Network Data Analysis: Activities

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This is an internal event for the Network Data Analysis Group at The Ohio State University. It will be held on Thursday, May 5th, 2022, 9:00 am to 4:10 pm EST. The location will be Cockins Hall CH240. Schedule:

Titles and abstracts:

Nate Clause, Fast computation of zigzag persistence, 30'
Mario Gomez Flores, Contractions in metric graphs, 30'
Woojin Kim, Computing generalized rank invariant via zigzag persistence, 20'
- The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval I of a Z^2-indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank over I by computing the barcode of the zigzag module obtained by restricting the bifiltration inducing M to that path. If the bifiltration and I have at most t simplices and points respectively, this computation takes O(t^ω) time where ω ∈ [2, 2.373) is the exponent of matrix multiplication. Among others, we apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M, determine whether M is interval decomposable and, if so, compute all intervals supporting its summands.
Facundo Mémoli, The Gromov-Hausdorff distance between spheres, 50'
- The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry, due the topology it generates, and also in applied geometry and topological data analysis, as a metric for expressing the stability of the persistent homology of geometric data (e.g. via the Vietoris-Rips filtration). Whereas it is often easy to estimate the value of the distance between two given metric spaces, its precise value is rarely easy to determine. Some of the best estimates follow from considerations actually related to both the stability of persistent homology and to Gromov's filling radius. However, these turn out to be non-sharp. In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between certain pairs of spheres (endowed with their geodesic distance). These results involve lower bounds, which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and from matching upper bounds which are induced from specialized constructions of "correspondences" between spheres.
Facundo Mémoli, Classical multidimensional scaling of metric measure spaces, 50'
- We study a generalization of the classical Multidimensional Scaling procedure (cMDS) to the setting of general metric measure spaces which can be seen as natural 'continuous limits' of finite data sets. We identify certain crucial spectral properties of the generalized cMDS operator thus providing a natural and rigorous formulation of cMDS in this setting. Furthermore, we characterize the cMDS output of several continuous exemplar metric measures spaces such as high dimensional spheres and tori (both with their geodesic distance). In particular, the case of spheres (with geodesic distance) requires that we establish that its cMDS operator is trace class, a condition which is natural in the context when the cMDS operator has infinite rank. Finally, we establish the stability of the generalized cMDS method with respect to the Gromov-Wasserstein distance.
Qingsong Wang, The persistent topology of optimal transport based metric thickenings, 30'
- A metric thickening of a given metric space X is any metric space admitting an isometric embedding of X. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the p-Vietoris–Rips and p-Čech metric thickenings for any p between 1 and infinity, which include all measures on X whose p-diameter or p-radius is bounded from above, equipped with an optimal transport metric. These families recover the previously studied Vietoris–Rips and Čech metric thickenings when p is infinity. As our main contribution, we prove a stability theorem for the persistent homology of p-Vietoris–Rips and p-Čech metric thickenings, which is novel even in the case p is infinity. In the specific case p equals 2, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the 2-Vietoris–Rips thickenings of the sphere as the scale increases. This is joint work with Henry Adams, Facundo Mémoli and Michael Moy.
Ling Zhou, Persistent cup-length, 20'
- Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual 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 useful for discriminating spaces. In this talk, we lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation. We furthermore show that this invariant is stable under suitable interleaving-type distances.

Network Data Analysis: Mini-workshop May 5th 2022