publications | Manuel Soriano Trigueros

2024

preprint
The Poset of Cancellations in a Filtered Complex

Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, and Manuel Soriano-Trigueros

arXiv, Nov 2024

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Motivated by questions about simplification and topology optimization, we take a discrete approach toward the dependency of topology simplifying operations and the reachability of perfect Morse functions. Representing the function by a filter on a Lefschetz complex, and its (non-essential) topological features by the pairing of its cells via persistence, we simplify using combinatorially defined cancellations. The main new concept is the depth poset on these pairs, whose linear extensions are schedules of cancellations that trim the Lefschetz complex to its essential homology. One such linear extensions is the cancellation of the pairs in the order of their persistence. An algorithm that constructs the depth poset in two passes of standard matrix reduction is given and proven correct.
@article{depth_poset, title = {The Poset of Cancellations in a Filtered Complex}, journal = {arXiv}, volume = {}, pages = {}, year = {2024}, month = nov, issn = {2311.14364}, doi = {}, author = {Edelsbrunner, Herbert and Lipiński, Michał and Mrozek, Marian and Soriano-Trigueros, Manuel}, keywords = {Lefschetz complexes, persistent homology, depth poset, shallow pairs}, }
preprint
Additive Partial Matchings Induced by Persistence Morphisms

Rocio Gonzalez-Diaz, Manuel Soriano-Trigueros, and Álvaro Torras-Casas

arXiv, Oct 2024

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Given a morphism of persistence modules (a.k.a. persistence morphism) f:V→U, we introduce a novel operator that determines a partial matching between the barcodes of V and U induced by f. We show that the proposed operator is additive with respect to the direct sum of persistence morphisms, and that it contains more information than fV and the rank invariant. We also illustrate some advantages of using our induced partial matching over the Bauer-Lesnick partial matching χf. Lastly, we provide a family of persistence morphisms that contain modules built from Morse filtrations, for which χf, the rank invariant and our proposed induced partial matching are equivalent.
@article{GONZALEZDIAZ2023ADDITIVE, title = {Additive Partial Matchings Induced by Persistence Morphisms}, journal = {arXiv}, volume = {}, pages = {}, year = {2024}, month = oct, issn = {2006.11100}, doi = {}, author = {Gonzalez-Diaz, Rocio and Soriano-Trigueros, Manuel and Torras-Casas, Álvaro}, keywords = {Partial matchings, Persistence modules, Topological data analysis, Block functions}, }

2023

A Survey of Vectorization Methods in Topological Data Analysis

Dashti Ali, Aras Asaad, Maria-Jose Jimenez, Vidit Nanda, Eduardo Paluzo-Hidalgo, and Manuel Soriano-Trigueros

IEEE Transactions on Pattern Analysis and Machine Intelligence, Dec 2023

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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.
@article{10235748, author = {Ali, Dashti and Asaad, Aras and Jimenez, Maria-Jose and Nanda, Vidit and Paluzo-Hidalgo, Eduardo and Soriano-Trigueros, Manuel}, journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, title = {A Survey of Vectorization Methods in Topological Data Analysis}, year = {2023}, month = dec, volume = {}, number = {}, pages = {1-14}, doi = {10.1109/TPAMI.2023.3308391}, }
Barcodes of Persistence Modules: from Summaries to Matchings

Manuel Soriano-Trigueros

Universidad de Sevilla. Thesis supervised by Rocío González-Díaz and María-José Jiménez, May 2023

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During recent decades, the application of topology to data analysis has experienced a revolution due to persistent homology. Persistent homology and its barcode representation offer many advantages to data analysis, such as the stability of the results and the possibility of quantifying qualitative geometric and topological features. The aim of this thesis is to enrich the existing knowledge about persistent homology and barcodes from three different perspectives. Each of these perspectives has led to a publication that together constitute the core of this thesis. First, from a statistical perspective, we study the stability properties of persistent entropy, a variable used to compare barcodes. In addition, we propose a new barcode vectorization method based on persistent entropy, which is more suitable for machine learning than persistent entropy itself. Second, from an applied perspective, we use barcode summaries to study the geometric and topological features of 2D images of epithelial cells. Lastly, from the algebraic perspective, we study how we can induce a partial matching between barcodes from a morphism between persistence modules, the algebraic structure underlying persistent homology. This study has led to the definition of an operator for such morphisms, called the induced block function. Moreover, we study some of its theoretical properties, such as linearity with respect to direct sums, and provide an algorithm to compute it in polynomial time.
journal = {Universidad de Sevilla. Thesis supervised by Rocío González-Díaz and María-José Jiménez}, volume = {}, pages = {}, year = {2023}, month = may, issn = {}, doi = {}, url = {https://hdl.handle.net/11441/164052}, author = {Soriano-Trigueros, Manuel}, keywords = {}, }

Partial matchings induced by morphisms between persistence modules

Rocio Gonzalez-Diaz, Manuel Soriano-Trigueros, and Álvaro Torras-Casas

Computational Geometry, Feb 2023

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We study how to obtain partial matchings using the block function Mf, induced by a morphism f between persistence modules. Mf is defined algebraically and is linear with respect to direct sums of morphisms. We study some interesting properties of Mf, and provide a way of obtaining Mf using matrix operations.

@article{GONZALEZDIAZ2023101985,
  title = {Partial matchings induced by morphisms between persistence modules},
  journal = {Computational Geometry},
  volume = {112},
  pages = {101985},
  year = {2023},
  month = feb,
  issn = {0925-7721},
  doi = {https://doi.org/10.1016/j.comgeo.2023.101985},
  url = {https://www.sciencedirect.com/science/article/pii/S0925772123000056},
  author = {Gonzalez-Diaz, Rocio and Soriano-Trigueros, Manuel and Torras-Casas, Álvaro},
  keywords = {Partial matchings, Persistence modules, Topological data analysis, Block functions},
}

2022

Can trans-S-manifolds be defined from the Gray-Hervella classification for almost Hermitian manifolds?

Pablo Alegre, Luis M. Fernández, and Manuel Soriano-Trigueros

Annali di Matematica Pura ed Applicata (1923 -), Apr 2022

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Recently, trans-S-manifolds have been defined as a wide class of metric f-manifolds which includes, for instance, f-Kenmotsu manifolds, S-manifolds and C-manifolds and generalize well-studied trans-Sasakian manifolds. The definition of trans-S-manifolds is formulated using the covariant derivative of the tensor f and although this formulation coincides with the characterization of trans-Sasakian manifolds in such a particular case, this latter type of manifolds were not initially defined in this way but using the Gray-Hervella classification of almost Hermitian manifolds. The aim of this paper is to study how (almost) trans-S-manifolds relate with the Gray-Hervella classification and to establish both similarities and differences with the trans-Sasakian case.
@article{Alegre2022, doi = {10.1007/s10231-022-01215-9}, url = {https://doi.org/10.1007/s10231-022-01215-9}, year = {2022}, month = apr, publisher = {Springer Science and Business Media {LLC}}, volume = {201}, number = {6}, pages = {2691--2706}, author = {Alegre, Pablo and Fern{\'{a}}ndez, Luis M. and Soriano-Trigueros, Manuel}, title = {Can trans-S-manifolds be defined from the Gray-Hervella classification for almost Hermitian manifolds?}, journal = {Annali di Matematica Pura ed Applicata (1923 -)}, }

2021

Stable Topological Summaries for Analyzing the Organization of Cells in a Packed Tissue

Nieves Atienza, Maria-Jose Jimenez, and Manuel Soriano-Trigueros

Mathematics, Jul 2021

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We use topological data analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify the lifetime of homology classes (persistent homology) along different filtrations (increasing nested sequences of simplicial complexes) that are built from the regions representing the cells in the tissue. We use a complete and well-grounded set of numerical variables over those persistence barcodes, also known as topological summaries. A novel combination of normalization methods for both the set of input segmented images and the produced barcodes allows for the proven stability results for those variables with respect to small changes in the input, as well as invariance to image scale. Our study provides new insights to this problem, such as a possible novel indicator for the development of the drosophila wing disc tissue or the importance of centroids’ distribution to differentiate some tissues from their CVT-path counterpart (a mathematical model of epithelia based on Voronoi diagrams). We also show how the use of topological summaries may improve the classification accuracy of epithelial images using a Random Forest algorithm.
@article{ATIENZA2021, title = {Stable Topological Summaries for Analyzing the Organization of Cells in a Packed Tissue}, volume = {9}, issn = {2227-7390}, doi = {http://dx.doi.org/10.3390/math9151723}, number = {15}, journal = {Mathematics}, publisher = {MDPI AG}, author = {Atienza, Nieves and Jimenez, Maria-Jose and Soriano-Trigueros, Manuel}, year = {2021}, month = jul, pages = {1723}, }

2020

On the stability of persistent entropy and new summary functions for topological data analysis

Nieves Atienza, Rocio Gonzalez-Diaz, and Manuel Soriano-Trigueros

Pattern Recognition, Nov 2020

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Persistent homology and persistent entropy have recently become useful tools for patter recognition. In this paper, we find requirements under which persistent entropy is stable to small perturbations in the input data and scale invariant. In addition, we describe two new stable summary functions combining persistent entropy and the Betti curve. Finally, we use the previously defined summary functions in a material classification task to show their usefulness in machine learning and pattern recognition.
@article{ATIENZA2020107509, title = {On the stability of persistent entropy and new summary functions for topological data analysis}, journal = {Pattern Recognition}, volume = {107}, pages = {107509}, year = {2020}, month = nov, issn = {0031-3203}, doi = {https://doi.org/10.1016/j.patcog.2020.107509}, url = {https://www.sciencedirect.com/science/article/pii/S0031320320303125}, author = {Atienza, Nieves and Gonzalez-Diaz, Rocio and Soriano-Trigueros, Manuel}, keywords = {Persistent homology, Persistent entropy, Stability, Dimensionality reduction}, }