2026
Efficient Sparsification of Simplicial Complexes via Local Densities of States
SIAM Journal on Mathematics of Data Science, 2026
Simplicial complexes (SCs), a generalization of graph models for relational data that account for higher-order relations between data items, have become a popular abstraction for analyzing complex data using tools from topological data analysis or topological signal processing. However, the analysis of many real-world …
2025
Algebraformer: A Neural Approach to Linear Systems
NeurIPS Workshop on Machine Learning and the Physical Sciences, 2025
Recent work in deep learning has opened new possibilities for solving classical algorithmic tasks using end-to-end learned models. In this work, we investigate the fundamental task of solving linear systems, particularly those that are ill-conditioned. Existing numerical methods for ill-conditioned systems often …
2024
Cholesky-like Preconditioner for Hodge Laplacians via Heavy Collapsible Subcomplex
SIAM Journal on Matrix Analysis and Applications, 2024
Techniques based on $k$-th order Hodge Laplacian operators $L_k$ are widely used to describe the topology as well as the governing dynamics of high-order systems modeled as simplicial complexes. In all of them, it is required to solve a number of least square problems with $L_k$ as coefficient matrix, for example in …
Optimizing network robustness via Krylov subspaces
ESAIM: Mathematical Modelling and Numerical Analysis, vol. 58, pp. 131–155, 2024
We consider the problem of attaining either the maximal increase or reduction of the robustness of a complex network by means of a bounded modification of a subset of the edge weights. We propose two novel strategies combining Krylov subspace approximations with a greedy scheme and with the limited-memory BFGS. The …
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 …
Extrapolation methods for fixed-point multilinear PageRank computations
Numerical Linear Algebra with Applications, vol. 27, pp. e2280, 2020
Nonnegative tensors arise very naturally in many applications that involve large and complex data flows. Due to the relatively small requirement in terms of memory storage and number of operations per step, the (shifted) higher-order power method is one of the most commonly used technique for the computation of …
2018
On the stability of network indices defined by means of matrix functions
SIAM Journal on Matrix Analysis and Applications, vol. 39, pp. 1521–1546, 2018
Identifying important components in a network is one of the major goals of network analysis. Popular and effective measures of importance of a node or a set of nodes are defined in terms of suitable entries of functions of matrices $f(A)$. These kinds of measures are particularly relevant as they are able to capture …
2017
Euler-Richardson method preconditioned by weakly stochastic matrix algebras: a potential contribution to Pagerank computation
Electronic Journal of Linear Algebra, vol. 32, pp. 254–272, 2017
Let $S$ be a column stochastic matrix with at least one full row. Then $S$ describes a Pagerank-like random walk since the computation of the Perron vector $x$ of $S$ can be tackled by solving a suitable M-matrix linear system $Mx = y$, where $M = I − \tau A$, $A$ is a column stochastic matrix and $\tau$ is a positive …
2016
Localization of dominant eigenpairs and planted communities by means of Frobenius inner products
Czechoslovak Mathematical Journal, vol. 66, pp. 881–893, 2016
We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$. The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$. Different choices for $X$ may be used, depending on the problem under investigation. In particular, we show …
Lower triangular Toeplitz-Ramanujan systems whose solution yields the Bernoulli numbers
Linear Algebra and its Applications, vol. 496, pp. 510–526, 2016
We prove that the Bernoulli numbers satisfy some special lower triangular Toeplitz systems of linear equations. One of these systems has a strong link with eleven Ramanujan’s linear equations satisfied by the first eleven Bernoulli numbers.
2015
Adaptive matrix algebras in unconstrained minimization
Linear Algebra and its Applications, vol. 471, pp. 544–568, 2015
In this paper we study adaptive $L(k)QN$ methods, involving special matrix algebras of low complexity, to solve general (non-structured) unconstrained minimization problems. These methods, which generalize the classical BFGS method, are based on an iterative formula which exploits, at each step, an ad hoc chosen matrix …
2013
Optimal Rank Matrix Algebras Preconditioners
Linear Algebra and its Applications, vol. 438, pp. 405–427, 2013
When a linear system $Ax = y$ is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner $P$. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. …
2011
A preconditioning approach to the PageRank computation problem
Linear Algebra and its Applications, vol. 435, pp. 2222–2246, 2011
Some spectral properties of the transition matrix of an oriented graph indicate the preconditioning of Euler-Richardson (ER) iterative scheme as a good way to compute efficiently the vertexrank vector associated with such graph. We choose the preconditioner from an algebra $\mathcal U$ of matrices, thereby obtaining an …