2026
Nonlinear Spectral Graph Theory
SIAM Review, 2026
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 …
The graph $\infty$-Laplacian eigenvalue problem
SIAM Journal on Mathematical Analysis, 2026
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 …
Nonlinear Spectral Duality
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2026
Nonlinear eigenvalue problems for pairs of homogeneous convex functions are particular nonlinear constrained optimization problems that arise in a variety of settings, including graph mining, machine learning, and network science. By considering different notions of duality transforms from both classical and recent …
2025
Nonlinear Joint Spectral Radius
We introduce a nonlinear extension of the joint spectral radius (JSR) for switched discrete-time dynamical systems. The classical JSR characterizes the exponential growth rate of linear switched systems, but its extension to nonlinear systems has remained an open problem. Our approach builds on the framework of …
2024
Subhomogeneous Deep Equilibrium Models
International Conference on Machine Learning (ICML), 2024
Implicit-depth neural networks have grown as powerful alternatives to traditional networks in various applications in recent years. However, these models often lack guarantees of existence and uniqueness, raising stability, performance, and reproducibility issues. In this paper, we present a new analysis of the …
A nonlinear spectral core-periphery detection method for multiplex networks
Proceedings of the Royal Society A, 2024
Core-periphery detection aims to separate the nodes of a complex network into two subsets: a core that is densely connected to the entire network and a periphery that is densely connected to the core but sparsely connected internally. The definition of core-periphery structure in multiplex networks that record …
2023
Nonlinear Perron-Frobenius theorems for nonnegative tensors
SIAM Review (SIGEST paper), vol. 65, pp. 495–536, 2023
We present a unifying Perron–Frobenius theory for nonlinear spectral problems defined in terms of nonnegative tensors. By using the concept of tensor shape partition, our results include, as a special case, a wide variety of particular tensor spectral problems considered in the literature and can be applied to a broad …
Core-periphery detection in hypergraphs
SIAM Journal on Mathematics of Data Science, vol. 5, 2023
Core-periphery detection is a key task in exploratory network analysis where one aims to find a core, a set of nodes well-connected internally and with the periphery, and a periphery, a set of nodes connected only (or mostly) with the core. In this work we propose a model of core-periphery for higher-order networks …
Nodal domain count for the generalized graph $p$-Laplacian
Applied and Computational Harmonic Analysis, vol. 64, pp. 1–32, 2023
Inspired by the linear Schrödinger 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 …
2021
The self-consistent field iteration for p-spectral clustering
The self-consistent field (SCF) iteration, combined with its variants, is one of the most widely used algorithms in quantum chemistry. We propose a procedure to adapt the SCF iteration for the p-Laplacian eigenproblem, which is an important problem in the field of unsupervised learning. We formulate the p-Laplacian …
The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms
Journal of Scientific Computing, vol. 88, pp. 21, 2021
We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $\ell^p$ matrix norms. In particular, …
2020
Shifted and extrapolated power methods for tensor $\ell^p$-eigenpairs
Electronic Transactions on Numerical Analysis (ETNA), vol. 53, pp. 1–27, 2020
This work is concerned with the computation of $\ell^p$-eigenvalues and eigenvectors of square tensors with $d$ modes. In the first part we propose two possible shifted variants of the popular (higher-order) power method for the computation of $\ell^p$-eigenpairs proving the convergence of both the schemes to the …
Total variation based community detection using a nonlinear optimization approach
SIAM Journal on Applied Mathematics, vol. 80, pp. 1392–1419, 2020
Maximizing the modularity of a network is a successful tool to identify an important community of nodes. However, this combinatorial optimization problem is known to be NP-hard. Inspired by recent nonlinear modularity eigenvector approaches, we introduce the modularity total variation $TV_Q$ and show that its …
2019
A fast and robust kernel optimization method for core–periphery detection in directed and weighted graphs
Applied Network Science, vol. 4:75, 2019
Many graph mining tasks can be viewed as classification problems on high dimensional data. Within this class we consider the issue of discovering core-periphery structure, which has wide applications in the economic and social sciences. In contrast to many current approaches, we allow for weighted and directed edges …
A nonlinear spectral method for core-periphery detection in networks
SIAM Journal on Mathematics of Data Science, vol. 1, pp. 269–292, 2019
We derive and analyse a new iterative algorithm for detecting network core–periphery structure. Using techniques in nonlinear Perron-Frobenius theory, we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales …
The contractivity of cone-preserving multilinear mappings
Nonlinearity, vol. 32, pp. 4713, 2019
With the notion of mode-$j$ Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff-Hopf and the Perron-Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of systems of nonlinear equations of the type $$f_i(x_1,…,x_ν)= λ_i x_i, ,$$ being $x_i$ …
A unifying Perron-Frobenius theorem for nonnegative tensors via multihomogeneous maps
SIAM Journal on Matrix Analysis and Applications, vol. 40, pp. 1206–1231, 2019
We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a nonnegative tensor $T$ in terms of the associated shape partition. We recast the …
The Perron-Frobenius theorem for multihomogeneous mappings
SIAM Journal on Matrix Analysis and Applications, vol. 40, pp. 1179–1205, 2019
The Perron-Frobenius theory for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative multilinear forms. We unify both approaches by introducing the concept of order-preserving multi-homogeneous mappings, their associated nonlinear spectral …
2018
Community detection in networks via nonlinear modularity eigenvectors
SIAM Journal on Applied Mathematics, vol. 78, pp. 2393–2419, 2018
Revealing a community structure in a network or dataset is a central problem arising in many scientific areas. The modularity function $Q$ is an established measure quantifying the quality of a community, being identified as a set of nodes having high modularity. In our terminology, a set of nodes with positive …
A nodal domain theorem and a higher-order Cheeger inequality for the graph p-Laplacian
EMS Journal of Spectral Theory, vol. 8, pp. 883–908, 2018
We consider the nonlinear graph $p$-Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational principle. We prove a nodal domain theorem for the graph $p$-Laplacian for any $p\geq 1$. While for $p>1$ the bounds on the number of weak and strong nodal domains are the …