Higher Interpolation and Extension for Persistence Modules

  title={Higher Interpolation and Extension for Persistence Modules},
  author={Peter Bubenik and Vin de Silva and Vidit Nanda},
  journal={SIAM J. Appl. Algebra Geom.},
The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of non-expansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions… 

Figures from this paper

Topological spaces of persistence modules and their properties
This work considers various classes of persistence modules, including many of those that have been previously studied, and describes the relationships between them, and undertake a systematic study of the resulting topological spaces and their basic topological properties.
Interleaving and Gromov-Hausdorff distance
One of the central notions to emerge from the study of persistent homology is that of interleaving distance. It has found recent applications in symplectic and contact geometry, sheaf theory,
Local Cohomology and Stratification
An algorithm to recover the canonical stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells, with the property that two cells are isomorphic in the last category if and only if they lie in the same canonical stratum.
Homotopy, homology, and persistent homology using closure spaces and filtered closure spaces
. We develop persistent homology in the setting of filtered (ˇCech) closure spaces. Examples of filtered closure spaces include filtered topological spaces, metric spaces, weighted graphs, and weighted
Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We
Generalized persistence diagrams
  • A. Patel
  • Mathematics
    J. Appl. Comput. Topol.
  • 2018
The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer is generalized to the setting of constructible persistence modules valued in a symmetric monoidal category and a second type of persistence diagram is defined, which enjoys a stronger stability theorem.
Computational Complexity of the Interleaving Distance
It is shown that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces, and it is obtained that the isomorphism problem for Reeb graphs is graph isomorphicism complete.
Topological Methods in Data Analysis
I develop algebraic-topological theories, algorithms and software for the analysis of nonlinear data and complex systems arising in various scientific contexts. In particular, I employ discrete
Positivity of Multiparameter Persistence Diagrams and Bottleneck Stability
Persistent homology studies the birth and death of cycles in a parameterized family of spaces. In this paper, we study the birth and death of cycles in a multifiltration of a chain complex with the
Connectedness and Lusternik-Schnirelmann categories of the spaces of persistence modules
The classes of various interval decomposable persistence modules in literature were analyzed, the sets were determined, and some topological characteristics that these sets gained through


Proximity of persistence modules and their diagrams
This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.
Geometry in the space of persistence modules
It is shown that the relationship between the Cech and Rips complexes is governed by certain `coherence' conditions on the corresponding families of interleavings or matchings in the spaces of persistence modules and diagrams.
Metrics for Generalized Persistence Modules
This work considers the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets, and introduces a distinction between ‘soft’ and ‘hard’ stability theorems.
The Structure and Stability of Persistence Modules
This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such
Stability of persistence diagrams
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram
Topological Signatures of Singularities in Simplicial Ricci Flow
We apply the methods of persistent homology to a selection of two and three--dimensional geometries evolved by simplicial Ricci flow. To implement persistent homology, we construct a triangular mesh
The observable structure of persistence modules
In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence
Topological Signals of Singularities in Ricci Flow
The results obtained suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formations (neckpinch) under Ricci flow.
Categorification of Persistent Homology
This work redevelops persistent homology (topological persistence) from a categorical point of view and gives a natural construction of a category of ε-interleavings of $\mathbf {(\mathbb {R},\leq)}$-indexed diagrams in some target category and shows that if the target category is abelian, so is this category of interleavments.
Statistical topological data analysis using persistence landscapes
A new topological summary for data that is easy to combine with tools from statistics and machine learning and obeys a strong law of large numbers and a central limit theorem is defined.