arXiv subjects

Miryam Gnazzo

Publications and source records attributed to Miryam Gnazzo.

8 recordsLinked to original sources

arXiv2026

Nearest Reversible Markov Chains with Sparsity Constraints: An Optimization Approach

Reversibility is a key property of Markov chains, central to algorithms such as Metropolis-Hastings and other MCMC methods. Yet many applications yield non-reversible chains, motivating the problem of approximating them by reversible ones with minimal modification. We formulate this task as a matrix nearness problem and focus on the practically relevant case of sparse transition matrices. The resulting optimization problem is a quadratic programming problem, and numerical experiments illustrate the effectiveness of the approach. This framework provides a principled way to enforce reversibility and sparsity patterns in Markov chains with applications in MCMC, computational chemistry, and data-driven modeling.

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arXiv2026

Kemeny's constant minimization for reversible Markov chains via structure-preserving perturbations

Kemeny's constant measures the efficiency of a Markov chain in traversing its states. We investigate whether structure-preserving perturbations to the transition probabilities of a reversible Markov chain can improve its connectivity while maintaining a fixed stationary distribution. Although the minimum achievable value for Kemeny's constant can be estimated, the required perturbations may be infeasible. We reformulate the problem as an optimization task, focusing on solution existence and efficient algorithms, with an emphasis on the problem of minimizing Kemeny's constant under sparsity constraints.

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arXiv2026

A Riemannian Optimization Approach for Finding the Nearest Reversible Markov Chain

We address the algorithmic problem of determining the reversible Markov chain $\tilde X$ that is closest to a given Markov chain $X$, with an identical stationary distribution. More specifically, $\tilde X$ is the reversible Markov chain with the closest transition matrix, in the Frobenius norm, to the transition matrix of $X$. To compute the transition matrix of $\tilde X$, we propose a novel approach based on Riemannian optimization. Our method introduces a modified multinomial manifold endowed with a prescribed stationary vector, while also satisfying the detailed balance conditions, all within the framework of the Fisher metric. We evaluate the performance of the proposed approach in comparison with an existing quadratic programming method and demonstrate its effectiveness through a series of synthetic experiments, as well as in the construction of a reversible Markov chain from transition count data obtained via direct estimation from a stochastic differential equation.

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arXiv2026

Structured distance to singularity as a nonlinear system of equations

In this article we study the structured distance to singularity for a nonsingular matrix $A\in\mathbb{C}^{n\times n}$, with a prescribed linear structure $\mathcal{S}$ (for instance, a sparsity pattern, or a real Toeplitz structure), i.e., the norm of the smallest perturbation $Δ\in \mathcal{S}$, such that $A + Δ$ is singular. This is an example of structured matrix nearness problem: a family of problems that arise in control and systems theory and in numerical analysis, when characterizing the robustness of a certain property of a system with respect to perturbations that are constrained to a certain structure (for example the structure of the nominal system). We start by highlighting the parallelism between two main tools which have been proposed in the literature: a gradient system approach for a functional in the eigenvalues, which requires the solution of certain low-rank matrix differential equations (see [Guglielmi, Lubich, Sicilia, SINUM 2023]), and a two-level optimization approach in which the inner linear least-squares problem is solved explicitly (see [Usevich, Markovsky, JCAM 2014] and [Gnazzo, Noferini, Nyman, Poloni, FoCM 2025]). In particular, these articles underline the remarkable property that $Δ$ is (at least generically) the orthogonal projection onto the structure $\mathcal{S}$ of a rank-1 matrix $uv^*$. This property and the parallelism suggest a new reformulation of the problem into a system of nonlinear equations in the two vector unknowns $u,v \in\mathbb{C}^n$. We study this new formulation, and propose an algorithm to solve these nonlinear equations directly with the multivariate Newton's method. We discuss how to avoid the singularity of such system of nonlinear equations, and how to ensure monotonic convergence. The resulting algorithm is faster than the existing ones for large matrices, and maintains comparable accuracy.

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arXiv2025

Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory

We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems.

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arXiv2025

On the numerical approximation of the distance to singularity for matrix-valued functions

Given a matrix-valued function $\mathcal{F}(λ)=\sum_{i=1}^d f_i(λ) A_i$, with complex matrices $A_i$ and $f_i(λ)$ entire functions for $i=1,\ldots,d$, we discuss a method for the numerical approximation of the distance to singularity of $\mathcal{F}(λ)$. The closest singular matrix-valued function $\widetilde{\mathcal{F}}(λ)$ with respect to the Frobenius norm is approximated using an iterative method. The property of singularity on the matrix-valued function is translated into a numerical constraint for a suitable minimization problem. Unlike the case of matrix polynomials, in the general setting of matrix-valued functions the main issue is that the function $\det ( \widetilde{\mathcal{F}}(λ) )$ may have an infinite number of roots. An important feature of the numerical method consists in the possibility of addressing different structures, such as sparsity patterns induced by the matrix coefficients, in which case the search of the closest singular function is restricted to the class of functions preserving the structure of the matrices.

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arXiv2025

Backward errors for multiple eigenpairs in structured and unstructured nonlinear eigenvalue problems

Given a nonlinear matrix-valued function $F(λ)$ and approximate eigenpairs $(λ_i, v_i)$, we discuss how to determine the smallest perturbation $δF$ such that $[F + δF](λ_i) v_i = 0$; we call the distance between the $F$ and $F + δF$ the backward error for this set of approximate eigenpairs. We focus on the case where $F(λ)$ is given as a linear combination of scalar functions multiplying matrix coefficients $F_i$, and the perturbation is done on the matrix coefficients. We provide inexpensive upper bounds, and a way to accurately compute the backward error by means of direct computations or through Riemannian optimization. We also discuss how the backward error can be determined when the $F_i$ have particular structures (such as symmetry, sparsity, or low-rank), and the perturbations are required to preserve them. For special cases (such as for symmetric coefficients), explicit and inexpensive formulas to compute the $δF_i$ are also given.

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arXiv2024

Approximating the closest structured singular matrix polynomial

Consider a matrix polynomial $P \left( λ\right)= A_0 + λA_1 + \ldots + λ^d A_d$, with $A_0,\ldots, A_d$ complex (or real) matrices with a certain structure. In this paper we discuss an iterative method to numerically approximate the closest structured singular matrix polynomial $\widetilde P\left( λ\right)$, using the distance induced by the Frobenius norm. An important peculiarity of the approach we propose is the possibility to include different types of structural constraints. The method also allows us to limit the perturbations to just a few matrices and also to include additional structures, such as the preservation of the sparsity pattern of one or more matrices $A_i$, and also collective-like properties, like a palindromic structure. The iterative method is based on the numerical integration of the gradient system associated with a suitable functional which quantifies the distance to singularity of a matrix polynomial.

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