TRIPTURES-TRITTURE

A NEW APPROACH TO THE MATCHING DISTANCE IN 2-PARAMETER PERSISTENT HOMOLOGY

A 2D persistence diagram D(f) associated with a function f:X → ℝ² is a function taking each positive slope line r in ℝ² to the persistence diagram associated with the natural filtration defined by r. Two 2D persistence diagrams D(f), D(g) can be compared by means of the 2D matching distance, which is defined as the supremum of the normalized bottleneck distances between the 1D persistence diagrams D_r(f), D_r(g) corresponding to the line r, varying r. In the papers [3], [4], [5] a new approach to the metric comparison of 2D persistence diagrams has been introduced and studied.

In particular, the paper [3] presents some new ideas that make it easier to compare 2D persistence diagrams:

1. The concept of extended Pareto grid Γ(f) has been introduced (Subsection 4.1 at pages 394-395). Γ(f) is a suitable collection of vertical lines, horizontal lines and arcs in ℝ² such that, when r varies, we can follow the motion of the points in the persistence diagram D_r(f) associated with the line r by computing the coordinates of the intersections of Γ(f) with r (Position Theorem 2 at page 398). The vertical lines correspond to critical values of the first component of f, while the horizontal lines correspond to critical values of the second component of f. The points in the arcs correspond to the images by f of the Pareto critical points of f.
   


2. We have observed that, when we make the slope of r approach 1, the bottleneck distance between the persistence diagrams D_r(f), D_r(g) does not decrease, provided that f and g are close enough and some regularity conditions hold. This is called the Maximum Principle for the coherent transport (Theorem 5 at page 417). As a consequence, if the matching realizing the bottleneck distance between persistence diagrams does not undergo combinatorial changes during the motion of r, then the maximum of the bottleneck distance between D_r(f) and D_r(g) is taken either at a line r with slope 1 or at a line r that is on the boundary of the region in which r varies (Theorem 6 at page 418). N.B.: In our paper slope 1 lines correspond to lines associated with the parameter a=1/2.
   


3. In order to manage the case where the combinatorial correspondence between persistence diagrams changes during the motion of r, we have introduced the concept of coherent transport of matchings , based on adding the constraint that the combinatorial correspondence between persistence diagrams is preserved during the motion of r (Subsection 5.1 at pages 404-408). The concept of coherent transport of matchings leads to the discover of the phenomenon of monodromy in persistent homology (page 384) and to the definition of coherent matching distance (Section 5.3 at pages 411-414).
   

Several technicalities are needed in order to manage the previous concepts. We refer the interested reader to [4] for details. As for the computational approximation of the matching distance we refer the reader to the papers [5] (for filtering functions taking values in ℝ² and persistence diagrams in degree 0) and [6] (for filtering functions taking values in ℝⁿ and persistence diagrams in degree k).

---

REFERENCES

[1] Marc Ethier, Patrizio Frosini, Nicola Quercioli, Francesca Tombari,
Geometry of the matching distance for 2D filtering functions,
Journal of Applied and Computational Topology, vol. 7 (2023), 815–830. DOI: 10.1007/s41468-023-00128-7. OPEN ACCESS URL: https://link.springer.com/article/10.1007/s41468-023-00128-7 .

[2] Andrea Cerri, Patrizio Frosini,
A new approximation algorithm for the matching distance in multidimensional persistence,
Journal of Computational Mathematics, vol. 38 (2020), n. 2, 291-309. DOI: 10.4208/jcm.1809-m2018-0043.
Preprint available at http://amsacta.unibo.it/2971/. WARNING! The first pdf file of this paper that was made available for download from the website of the Journal of Computational Mathematics was not the correct one and contained many misprints. In case you are interested in this paper, please download the last version from this webpage: https://www.global-sci.org/intro/article_detail/jcm/14518.html.

[3] Andrea Cerri, Marc Ethier, Patrizio Frosini, A study of monodromy in the computation of multidimensional persistence, Proceedings of the 17th IAPR International Conference on Discrete Geometry for Computer Imagery, Seville, Spain, March 20-22, 2013, Lecture Notes in Computer Science 7749, 2013, 192-202. Preprint

[4] Andrea Cerri, Marc Ethier, Patrizio Frosini, The coherent matching distance in 2D persistent homology, Lecture Notes in Computer Science, Proceedings of the 6th International Workshop on Computational Topology in Image Context, Marseille, France, June 15-17, 2016, Springer International Publishing Switzerland, A. Bac and J.-L. Mari (Eds.), LNCS 9667, 216-227, 2016. DOI: 10.1007/978-3-319-39441-1_20. Preprint available at http://arxiv.org/pdf/1603.03886v1.pdf.

[5] Andrea Cerri, Marc Ethier, Patrizio Frosini,
On the geometrical properties of the coherent matching distance in 2D persistent homology, Journal of Applied and Computational Topology, vol. 3 (2019), n. 4, 381–422. DOI: 10.1007/s41468-019-00041-y
Preprint available at arXiv:1801.06636.

[6] Silvia Biasotti, Andrea Cerri, Patrizio Frosini, Daniela Giorgi, A new algorithm for computing the 2-dimensional matching distance between size functions, Pattern Recognition Letters, vol. 32 (2011), n. 14, 1735-1746. DOI: 10.1016/j.patrec.2011.07.014, preprint available here.

[7] Andrea Cerri, Patrizio Frosini, A new approximation algorithm for the matching distance in multidimensional persistence, AMS Acta (Institutional Research Repository of the University of Bologna), 2971 (2011). DOI: 10.6092/unibo/amsacta/2971. Preprint available at http://amsacta.unibo.it/2971/. To appear in the Journal of Computational Mathematics.