arXiv subjects

Marco Guerra

Publications and source records attributed to Marco Guerra.

7 recordsLinked to original sources

arXiv2026

Hook-decomposable modules and their resolutions

We compare several classes of biparameter persistence modules: $\gamma$-products of monoparameter modules, hook-decomposable modules, modules admitting a Smith-type structure theorem, and modules of projective dimension at most 1. We determine all logical implications among these classes, providing explicit counterexamples showing that the converses fail when appropriate. In particular, $\gamma$-products (i.e., hook-decomposable modules) form a very small subclass of biparameter modules, precisely the ones for which a structure theorem still holds, thus making explicit the richer structural complexity of the biparameter setting compared to the monoparameter one.

math.AT↗

arXiv2025

SHREC 2025: Protein surface shape retrieval including electrostatic potential

This SHREC 2025 track dedicated to protein surface shape retrieval involved 9 participating teams. We evaluated the performance in retrieval of 15 proposed methods on a large dataset of 11,555 protein surfaces with calculated electrostatic potential (a key molecular surface descriptor). The performance in retrieval of the proposed methods was evaluated through different metrics (Accuracy, Balanced accuracy, F1 score, Precision and Recall). The best retrieval performance was achieved by the proposed methods that used the electrostatic potential complementary to molecular surface shape. This observation was also valid for classes with limited data which highlights the importance of taking into account additional molecular surface descriptors.

cs.CV↗

arXiv2025

Retrieving biparameter persistence modules from monoparameter ones: a characterization of hook-decomposable persistence modules

Motivated by the need to relate the biparameter persistence module induced by a pair of scalar functions with the monoparameter persistence modules induced by each function separately, we introduce a construction that defines a kind of product between two monoparameter persistence modules. While originally conceived to serve this comparative purpose, our construction unexpectedly reveals a deeper structural property: it also characterizes a class of biparameter modules known as hook-decomposable modules.

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arXiv2024

A topology-based algorithm for the isomorphism check of 2-level Orthogonal Arrays

We introduce a construction and an algorithm, both based on Topological Data Analysis (TDA), to tackle the problem of the isomorphism check of Orthogonal Arrays (OAs). Specifically, we associate to any binary OA a persistence diagram, one of the main tools in TDA, and explore how the Wasserstein distance between persistence diagrams can be used to inform whether two designs are isomorphic.

stat.CO↗

arXiv2024

Scale-Free Image Keypoints Using Differentiable Persistent Homology

In computer vision, keypoint detection is a fundamental task, with applications spanning from robotics to image retrieval; however, existing learning-based methods suffer from scale dependency and lack flexibility. This paper introduces a novel approach that leverages Morse theory and persistent homology, powerful tools rooted in algebraic topology. We propose a novel loss function based on the recent introduction of a notion of subgradient in persistent homology, paving the way toward topological learning. Our detector, MorseDet, is the first topology-based learning model for feature detection, which achieves competitive performance in keypoint repeatability and introduces a principled and theoretically robust approach to the problem.

cs.CV↗

arXiv2021

Parallel computation of interval bases for persistence module decomposition

A persistence module $M$, with coefficients in a field $\mathbb{F}$, is a finite-dimensional linear representation of an equioriented quiver of type $A_n$ or, equivalently, a graded module over the ring of polynomials $\mathbb{F}[x]$. It is well-known that $M$ can be written as the direct sum of indecomposable representations or as the direct sum of cyclic submodules generated by homogeneous elements. An interval basis for $M$ is a set of homogeneous elements of $M$ such that the sum of the cyclic submodules of $M$ generated by them is direct and equal to $M$. We introduce a novel algorithm to compute an interval basis for $M$. Based on a flag of kernels of the structure maps, our algorithm is suitable for parallel or distributed computation and does not rely on a presentation of $M$. This algorithm outperforms the approach via the presentation matrix and Smith Normal Form. We specialize our parallel approach to persistent homology modules, and we close by applying the proposed algorithm to tracking harmonics via Hodge decomposition.

math.AT↗

arXiv2021

Homological Scaffold via Minimal Homology Bases

The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global features onto individual network components, unless one provides a principled way to make such a choice. In this paper, we apply recent advances in the computation of minimal homology bases to introduce a quasi-canonical version of the scaffold, called minimal, and employ it to analyze data both real and in silico. At the same time, we verify that, statistically, the standard scaffold is a good proxy of the minimal one for sufficiently complex networks.

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