arXiv subjects

Andrea Guidolin

Publications and source records attributed to Andrea Guidolin.

7 recordsLinked to original sources

arXiv2025

Certifying Robustness via Topological Representations

We propose a neural network architecture that can learn discriminative geometric representations of data from persistence diagrams, common descriptors of Topological Data Analysis. The learned representations enjoy Lipschitz stability with a controllable Lipschitz constant. In adversarial learning, this stability can be used to certify $\epsilon$-robustness for samples in a dataset, which we demonstrate on the ORBIT5K dataset representing the orbits of a discrete dynamical system.

stat.ML↗

arXiv2024

Koszul complexes and relative homological algebra of functors over posets

Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

math.AT↗

arXiv2024

Morse theory of Euclidean distance functions from algebraic hypersurfaces

Let $Y\subseteq \mathbb{R}^n$ be a closed definable subset and $X\subseteq \mathbb{R}^n$ be a smooth manifold. We construct a version of Morse theory for the restriction to $X$ of the Euclidean distance function from $Y$. This is done using the notion of critical points of Lipschitz functions and applying the theory of continuous selections. In this theory, nondegenerate critical points have two indices: a quadratic index (as in classical Morse theory), and a piecewise linear index (that relates to the notion of bottlenecks). This framework is flexible enough to simultaneously treat and unify the study of two cases of interest for computational algebraic geometry: bottlenecks and nearest point problems. We provide a technical toolset guaranteeing the applicability of the theory to the case where $X, Y$ are generic algebraic hypersurfaces and use it to bound the number of critical points of the distance from $Y$ restricted to $X$, among other applications.

math.AG↗

arXiv2023

On the support of Betti tables of multiparameter persistent homology modules

Persistent homology encodes the evolution of homological features of a multifiltered cell complex in the form of a multigraded module over a polynomial ring, called a multiparameter persistence module, and quantifies it through invariants suitable for topological data analysis. In this paper, we establish relations between the Betti tables, a standard invariant for multigraded modules commonly used in multiparameter persistence, and the multifiltered cell complex. In particular, we show that the grades at which cells of specific dimensions first appear in the filtration reveal all positions in which the Betti tables are possibly nonzero. This result can be used in combination with discrete Morse theory on the multifiltered cell complex originating the module to obtain a better approximation of the support of the Betti tables. In the case of bifiltrations, we refine our results by considering homological critical grades of a filtered chain complex instead of entrance grades of cells.

math.AT↗

arXiv2023

Morse inequalities for the Koszul complex of multi-persistence

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of relative homology of nested filtration pieces, and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.

math.AT↗

arXiv2023

Algebraic Wasserstein distances and stable homological invariants of data

Distances have a ubiquitous role in persistent homology, from the direct comparison of homological representations of data to the definition and optimization of invariants. In this article we introduce a family of parametrized pseudometrics between persistence modules based on the algebraic Wasserstein distance defined by Skraba and Turner, and phrase them in the formalism of noise systems. This is achieved by comparing $p$-norms of cokernels (resp. kernels) of monomorphisms (resp. epimorphisms) between persistence modules and corresponding bar-to-bar morphisms, a novel notion that allows us to bridge between algebraic and combinatorial aspects of persistence modules. We use algebraic Wasserstein distances to define invariants, called Wasserstein stable ranks, which are 1-Lipschitz stable with respect to such pseudometrics. We prove a low-rank approximation result for persistence modules which allows us to efficiently compute Wasserstein stable ranks, and we propose an efficient algorithm to compute the interleaving distance between them. Importantly, Wasserstein stable ranks depend on interpretable parameters which can be learnt in a machine learning context. Experimental results illustrate the use of Wasserstein stable ranks on real and artificial data and highlight how such pseudometrics could be useful in data analysis tasks.

math.AT↗

arXiv2021

Computing higher Leray-Serre spectral sequences of towers of fibrations

The higher Leray-Serre spectral sequence associated with a tower of fibrations represents a generalization of the classical Leray-Serre spectral sequence of a fibration. In this work, we present algorithms to compute higher Leray-Serre spectral sequences leveraging the effective homology technique, which allows to perform computations involving chain complexes of infinite type associated with interesting objects in algebraic topology. In order to develop the programs, implemented as a new module for the Computer Algebra system Kenzo, we translated the original construction of the higher Leray-Serre spectral sequence in a simplicial framework and studied some of its fundamental properties.

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