arXiv subjects

Martina Scolamiero

Publications and source records attributed to Martina Scolamiero.

13 recordsLinked to original sources

arXiv2026

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↗

arXiv2025

Relative Representations: Topological and Geometric Perspectives

Relative representations are an established approach to zero-shot model stitching, consisting of a non-trainable transformation of the latent space of a deep neural network. Based on insights of topological and geometric nature, we propose two improvements to relative representations. First, we introduce a normalization procedure in the relative transformation, resulting in invariance to non-isotropic rescalings and permutations. The latter coincides with the symmetries in parameter space induced by common activation functions. Second, we propose to deploy topological densification when fine-tuning relative representations, a topological regularization loss encouraging clustering within classes. We provide an empirical investigation on a natural language task, where both the proposed variations yield improved performance on zero-shot model stitching.

cs.LG↗

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 $ε$-robustness for samples in a dataset, which we demonstrate on the ORBIT5K dataset representing the orbits of a discrete dynamical system.

stat.ML↗

arXiv2024

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↗

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↗

arXiv2020

Homotopical decompositions of simplicial and Vietoris Rips complexes

Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recover the first and zero-th homology of the gluing from the respective homologies of the components.

math.AT↗

arXiv2016

Multidimensional Persistence and Noise

In this paper we study multidimensional persistence modules [5,13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

math.AT↗

arXiv2016

Quantifying topological invariants of neuronal morphologies

Nervous systems are characterized by neurons displaying a diversity of morphological shapes. Traditionally, different shapes have been qualitatively described based on visual inspection and quantitatively described based on morphometric parameters. Neither process provides a solid foundation for categorizing the various morphologies, a problem that is important in many fields. We propose a stable topological measure as a standardized descriptor for any tree-like morphology, which encodes its skeletal branching anatomy. More specifically it is a barcode of the branching tree as determined by a spherical filtration centered at the root or neuronal soma. This Topological Morphology Descriptor (TMD) allows for the discrimination of groups of random and neuronal trees at linear computational cost.

q-bio.NC↗

arXiv2016

Topological analysis of the connectome of digital reconstructions of neural microcircuits

A recent publication provides the network graph for a neocortical microcircuit comprising 8 million connections between 31,000 neurons (H. Markram, et al., Reconstruction and simulation of neocortical microcircuitry, Cell, 163 (2015) no. 2, 456-492). Since traditional graph-theoretical methods may not be sufficient to understand the immense complexity of such a biological network, we explored whether methods from algebraic topology could provide a new perspective on its structural and functional organization. Structural topological analysis revealed that directed graphs representing connectivity among neurons in the microcircuit deviated significantly from different varieties of randomized graph. In particular, the directed graphs contained in the order of $10^7$ simplices Ð groups of neurons with all-to-all directed connectivity. Some of these simplices contained up to 8 neurons, making them the most extreme neuronal clustering motif ever reported. Functional topological analysis of simulated neuronal activity in the microcircuit revealed novel spatio-temporal metrics that provide an effective classification of functional responses to qualitatively different stimuli. This study represents the first algebraic topological analysis of structural connectomics and connectomics-based spatio-temporal activity in a biologically realistic neural microcircuit. The methods used in the study show promise for more general applications in network science.

q-bio.NC↗

arXiv2015

Combinatorial resolutions of multigraded modules and multipersistent homology

Let $ R=k[x_1...x_r]$ and $M$ a multigraded $R-$module. In this work we interpret $M$ as a multipersistent homology module and give a multigraded resolution of it. The construction involves cellular resolutions of monomial ideals and reflects the combinatorial structure of multipersistence homology modules. In the one critical case, a multifiltration is represented by a labelled cellular complex. A multipersistence homology module measures the defect of acyclicity of the associated multigraded cellular chain complex.

math.AT↗

arXiv2015

A presentation of general multipersistence modules computable in polynomial time?

Multipersistence homology modules were introduced by G.Carlsson and A.Zomorodian which gave, together with G.Singh, an algorithm to compute their Groebner bases. Although their algorithm has polynomial complexity when the chain modules are free, i.e. in the one-critical case, it might be exponential in general. We give a new presentation of multipersistence homology modules, which allows us to design an algorithm to compute their Groebner bases always in polynomial time by avoiding the mapping telescope.

math.AT↗

arXiv2014

Combinatorial presentation of multidimensional persistent homology

A multifiltration is a functor indexed by $\mathbb{N}^r$ that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-modules that can occur as $R$-spans of multifiltrations of sets are the direct sums of monomial ideals.

math.AT↗

arXiv2013

Topological strata of weighted complex networks

The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and --more recently-- correlations between neighboring nodes. However, statistical methods quickly become cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.

cond-mat.stat-mech↗