arXiv subjects

Piero Deidda

Publications and source records attributed to Piero Deidda.

8 recordsLinked to original sources

arXiv2025

Stuart-Landau Oscillatory Graph Neural Network

Oscillatory Graph Neural Networks (OGNNs) are an emerging class of physics-inspired architectures designed to mitigate oversmoothing and vanishing gradient problems in deep GNNs. In this work, we introduce the Complex-Valued Stuart-Landau Graph Neural Network (SLGNN), a novel architecture grounded in Stuart-Landau oscillator dynamics. Stuart-Landau oscillators are canonical models of limit-cycle behavior near Hopf bifurcations, which are fundamental to synchronization theory and are widely used in e.g. neuroscience for mesoscopic brain modeling. Unlike harmonic oscillators and phase-only Kuramoto models, Stuart-Landau oscillators retain both amplitude and phase dynamics, enabling rich phenomena such as amplitude regulation and multistable synchronization. The proposed SLGNN generalizes existing phase-centric Kuramoto-based OGNNs by allowing node feature amplitudes to evolve dynamically according to Stuart-Landau dynamics, with explicit tunable hyperparameters (such as the Hopf-parameter and the coupling strength) providing additional control over the interplay between feature amplitudes and network structure. We conduct extensive experiments across node classification, graph classification, and graph regression tasks, demonstrating that SLGNN outperforms existing OGNNs and establishes a novel, expressive, and theoretically grounded framework for deep oscillatory architectures on graphs.

cs.LG↗

arXiv2025

Nonlinear Joint Spectral Radius

We introduce a nonlinear extension of the joint spectral radius (JSR) for switched discrete-time dynamical systems governed by sub-homogeneous and order-preserving maps acting on cones. We show that this nonlinear JSR characterizes both the asymptotic stability of the system and the divergence or convergence rate of trajectories originating from different points within the cone. Our analysis establishes upper and lower bounds on the nonlinear JSR of a sub-homogeneous family via the JSRs of two associated homogeneous families obtained through asymptotic scaling. In the homogeneous case, we develop a dual formulation of the JSR and investigate the equality between the joint spectral radius and the generalized joint spectral radius, extending classical results from linear theory to the nonlinear setting. We also propose a polytopal-type algorithm to approximate the nonlinear JSR and provide conditions ensuring its finite-time convergence. The proposed framework is motivated by applications such as the analysis of deep neural networks, which can be modeled as switched systems with structured nonlinear layers. Our results offer new theoretical tools for studying the stability, robustness, and convergence behavior of such models.

math.DS↗

arXiv2025

Nonlinear spectral graph theory

Nonlinear spectral graph theory is an extension of the traditional (linear) spectral graph theory and studies relationships between spectral properties of nonlinear operators defined on a graph and topological properties of the graph itself. Many of these relationships get tighter when going from the linear to the nonlinear case. In this manuscript, we discuss the spectral theory of the graph $p$-Laplacian operator. In particular we report links between the $p$-Laplacian spectrum and higher-order Cheeger (or isoperimetric) constants, sphere packing constants, independence and matching numbers of the graph. The main aim of this paper is to present a complete and self-contained introduction to the problem accompanied by a discussion of the main results and the proof of new results that fill some gaps in the theory. The majority of the new results are devoted to the study of the graph infinity Laplacian spectrum and the information that it yields about the packing radii, the independence numbers and the matching number of the graph. This is accompanied by a novel discussion about the nodal domains induced by the infinity eigenfunctions. There are also new results about the variational spectrum of the $p$-Laplacian, the regularity of the $p$-Laplacian spectrum varying $p$, and the relations between the $1$-Laplacian spectrum and new Cheeger constants.

math.SP↗

arXiv2025

Are We Measuring Oversmoothing in Graph Neural Networks Correctly?

Oversmoothing is a fundamental challenge in graph neural networks (GNNs): as the number of layers increases, node embeddings become increasingly similar, and model performance drops sharply. Traditionally, oversmoothing has been quantified using metrics that measure the similarity of neighbouring node features, such as the Dirichlet energy. We argue that these metrics have critical limitations and fail to reliably capture oversmoothing in realistic scenarios. For instance, they provide meaningful insights only for very deep networks, while typical GNNs show a performance drop already with as few as 10 layers. As an alternative, we propose measuring oversmoothing by examining the numerical or effective rank of the feature representations. We provide extensive numerical evaluation across diverse graph architectures and datasets to show that rank-based metrics consistently capture oversmoothing, whereas energy-based metrics often fail. Notably, we reveal that drops in the rank align closely with performance degradation, even in scenarios where energy metrics remain unchanged. Along with the experimental evaluation, we provide theoretical support for this approach, clarifying why Dirichlet-like measures may fail to capture performance drop and proving that the numerical rank of feature representations collapses to one for a broad family of GNN architectures.

cs.LG↗

arXiv2024

The graph $\infty$-Laplacian eigenvalue problem

We analyze various formulations of the $\infty$-Laplacian eigenvalue problem on graphs, comparing their properties and highlighting their respective advantages and limitations. First, we investigate the graph $\infty$-eigenpairs arising as limits of $p$-Laplacian eigenpairs, extending key results from the continuous setting to the discrete domain. We prove that every limit of $p$-Laplacian eigenpair, for $p$ going to $\infty$, satisfies a limit eigenvalue equation and establish that the corresponding eigenvalue can be bounded from below by the packing radius of the graph, indexed by the number of nodal domains induced by the eigenfunction. Additionally, we show that the limits, for $p$ going to $\infty$, of the variational $p$-Laplacian eigenvalues are bounded both from above and from below by the packing radii, achieving equality for the smallest two variational eigenvalues and corresponding packing radii of the graph. In the second part of the paper, we introduce generalized $\infty$-Laplacian eigenpairs as generalized critical points and values of the $\infty$-Rayleigh quotient. We prove that the generalized variational $\infty$-eigenvalues equal the limit of the $p$-Laplacian variational eigenvalues and so satisfy the same upper bounds in terms of packing radii. Finally, we establish that any solution to the limit eigenvalue equation is also a generalized eigenpair, while any generalized eigenpair satisfies the limit eigenvalue equation on a suitable subgraph.

math.SP↗

arXiv2024

Graph $p$-Laplacian eigenpairs as saddle points of a family of spectral energy functions

We address the problem of computing the graph $p$-Laplacian eigenpairs for $p\in (2,\infty)$. We propose a reformulation of the graph $p$-Laplacian eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue problem and discuss theoretical and computational advantages. We provide a correspondence between $p$-Laplacian eigenpairs and linear eigenpair of a constrained generalized weighted Laplacian eigenvalue problem. As a result, we can assign an index to any $p$-Laplacian eigenpair that matches the Morse index of the $p$-Rayleigh quotient evaluated at the eigenfunction. In the second part of the paper we introduce a class of spectral energy functions that depend on edge and node weights. We prove that differentiable saddle points of the $k$-th energy function correspond to $p$-Laplacian eigenpairs having index equal to $k$. Moreover, the first energy function is proved to possess a unique saddle point which corresponds to the unique first $p$-Laplacian eigenpair. Finally we develop novel gradient-based numerical methods suited to compute $p$-Laplacian eigenpairs for any $p\in(2,\infty)$ and present some experiments.

math.NA↗

arXiv2024

Provable Emergence of Deep Neural Collapse and Low-Rank Bias in $L^2$-Regularized Nonlinear Networks

We present a unified theoretical framework connecting the first property of Deep Neural Collapse (DNC1) to the emergence of implicit low-rank bias in nonlinear networks trained with $L^2$ weight decay regularization. Our main contributions are threefold. First, we derive a quantitative relation between the Total Cluster Variation (TCV) of intermediate embeddings and the numerical rank of stationary weight matrices. In particular, we establish that, at any critical point, the distance from a weight matrix to the set of rank-$K$ matrices is bounded by a constant times the TCV of earlier-layer features, scaled inversely with the weight-decay parameter. Second, we prove global optimality of DNC1 in a constrained representation-cost setting for both feedforward and residual architectures, showing that zero TCV across intermediate layers minimizes the representation cost under natural architectural constraints. Third, we establish a benign landscape property: for almost every interpolating initialization there exists a continuous, loss-decreasing path from the initialization to a globally optimal, DNC1-satisfying configuration. Our theoretical claims are validated empirically; numerical experiments confirm the predicted relations among TCV, singular-value structure, and weight decay. These results indicate that neural collapse and low-rank bias are intimately linked phenomena arising from the optimization geometry induced by weight decay.

cs.LG↗

arXiv2022

Nodal domain count for the generalized graph $p$-Laplacian

Inspired by the linear Schr\"odinger operator, we consider a generalized $p$-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous $p$-Laplacian, we prove that the variational spectrum of the discrete generalized $p$-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl's inequalities for the Laplacian operator to the nonlinear case and prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized $p$-Laplacian on generic graphs, including variational eigenpairs. In particular, when applied to the linear case $p=2$, in addition to recovering well-known features, the new results provide novel properties of the linear Schr\"odinger operator.

math.NA↗