscience, and signal processing, persistent homology produces easily-visualized algebraic invariants, called barcodes, which convey information about the topological structure of data. A multiparameter variant of persistent homology is particularly desirable for working with data simultaneously indexed by multiple parameters, but its algebraically complexity poses challenges in practice.

This talk with introduce multiparameter persistent homology, with emphasis on mathematical foundations of this subject. We will see how multifiltered topological spaces arise from real-world data scenarios. We will introduce multiparameter persistence modules and see how their algebraic complexity poses challenges for their use in practice. We will briefly consider recent work, applications, and open questions in multiparameter persistent homology.

 

Computing minimal presentations and bigraded Betti numbers of 2-parameter persistent homology

Michael Lesnick
University of Albany

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm assumes that $M$ is given implicitly: It takes as input a short chain complex of free bipersistence modules [Faxrightarrow{ma} Fb xrightarrow{mb} Fc] such that $Mcong ker{mb}/im{ma}$. The algorithm runs in time $O(sum_i |F^i|^3)$ and requires $O(sum_i |F^i|^2)$ storage, where $|F^i|$ denotes the size of a basis of $F^i$. Given the presentation, the bigraded Betti numbers of the module are readily computed. We also present a different but related algorithm, based on Koszul homology, which computes the bigraded Betti numbers without computing a presentation, with these same complexity bounds.

These algorithms have been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In preliminary experiments on topological data analysis problems, our approach outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

 

A kernel for multi-parameter persistent homology and its computation

René Corbet
TU Graz

Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. In this talk, we discuss a kernel construction for multi-parameter persistence and why this kernel can provably be useful in applications.

 

Morse inequalities for multiparameter persistence

Andrea Guidolin1, Claudia Landi2
1Basque Center for Applied Mathematics, 2Università di Modena e Reggio Emilia

Discrete Morse theory is a combinatorial version of Morse theory which has proved to be an incredibly useful tool with applications in a large variety contexts. Intuitively speaking, discrete Morse theory allows to reduce a combinatorial cell complex (for example, a simplicial or cubical complex) to a subset of its cells, called critical, that carry all the homological information. The relation between the number of critical cells and the Betti numbers is described by the so-called Morse inequalities.

So far, the connection between persistent homology and discrete Morse theory has been studied mainly with the purpose of simplifying complexes and speeding up the algorithms that compute the persistence modules. In this sense, only the reduction aspect of discrete Morse theory has been leveraged in connection to persistence.

In this talk we show the possibility of establishing Morse inequalities for persistence. To this aim, we consider a filtration of a cell complex that varies according to multiple parameters, the associated multiparameter persistence module, and the critical cells of a discrete gradient vector field compatible with the multifiltration. Our goal is to derive Morse inequalities relating the number of critical cells of the given vector field to the multigraded Betti numbers of the persistence module. Thisrequires the use of specific tools from homological algebra, which we briefly illustrate.