Persistence Modules and Quiver Representations

The algebraic structure of persistent homology was originally understood as modules over a polynomial ring in one variable. More recently, it was shown that this theoretical foundation can be phrased in terms of the representation theory of (bound) quivers, or more generally, as functor categories. This enabled the use of representation and category-theoretic tools in the study of persistence. Consequently, TDA can be applied to a much wider range of applications, e.g., spatiotemporal analysis, multi-parameter persistence, etc.

On commutative ladders

In [5], we introduced and studied persistence modules over commutative ladders of finite type. In [2], we developed a matrix method for computing indecomposable decompositions, from which we can obtain generalized persistence diagrams in the finite type case.

In [4], we provided a construction of some indecomposable persistence modules with arbitrarily large dimension together with relatively simple geometric realizations. From this, we argued that large indecomposables may encode important geometric information and should not be ignored.

On 2D persistence modules

In [3], we introduced pre-interval, interval, and indecomposable thin representations and studied the relationships among them. Moreover, we provided an algorithm for determining whether or not a generalized persistence module is interval/pre-interval/thin-decomposable.

In [1], we showed that each 1D persistence module can be found as a line restriction of some indecomposable 2D persistence module. This is yet another expression of how complicated 2D indecomposables can be. Our construction can generate indecomposable 2D persistence modules whose support has holes, which can be seen below.

On metrics between zigzag persistence modules

We introduced a bottleneck metric between arbitrary zigzag persistence modules that relies on the structure and shape of the Auslander-Reiten quiver of the underlying zigzag structure. We proposed a refinement of the often used "pure zigzag" structure, over which this new metric behaves very well, and emphasizes different features than those of the other metrics common in the literature.

literature

[1] M. Buchet and E.G. Escolar. Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module. arXiv:1902.07405.

[2] H. Asashiba, E.G. Escolar, Y. Hiraoka, H. Takeuchi. Matrix Method for Persistence Modules on Commutative Ladders of Finite Type. Japan J. Indust. Appl. Math. 36 (1), 97--130 (2019).

[3] H. Asashiba, M. Buchet, E.G. Escolar, K. Nakashima and M. Yoshiwaki, On Interval Decomposability of 2D Persistence Modules. arXiv:1812.05261.

[4] M. Buchet and E.G. Escolar. Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension. 34th International Symposium on Computational Geometry (SoCG 2018), Leibniz International Proceedings in Informatics (LIPIcs) 15, 1--13 (2018).

[5] E.G. Escolar and Y. Hiraoka. Persistence modules on commutative ladders of finite type. Discrete Comput. Geom. 55, 100--157 (2016).