Stratifying Multiparameter Persistent Homology

Abstract

A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.2000 Mathematics Subject Classification. 55B55, 68U05, 68Q17, 13P25 (primary) .

A Topological Perspective on Weather Regimes

“…In particular, there is no generalisation of the barcode, as described in Section 3.1 and illustrated in Figure 3, for multifiltrations. Finding appropriate ways to quantify the 'persistence' of topological invariants, such as the number of components or holes, is currently one of the most active areas of research in TDA, and several researchers have proposed invariants that are computable, and capture in an appropriate sense what it means for topological features to be 'persistent', see, for instance, [56,57,58].…”

Computing Multipersistence by Means of Spectral Systems

“…Even if in the present work we will focus mainly on the rank invariant, we want to recall that other invariants have been proposed for multipersistence [2,5,4,16,31,30,13,6,21].…”

“…This idea has been further developed in [16], together with the theoretical bases of the software RIVET for visualizing 2-parameter persistence. The paper [13] presents an interesting approach via commutative algebra. Efficient methods to deal with a particular class of 2-parameter persistence modules are introduced in [6].…”

“…For multipersistence, the concepts of birth and death of a homology class cannot be immediately generalized from the 1-parameter case (but see [13,21] for interesting approaches). For example, in Figure 1 we cannot say that the 1homology class (1-hole) corresponding to the generator 1 * bc−1 * bd+1 * cd is born at a unique particular position of Z 2 (because it is present at both positions (1,2) and (2, 1) and for both of them it is not present at a previous step).…”

Euler characteristic surfaces

“…Beginning with the work by Carlsson and Zomorodian [11], there has been a large number of approaches proposed to deal with multi-parameter persistence. Approaches include directions such as the rank invariant [11,10,59], microlocal analysis [37], higher dimensional analogues of persistence diagrams [46,29], just to name a few. These all capture related but somewhat different papers, and to the best of our knowledge, the rank invariant is the only case where implementations exist which although perform well -do not scale in the same way as one-dimensional persistence.…”