@article{22291, abstract = {Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and fast-to-compute invariant known as the persistence diagram. However, this is no longer the case for maps between persistence modules (i.e. persistence maps). We propose a new invariant for persistence maps, consisting of a partial matching between the persistence diagrams of the domain and codomain modules. We show that this invariant is additive with respect to the direct sum decomposition of persistence maps, is more discriminative than the image module, and is computable in cubic time. Furthermore, we provide an implementation and demonstrate its efficiency by integrating it with edge collapse techniques for flag complexes (e.g., Vietoris–Rips complexes). As a key technical contribution, we describe how to induce a persistence map between two flag complexes that have been independently simplified via edge collapses, even when a direct simplicial map between them is no longer available.}, author = {Gonzalez-Diaz, Rocio and Soriano Trigueros, Manuel and Torras-Casas, Alvaro}, issn = {1095-855X}, journal = {Journal of Symbolic Computation}, keywords = {Persistence module, Persistence map, Persistent homology}, publisher = {Elsevier}, title = {{Additive partial matchings induced by persistence maps}}, doi = {10.1016/j.jsc.2026.102598}, volume = {138}, year = {2026}, } @article{22648, abstract = {Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems—a young but already well-established theory. We introduce the Conley–Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of the Conley–Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose.}, author = {Dey, Tamal K. and Lipiński, Michał and Soriano Trigueros, Manuel}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics}, keywords = {Multivector field, Conley index, Morse decomposition, Bifurcation, Continuation, Zigzag persistence, Persistence barcode, Gentle algebra}, publisher = {Springer}, title = {{Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations}}, doi = {10.1007/s10208-026-09766-6}, year = {2026}, } @inproceedings{22299, abstract = {The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.}, author = {Edelsbrunner, Herbert and Lipiński, Michał and Mrozek, Marian and Soriano Trigueros, Manuel and Zimin, Fedor}, booktitle = {42nd International Symposium on Computational Geometry}, isbn = {9783959774185}, issn = {1868-8969}, keywords = {Algebraic topology, Lefschetz complexes, persistent homology, vines and vineyards, birth-death pairs, shallow pairs, relations, partial orders, transpositions, Theory of computation → Computational geometry}, location = {New Brunswick, NJ, United States}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The depth poset under transpositions in the filter}}, doi = {10.4230/LIPICS.SOCG.2026.41}, volume = {367}, year = {2026}, } @inproceedings{20729, abstract = {Persistence modules (defined as a sequence of vector spaces and linear maps between them) are a key tool in topological data analysis. They are easy to interpret and fast to compute. However, when considering persistence maps (i.e. maps between persistence modules), these properties are lost. We propose a new invariant for persistence maps consisting of a partial matching such that: it is easy to interpret, it is more discriminative than the image of the persistence map, and can be calculated with cubical complexity.}, author = {Gonzalez-Diaz, Rocio and Soriano Trigueros, Manuel and Torras-Casas, Alvaro}, booktitle = {Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation}, isbn = {9798400720758}, location = {Guanajuato, Mexico}, pages = {188--196}, publisher = {Association for Computing Machinery}, title = {{Additive partial matchings for persistent homology}}, doi = {10.1145/3747199.3747561}, year = {2025}, } @article{14739, abstract = {Attempts to incorporate topological information in supervised learning tasks have resulted in the creation of several techniques for vectorizing persistent homology barcodes. In this paper, we study thirteen such methods. Besides describing an organizational framework for these methods, we comprehensively benchmark them against three well-known classification tasks. Surprisingly, we discover that the best-performing method is a simple vectorization, which consists only of a few elementary summary statistics. Finally, we provide a convenient web application which has been designed to facilitate exploration and experimentation with various vectorization methods.}, author = {Ali, Dashti and Asaad, Aras and Jimenez, Maria-Jose and Nanda, Vidit and Paluzo-Hidalgo, Eduardo and Soriano Trigueros, Manuel}, issn = {1939-3539}, journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, keywords = {Applied Mathematics, Artificial Intelligence, Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Software}, number = {12}, pages = {14069--14080}, publisher = {IEEE}, title = {{A survey of vectorization methods in topological data analysis}}, doi = {10.1109/tpami.2023.3308391}, volume = {45}, year = {2023}, }