arXiv subjects

Francesco Vaccarino

Publications and source records attributed to Francesco Vaccarino.

At least 19 recordsLinked to original sources

arXiv2026

Statistical models as natural transformations: meaningfulness, coherence and priors as states in Markov categories

We show that a statistical model in the sense of McCullagh, in the form given by Brøns, is a natural transformation between two functors from the category of designs to the Kleisli category Stoch of the Giry monad, provided that its components are measurable in the parameter. The condition is empty for finite models. A design-indexed quantity is a family of morphisms of Stoch defined on the parameter objects, called meaningful if it is natural. We prove that Tjur's criterion, imposed on parameter functions indexed by finite samples with multiplicities, forces the indexing by the support and then coincides with naturality over the insertions. For finite designs we show that a quantity can be corrected to a natural one within a given class of corrections if and only if a class vanishes in the first cohomology group of a Baues-Wirsching complex relative to that class, while its image in the absolute group is always zero. In the one-way layout, marginal dispersion is not meaningful, and within-group dispersion is the unique correction that leaves the merged design unchanged. A prior is a family of states on the parameter objects, called coherent over a class of design morphisms if it is natural over that class. We show that coherence at a merge confines the prior to the image of the corresponding parameter map, that coherence over the insertions is Kolmogorov consistency, and that coherence over the injections adds the exchangeability assumed by the categorical de Finetti theorem. In the finite one-way scheme, the coherent priors form polytopes of known dimension. The analogue of Jeffreys' general rule is not coherent, while the analogue for location-scale families is. Finally, we show that ridge regression is the Bayesian inversion of the Gaussian linear model with respect to a Gaussian prior, which is coherent over the insertions and never over the injections.

math.ST↗

arXiv2026

Topology and Geometry of the Learning Space of ReLU Networks: Connectivity and Singularities

Understanding the properties of the parameter space in feed-forward ReLU networks is critical for effectively analyzing and guiding training dynamics. After initialization, training under gradient flow decisively restricts the parameter space to an algebraic variety that emerges from the homogeneous nature of the ReLU activation function. In this study, we examine two key challenges associated with feed-forward ReLU networks built on general directed acyclic graph (DAG) architectures: the (dis)connectedness of the parameter space and the existence of singularities within it. We extend previous results by providing a thorough characterization of connectedness, highlighting the roles of bottleneck nodes and balance conditions associated with specific subsets of the network. Our findings clearly demonstrate that singularities are intricately connected to the topology of the underlying DAG and its induced sub-networks. We discuss the reachability of these singularities and establish a principled connection with differentiable pruning. We validate our theory with simple numerical experiments.

cs.LG↗

arXiv2025

Diagrammatic Hochschild cohomology via cohomology of categories, and incidence algebras

In this paper, we study the Hochschild cohomology of diagrams of algebras introduced by Gerstenhaber and Schack and provide computations for filtrations of incidence algebras. Our aims are threefold: firstly, we revisit and explore the connection between the Gerstenhaber-Schack complexes and the Baues-Wirsching cohomology of categories. Secondly, we analyse the behaviour of (diagrammatic) Hochschild cohomology in the context of homological epimorphisms of algebras. Thirdly, we study diagrams of incidence algebras. In particular, as a main application, we compute the diagrammatic Hochschild cohomology of diagrams of incidence algebras associated to finite filtrations of simplicial complexes.

math.AT↗

arXiv2025

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↗

arXiv2025

Attributes Shape the Embedding Space of Face Recognition Models

Face Recognition (FR) tasks have made significant progress with the advent of Deep Neural Networks, particularly through margin-based triplet losses that embed facial images into high-dimensional feature spaces. During training, these contrastive losses focus exclusively on identity information as labels. However, we observe a multiscale geometric structure emerging in the embedding space, influenced by interpretable facial (e.g., hair color) and image attributes (e.g., contrast). We propose a geometric approach to describe the dependence or invariance of FR models to these attributes and introduce a physics-inspired alignment metric. We evaluate the proposed metric on controlled, simplified models and widely used FR models fine-tuned with synthetic data for targeted attribute augmentation. Our findings reveal that the models exhibit varying degrees of invariance across different attributes, providing insight into their strengths and weaknesses and enabling deeper interpretability. Code available here: https://github.com/mantonios107/attrs-fr-embs}{https://github.com/mantonios107/attrs-fr-embs

cs.CV↗

arXiv2025

Bound by semanticity: universal laws governing the generalization-identification tradeoff

Intelligent systems must deploy internal representations that are simultaneously structured -- to support broad generalization -- and selective -- to preserve input identity. We expose a fundamental limit on this tradeoff. For any model whose representational similarity between inputs decays with finite semantic resolution $\varepsilon$, we derive closed-form expressions that pin its probability of correct generalization $p_S$ and identification $p_I$ to a universal Pareto front independent of input space geometry. Extending the analysis to noisy, heterogeneous spaces and to $n>2$ inputs predicts a sharp $1/n$ collapse of multi-input processing capacity and a non-monotonic optimum for $p_S$. A minimal ReLU network trained end-to-end reproduces these laws: during learning a resolution boundary self-organizes and empirical $(p_S,p_I)$ trajectories closely follow theoretical curves for linearly decaying similarity. Finally, we demonstrate that the same limits persist in two markedly more complex settings -- a convolutional neural network and state-of-the-art vision-language models -- confirming that finite-resolution similarity is a fundamental emergent informational constraint, not merely a toy-model artifact. Together, these results provide an exact theory of the generalization-identification trade-off and clarify how semantic resolution shapes the representational capacity of deep networks and brains alike.

cs.LG↗

arXiv2025

Edge-Wise Graph-Instructed Neural Networks

The problem of multi-task regression over graph nodes has been recently approached through Graph-Instructed Neural Network (GINN), which is a promising architecture belonging to the subset of message-passing graph neural networks. In this work, we discuss the limitations of the Graph-Instructed (GI) layer, and we formalize a novel edge-wise GI (EWGI) layer. We discuss the advantages of the EWGI layer and we provide numerical evidence that EWGINNs perform better than GINNs over some graph-structured input data, like the ones inferred from the Barabasi-Albert graph, and improve the training regularization on graphs with chaotic connectivity, like the ones inferred from the Erdos-Renyi graph.

cs.LG↗

arXiv2024

Topological obstruction to the training of shallow ReLU neural networks

Studying the interplay between the geometry of the loss landscape and the optimization trajectories of simple neural networks is a fundamental step for understanding their behavior in more complex settings. This paper reveals the presence of topological obstruction in the loss landscape of shallow ReLU neural networks trained using gradient flow. We discuss how the homogeneous nature of the ReLU activation function constrains the training trajectories to lie on a product of quadric hypersurfaces whose shape depends on the particular initialization of the network's parameters. When the neural network's output is a single scalar, we prove that these quadrics can have multiple connected components, limiting the set of reachable parameters during training. We analytically compute the number of these components and discuss the possibility of mapping one to the other through neuron rescaling and permutation. In this simple setting, we find that the non-connectedness results in a topological obstruction, which, depending on the initialization, can make the global optimum unreachable. We validate this result with numerical experiments.

cs.LG↗

arXiv2024

ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain

This paper describes the 2nd edition of the ICML Topological Deep Learning Challenge that was hosted within the ICML 2024 ELLIS Workshop on Geometry-grounded Representation Learning and Generative Modeling (GRaM). The challenge focused on the problem of representing data in different discrete topological domains in order to bridge the gap between Topological Deep Learning (TDL) and other types of structured datasets (e.g. point clouds, graphs). Specifically, participants were asked to design and implement topological liftings, i.e. mappings between different data structures and topological domains --like hypergraphs, or simplicial/cell/combinatorial complexes. The challenge received 52 submissions satisfying all the requirements. This paper introduces the main scope of the challenge, and summarizes the main results and findings.

cs.LG↗

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↗

arXiv2024

Higher-order Laplacian Renormalization

We propose a cross-order Laplacian renormalization group (X-LRG) scheme for arbitrary higher-order networks. The renormalization group is a pillar of the theory of scaling, scale-invariance, and universality in physics. An RG scheme based on diffusion dynamics was recently introduced for complex networks with dyadic interactions. Despite mounting evidence of the importance of polyadic interactions, we still lack a general RG scheme for higher-order networks. Our approach uses a diffusion process to group nodes or simplices, where information can flow between nodes and between simplices (higher-order interactions). This approach allows us (i) to probe higher-order structures, defining scale-invariance at various orders, and (ii) to propose a coarse-graining scheme. We demonstrate our approach on controlled synthetic higher-order systems and then use it to detect the presence of order-specific scale-invariant profiles of real-world complex systems from multiple domains.

cond-mat.stat-mech↗

arXiv2023

A unified framework for Simplicial Kuramoto models

Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. Our framework is based on topology, discrete differential geometry as well as gradient flows and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models.

nlin.AO↗

arXiv2023

The Shapley-Hodge Associated Game

In cooperative game theory, associated games allow for providing meaningful characterizations of solution concepts. Moreover, generalized values allow computing an influence or power index of each coalition in a game. In this paper, we view associated games through the lens of game maps and we define the novel Shapley-Hodge Game, briefly ``SHoGa''. We characterize SHoGa via an axiomatic approach as a generalized value, and we thoroughly discuss the consistency properties of its associated game. Furthermore, we describe the Hodge decomposition of an oriented graph representing the transitive closure of the Hasse diagram of coalitions in the game. Finally, we show how SHoGa is linked to the solution of the Poisson equation derived from such a decomposition.

cs.GT↗

arXiv2021

The physics of higher-order interactions in complex systems

Complex networks have become the main paradigm for modelling the dynamics of interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by higher-order interactions involving groups of three or more units. Higher-order structures, such as hypergraphs and simplicial complexes, are therefore a better tool to map the real organization of many social, biological and man-made systems. Here, we highlight recent evidence of collective behaviours induced by higher-order interactions, and we outline three key challenges for the physics of higher-order systems.

physics.soc-ph↗

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.

math.AT↗

arXiv2020

On the notion of weak isometry for finite metric spaces

Finite metric spaces are the object of study in many data analysis problems. We examine the concept of weak isometry between finite metric spaces, in order to analyse properties of the spaces that are invariant under strictly increasing rescaling of the distance functions. In this paper, we analyse some of the possible complete and incomplete invariants for weak isometry and we introduce a dissimilarity measure that asses how far two spaces are from being weakly isometric. Furthermore, we compare these ideas with the theory of persistent homology, to study how the two are related.

math.MG↗

arXiv2017

Construction of and efficient sampling from the simplicial configuration model

Simplicial complexes are now a popular alternative to networks when it comes to describing the structure of complex systems, primarily because they encode multi-node interactions explicitly. With this new description comes the need for principled null models that allow for easy comparison with empirical data. We propose a natural candidate, the simplicial configuration model. The core of our contribution is an efficient and uniform Markov chain Monte Carlo sampler for this model. We demonstrate its usefulness in a short case study by investigating the topology of three real systems and their randomized counterparts (using their Betti numbers). For two out of three systems, the model allows us to reject the hypothesis that there is no organization beyond the local scale.

physics.soc-ph↗

arXiv2017

Temporal Characterization of the Remote Sensors Response to Radiation Damage in L2

Remote sensors on spacecrafts acquire huge volumes of data that can be processed for other purposes in addition to those they were designed for. The project TECSEL2 was born for the usage of the Gaia AIM/AVU daily pipeline output and solar events data to characterize the response of detectors subjected to strong radiation damage within an environment not protected by the terrestrial magnetic field, the Lagrangian point L2, where Gaia operates. The project also aims at identifying anomalies in the scientific output parameters and relate them to detectors malfunctioning due to radiation damage issues correlating with solar events occurred in the same time range. TECSEL2 actually designs and implements a system based on big data technologies which are the state of art in the fields of data processing and data storage. The final goal of TECSEL2 is not only related to the Gaia project, because it provides useful analysis techniques for generic and potentially huge time series datasets.

astro-ph.EP↗