arXiv subjects

Marco Pacini

Publications and source records attributed to Marco Pacini.

At least 19 recordsLinked to original sources

arXiv2026

Graph Hierarchical Recurrence for Long-Range Generalization

Graph Neural Networks (GNNs) and Graph Transformers (GTs) are now a fundamental paradigm for graph learning, combining the representation-learning capabilities of deep models with the sample efficiency induced by their inductive biases. Despite their effectiveness, a large body of work has shown that these models still face fundamental limitations in tasks that require capturing correlations between distant regions of a graph. To address this issue, we introduce Graph Hierarchical Recurrence (GHR), a novel framework that operates jointly on the input graph and on a hierarchical abstraction obtained through pooling. We also show that the limitations of existing models are even more pronounced in out-of-range generalization, where test instances involve interactions over distances longer than those observed during training. By contrast, despite its simple design, GHR provides three key advantages: strong performance on long-range dependencies, improved out-of-range generalization, and high parameter efficiency. To corroborate these claims, we show that across a broad set of long-range benchmarks, GHR consistently outperforms existing graph models while using as little as 1% of the parameters of current state-of-the-art models. These results suggest a complementary direction to the current trend of scaling architectures to obtain graph foundation models, indicating that increased model capacity alone may not be sufficient for generalization.

cs.LG↗

arXiv2025

On Uncertainty Calibration for Equivariant Functions

Data-sparse settings such as robotic manipulation, molecular physics, and galaxy morphology classification are some of the hardest domains for deep learning. For these problems, equivariant networks can help improve modeling across undersampled parts of the input space, and uncertainty estimation can guard against overconfidence. However, until now, the relationships between equivariance and model confidence, and more generally equivariance and model calibration, has yet to be studied. Since traditional classification and regression error terms show up in the definitions of calibration error, it is natural to suspect that previous work can be used to help understand the relationship between equivariance and calibration error. In this work, we present a theory relating equivariance to uncertainty estimation. By proving lower and upper bounds on uncertainty calibration errors (ECE and ENCE) under various equivariance conditions, we elucidate the generalization limits of equivariant models and illustrate how symmetry mismatch can result in miscalibration in both classification and regression. We complement our theoretical framework with numerical experiments that clarify the relationship between equivariance and uncertainty using a variety of real and simulated datasets, and we comment on trends with symmetry mismatch, group size, and aleatoric and epistemic uncertainties.

cs.LG↗

arXiv2025

On Universality of Deep Equivariant Networks

Universality results for equivariant neural networks remain rare. Those that do exist typically hold only in restrictive settings: either they rely on regular or higher-order tensor representations, leading to impractically high-dimensional hidden spaces, or they target specialized architectures, often confined to the invariant setting. This work develops a more general account. For invariant networks, we establish a universality theorem under separation constraints, showing that the addition of a fully connected readout layer secures approximation within the class of separation-constrained continuous functions. For equivariant networks, where results are even scarcer, we demonstrate that standard separability notions are inadequate and introduce the sharper criterion of $\textit{entry-wise separability}$. We show that with sufficient depth or with the addition of appropriate readout layers, equivariant networks attain universality within the entry-wise separable regime. Together with prior results showing the failure of universality for shallow models, our findings identify depth and readout layers as a decisive mechanism for universality, additionally offering a unified perspective that subsumes and extends earlier specialized results.

stat.ML↗

arXiv2025

On Universality Classes of Equivariant Networks

Equivariant neural networks provide a principled framework for incorporating symmetry into learning architectures and have been extensively analyzed through the lens of their separation power, that is, the ability to distinguish inputs modulo symmetry. This notion plays a central role in settings such as graph learning, where it is often formalized via the Weisfeiler-Leman hierarchy. In contrast, the universality of equivariant models-their capacity to approximate target functions-remains comparatively underexplored. In this work, we investigate the approximation power of equivariant neural networks beyond separation constraints. We show that separation power does not fully capture expressivity: models with identical separation power may differ in their approximation ability. To demonstrate this, we characterize the universality classes of shallow invariant networks, providing a general framework for understanding which functions these architectures can approximate. Since equivariant models reduce to invariant ones under projection, this analysis yields sufficient conditions under which shallow equivariant networks fail to be universal. Conversely, we identify settings where shallow models do achieve separation-constrained universality. These positive results, however, depend critically on structural properties of the symmetry group, such as the existence of adequate normal subgroups, which may not hold in important cases like permutation symmetry.

cs.LG↗

arXiv2024

Separation Power of Equivariant Neural Networks

The separation power of a machine learning model refers to its ability to distinguish between different inputs and is often used as a proxy for its expressivity. Indeed, knowing the separation power of a family of models is a necessary condition to obtain fine-grained universality results. In this paper, we analyze the separation power of equivariant neural networks, such as convolutional and permutation-invariant networks. We first present a complete characterization of inputs indistinguishable by models derived by a given architecture. From this results, we derive how separability is influenced by hyperparameters and architectural choices-such as activation functions, depth, hidden layer width, and representation types. Notably, all non-polynomial activations, including ReLU and sigmoid, are equivalent in expressivity and reach maximum separation power. Depth improves separation power up to a threshold, after which further increases have no effect. Adding invariant features to hidden representations does not impact separation power. Finally, block decomposition of hidden representations affects separability, with minimal components forming a hierarchy in separation power that provides a straightforward method for comparing the separation power of models.

cs.LG↗

arXiv2024

A Characterization Theorem for Equivariant Networks with Point-wise Activations

Equivariant neural networks have shown improved performance, expressiveness and sample complexity on symmetrical domains. But for some specific symmetries, representations, and choice of coordinates, the most common point-wise activations, such as ReLU, are not equivariant, hence they cannot be employed in the design of equivariant neural networks. The theorem we present in this paper describes all possible combinations of finite-dimensional representations, choice of coordinates and point-wise activations to obtain an exactly equivariant layer, generalizing and strengthening existing characterizations. Notable cases of practical relevance are discussed as corollaries. Indeed, we prove that rotation-equivariant networks can only be invariant, as it happens for any network which is equivariant with respect to connected compact groups. Then, we discuss implications of our findings when applied to important instances of exactly equivariant networks. First, we completely characterize permutation equivariant networks such as Invariant Graph Networks with point-wise nonlinearities and their geometric counterparts, highlighting a plethora of models whose expressive power and performance are still unknown. Second, we show that feature spaces of disentangled steerable convolutional neural networks are trivial representations.

cs.LG↗

arXiv2023

A Torelli theorem for graphs via quasistable divisors

The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine compactified Jacobian. The fine compactified Jacobian of a curve comes with a natural stratification that can be regarded as a poset. Furthermore, this poset is entirely determined by the dual graph of the curve and is referred to as the poset of quasistable divisors on the graph. We present a combinatorial version of the Torelli theorem, which demonstrates that the poset of quasistable divisors of a graph completely determines the biconnected components of the graph (up to contracting separating edges). Moreover, we achieve a natural extension of this theorem to tropical curves.

math.CO↗

arXiv2022

On moduli spaces of roots in algebraic and tropical geometry

In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs.

math.AG↗

arXiv2022

Degree-2 Abel maps and hyperelleptic curves

In this paper we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an application, we characterize when the (symmetrized) degree-2 Abel map is not injective, a property that, for a smooth curve, is equivalent to the curve being hyperelliptic.

math.AG↗

arXiv2020

The moduli space of quasistable spin curves

We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones.

math.AG↗

arXiv2020

Abel Maps for nodal curves via tropical geometry

We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one.

math.AG↗

arXiv2020

Theta characteristics on tropical curves

We give an explicit description of theta-characteristics on tropical curves and characterize the effective ones. We construct the moduli space for tropical theta-characteristics of genus g as a generalized cone complex. We describe the fibers of the specialization map from the classical moduli space of theta-characteristics on algebraic curves to our moduli space of tropical theta-characteristics.

math.AG↗

arXiv2020

Complex and tropical counts via positive characteristic

We reconcile the discrepancy between the complex and tropical counts of some enumerative problems reducing to positive characteristic. Each problem that we consider suggests a prime with special behaviour. Modulo this prime, the solutions coalesce in uniform clusters: at this special prime, the geometric and the tropical behaviours match. As examples, we concentrate on inflection points of plane curves and theta-hyperplanes of canonical curves.

math.AG↗

arXiv2019

A universal tropical Jacobian over $M_g^{\trop}$

We introduce and study polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion-free rank-1 sheaves on a nodal curve. We construct a universal tropical Jacobian over the moduli space of tropical curves of genus $g$. This space parametrizes equivalence classes of tropical curves of genus $g$ together with a $\mu$-polystable divisor, and can be seen as a tropical counterpart of Caporaso universal Picard scheme. We describe polyhedral decompositions of the Jacobian of a tropical curve via polystable divisors, relating them with other known polyhedral decompositions.

math.AG↗

arXiv2019

The resolution of the universal Abel map via tropical geometry and applications

Let $g$ and $n$ be nonnegative integers and $\mathcal A=(a_0,\dots,a_n)$ a sequence of $n+1$ integers summing up to $d$. Let $\overline{\mathcal M}_{g,n+1}$ be the moduli space of $(n+1)$-pointed stable curves of genus $g$ and $\overline{\mathcal J}_{\mu,g}\rightarrow \overline{\mathcal M}_{g,1}$ be the Esteves' universal Jacobian, where $\mu$ is a universal genus-$g$ polarization of degree $d$. We give an explicit resolution of the universal Abel map $\alpha_{\mathcal A,\mu}\colon \overline{\mathcal M}_{g,n+1}\dashrightarrow \overline{\mathcal J}_{\mu,g}$, taking a pointed curve $(X,p_0,\dots,p_n)$ to $\mathcal{O}_X(\sum_{0\le i\le n} a_ip_i)$. The blowup of $\overline{\mathcal M}_{g,n+1}$ giving rise to the resolution is inspired by the resolution of the tropical analogue of the map $\alpha_{\mathcal A,\mu}$ (in the category of generalized cone complexes). As an application, we describe the double ramification cycle in terms of the universal sheaf inducing the resolution of the map $\alpha_{\mathcal A,\mu}$.

math.AG↗

arXiv2019

Tropicalizing the moduli space of spin curves

We study the tropicalization of the moduli space of algebraic spin curves, exhibit its combinatorial stratification and prove that the strata are irreducible. We construct the moduli space of tropical spin curves, prove that it is isomorphic to the skeleton of the Berkovich analytification of the moduli space of algebraic spin curves, and give a geometric interpretation of the retraction map in terms of a tropicalization map.

math.AG↗

arXiv2018

The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian

For each universal genus-$g$ polarization $\mu$ of degree $d$, we construct a universal tropical Jacobian $J_{\mu,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several properties of the space $J_{\mu,g}^{trop}$. In particular, we prove that the natural compactification of $J_{\mu,g}^{trop}$ is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus-$g$ curves.

math.AG↗

arXiv2018

On a question of Dolgachev

For each even, positive integer $n$, we define a rational self-map on the space of plane curves of degree $n$, using classical contravariants. In the case of plane quartics, we show that the degree of this map is 15. This answers a question of Dolgachev on the moduli space of curves of genus 3.

math.AG↗