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Register a new userPersistent Laplacians: properties, algorithms and implications
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Added by
Zhengchao Wan
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We present a thorough study of the theoretical properties and devise efficient algorithms for the emph{persistent Laplacian}, an extension of the standard combinatorial Laplacian to the setting of pairs (or, in more generality, sequences) of simplicial complexes $K hookrightarrow L$, which was recently introduced by Wang, Nguyen, and Wei. In particular, in analogy with the non-persistent case, we first prove that the nullity of the $q$-th persistent Laplacian $Delta_q^{K,L}$ equals the $q$-th persistent Betti number of the inclusion $(K hookrightarrow L)$. We then present an initial algorithm for finding a matrix representation of $Delta_q^{K,L}$, which itself helps interpret the persistent Laplacian. We exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix which has several important implications. In the graph case, it both uncovers a link with the notion of effective resistance and leads to a persistent version of the Cheeger inequality. This relationship also yields an additional, very simple algorithm for finding (a matrix representation of) the $q$-th persistent Laplacian which in turn leads to a novel and fundamentally different algorithm for computing the $q$-th persistent Betti number for a pair $(K,L)$ which can be significantly more efficient than standard algorithms. Finally, we study persistent Laplacians for simplicial filtrations and present novel stability results for their eigenvalues. Our work brings methods from spectral graph theory, circuit theory, and persistent homology together with a topological view of the combinatorial Laplacian on simplicial complexes.
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In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including its stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a normal filtering function as well as of transport of a matching. As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.
Cohomological ideas have recently been injected into persistent homology and have been utilized for both enriching and accelerating the calculation of persistence diagrams. For instance, the software Ripser fundamentally exploits the computational advantages offered by cohomological ideas. The cup product operation which is available at cohomology level gives rise to a graded ring structure which extends the natural vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the Cup-Length, which is efficient at discriminating spaces. In this paper, we lift the cup-length into the Persistent Cup-Length invariant for the purpose of extracting non-trivial information about the evolution of the cohomology ring structure across a filtration. We show that the Persistent Cup-Length can be computed from a family of representative cocycles and devise a polynomial time algorithm for the computation of the Persistent Cup-Length invariant. We furthermore show that this invariant is stable under suitable interleaving-type distances. Along the way, we identify an invariant which we call the Cup-Length Diagram, which is stronger than persistent cup-length but can still be computed efficiently. In addition, by considering the $ell$-fold product of persistent cohomology rings, we identify certain persistence modules, which are also stable and can be used to evaluate the persistent cup-length.
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