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Register a new userRectangular Approximation and Stability of $2$-parameter Persistence Modules
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Added by
Cheng Xin
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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One of the main reasons for topological persistence being useful in data analysis is that it is backed up by a stability (isometry) property: persistence diagrams of $1$-parameter persistence modules are stable in the sense that the bottleneck distance between two diagrams equals the interleaving distance between their generating modules. However, in multi-parameter setting this property breaks down in general. A simple special case of persistence modules called rectangle decomposable modules is known to admit a weaker stability property. Using this fact, we derive a stability-like property for $2$-parameter persistence modules. For this, first we consider interval decomposable modules and their optimal approximations with rectangle decomposable modules with respect to the bottleneck distance. We provide a polynomial time algorithm to exactly compute this optimal approximation which, together with the polynomial-time computable bottleneck distance among interval decomposable modules, provides a lower bound on the interleaving distance. Next, we leverage this result to derive a polynomial-time computable distance for general multi-parameter persistence modules which enjoys similar stability-like property. This distance can be viewed as a generalization of the matching distance defined in the literature.
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We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still to work. The constructions and proofs enabled by our framework are, we claim, cleaner and simpler.
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under $mathbb{Z}_2$ coefficients for a sequence of general simplicial maps and show how these maps arise naturally in some applications of topological data analysis. First, we observe that it is not hard to simulate simplicial maps by inclusion maps but not necessarily in a monotone direction. This, combined with the known algorithms for zigzag persistence, provides an algorithm for computing the persistence induced by simplicial maps. Our main result is that the above simple minded approach can be improved for a sequence of simplicial maps given in a monotone direction. A simplicial map can be decomposed into a set of elementary inclusions and vertex collapses--two atomic operations that can be supported efficiently with the notion of simplex annotations for computing persistent homology. A consistent annotation through these atomic operations implies the maintenance of a consistent cohomology basis, hence a homology basis by duality. While the idea of maintaining a cohomology basis through an inclusion is not new, maintaining them through a vertex collapse is new, which constitutes an important atomic operation for simulating simplicial maps. Annotations support the vertex collapse in addition to the usual inclusion quite naturally. Finally, we exhibit an application of this new tool in which we approximate the persistence diagram of a filtration of Rips complexes where vertex collapses are used to tame the blow-up in size.
The classical persistence algorithm virtually computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological data analysis. Its input is a simplicial filtration defined over the integers $mathbb{Z}$ giving rise to a $1$-parameter persistence module. It has been recognized that multi-parameter version of persistence modules given by simplicial filtrations over $d$-dimensional integer grids $mathbb{Z}^d$ is equally or perhaps more important in data science applications. However, in the multi-parameter setting, one of the main challenges is that topological summaries based on algebraic structure such as decompositions and bottleneck distances cannot be as efficiently computed as in the $1$-parameter case because there is no known extension of the persistence algorithm to multi-parameter persistence modules. We present an efficient algorithm to compute the unique decomposition of a finitely presented persistence module $M$ defined over the multiparameter $mathbb{Z}^d$.The algorithm first assumes that the module is presented with a set of $N$ generators and relations that are emph{distinctly graded}. Based on a generalized matrix reduction technique it runs in $O(N^{2omega+1})$ time where $omega<2.373$ is the exponent for matrix multiplication. This is much better than the well known algorithm called Meataxe which runs in $tilde{O}(N^{6(d+1)})$ time on such an input. In practice, persistence modules are usually induced by simplicial filtrations. With such an input consisting of $n$ simplices, our algorithm runs in $O(n^{2omega+1})$ time for $d=2$ and in $O(n^{d(2omega + 1)})$ time for $d>2$.
Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one instrument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general $O(m^omega)$ time complexity are not known, where $omega< 2.37286$ is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with $m$ additions and deletions on a graph with $n$ vertices and edges, the algorithm for $0$-dimension runs in $O(mlog^2 n+mlog m)$ time and the algorithm for 1-dimension runs in $O(mlog^4 n)$ time. The algorithm for $0$-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on $2$-manifolds. The algorithm for $1$-dimension pairs a negative edge with the earliest positive edge so that a $1$-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for $0$-dimension to compute the $(p-1)$-dimensional zigzag persistence for $mathbb{R}^p$-embedded complexes in $O(mlog^2 n+mlog m+nlog n)$ time.
The Wasserstein distances are a family of $L^p$ distances, with $1 leq p leq infty$, for persistence diagrams. We define Wasserstein distances for persistence modules, the algebraic counterpart to persistence diagrams, and prove the following isometry theorem. The $p$-Wasserstein distance of a persistence module and its persistence diagram agree. Since our algebraic Wasserstein distances do not require computing a persistence diagram, they apply to persistence modules that are not interval decomposable and also to generalized persistence modules, such as multi-parameter persistence modules. We also prove structure theorems for maps from an interval module and maps to an interval module and show that for monomorphisms and epimorphisms of persistence modules there is an induced algebraic matching.
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