arXiv subjects

Augusto Ferrante

Publications and source records attributed to Augusto Ferrante.

At least 19 recordsLinked to original sources

arXiv2026

On the Lyapunov equation with the state matrix in companion form

We study the continuous-time Lyapunov equation under the assumption that the state matrix is a Hurwitz companion matrix. The standard Lyapunov theory implies that the unique solution $X$ is positive semidefinite. Motivated by positive systems, we investigate the question of whether $X$ is entrywise nonnegative. We prove that this is the case when the companion matrix has only real eigenvalues. The proof reduces each entry of $X$ to a quadratic form associated with a class of Cauchy-like matrices whose entries are expressed in terms of elementary symmetric polynomials. The required nonnegativity then follows from the positive semidefiniteness of these Cauchy-like matrices. We also discuss a stronger total-positivity property: total nonnegativity does not hold in general, but it is recovered under an additional sign condition on the expansion of the forcing vector in the eigenbasis of $A^\top$.

eess.SY↗

arXiv2026

On the Positivity of a Class of Cauchy-Like Matrices

Let $0<\lambda_1<\cdots<\lambda_n$. Motivated by a problem related to Lyapunov equations we consider a class of Cauchy-like matrices whose elements have the form $C_{ij}=\frac{r_i(k,l)+r_j(k,l)}{\lambda_i+\lambda_j},$ where for any pair $1\le k,l\le n$, $r_i(k,l)$ are functions of $\{\lambda_1,\cdots,\lambda_n\}\setminus \lambda_i$. We show that these matrices are positive semidefinite for every pair $1\le k,l\le n$. After passing to the reciprocal variables $x_i=1/\lambda_i$, the problem is reduced by a diagonal congruence to the positivity of a two-parameter family $A_n^{(p,q)}(x)$. The proof introduces a singular augmented matrix $\mathcal H_n^{(p,q)}(x)$, proves its singularity by Cauchy-kernel generating function identities, and then proves positive semidefiniteness by induction on $n$ using the principal-minor criterion.

math.RA↗

arXiv2025

Context-Selective State Space Models: Feedback is All You Need

Transformers, powered by the attention mechanism, are the backbone of most foundation models, yet they suffer from quadratic complexity and difficulties in dealing with long-range dependencies in the input sequence. Recent work has shown that state space models (SSMs) provide a promising alternative. In this paper, we introduce the COFFEE (COntext From FEEdback) model, a novel time-varying SSM that incorporates state feedback to enable context-dependent selectivity, while still allowing for parallel implementation. This idea allows the model to regulate its dynamics based on the context described by the internal state, which embodies a compact representation of the input history. State feedback allows COFFEE to improve its ability to capture long-range dependencies: on the induction head task, it achieves near-perfect accuracy with two orders of magnitude fewer parameters and training sequences compared to S6 (the SSM of Mamba). On MNIST, COFFEE largely outperforms S6 within the same architecture, reaching 97% accuracy with only 3585 parameters. These results showcase the role of state feedback as a key mechanism for building scalable and efficient sequence models.

cs.LG↗

arXiv2025

Distributionally Robust LQG with Kullback-Leibler Ambiguity Sets

The Linear Quadratic Gaussian (LQG) controller is known to be inherently fragile to model misspecifications common in real-world situations. We consider discrete-time partially observable stochastic linear systems and provide a robustification of the standard LQG against distributional uncertainties on the process and measurement noise. Our distributionally robust formulation specifies the admissible perturbations by defining a relative entropy based ambiguity set individually for each time step along a finite-horizon trajectory, and minimizes the worst-case cost across all admissible distributions. We prove that the optimal control policy is still linear, as in standard LQG, and derive a computational scheme grounded on iterative best response that provably converges to the set of saddle points. Finally, we consider the case of endogenous uncertainty captured via decision-dependent ambiguity sets and we propose an approximation scheme based on dynamic programming.

math.OC↗

arXiv2024

Robust, positive and exact model reduction via monotone matrices

This work focuses on the problem of exact model reduction of positive linear systems, by leveraging minimal realization theory. While determining the existence of a positive reachable realization remains in general an open problem, we are able to fully characterize the cases in which the new model is obtained with non-negative reduction matrices, and hence positivity of the reduced model is robust with respect to small perturbations of the original system. The characterization is obtained by specializing monotone matrix theory to positive matrices. In addition, we provide a systematic method to construct positive reductions also when minimal ones are not available, by exploiting algebraic techniques.

eess.SY↗

arXiv2023

Distributionally Robust LQG control under Distributed Uncertainty

A new paradigm is proposed for the robustification of the LQG controller against distributional uncertainties on the noise process. Our controller optimizes the closed-loop performances in the worst possible scenario under the constraint that the noise distributional aberrance does not exceed a certain threshold limiting the relative entropy pseudo-distance between the actual noise distribution the nominal one. The main novelty is that the bounds on the distributional aberrance can be arbitrarily distributed along the whole disturbance trajectory. We discuss why this can, in principle, be a substantial advantage and we provide simulation results that substantiate such a principle.

eess.SY↗

arXiv2022

Hidden Factor estimation in Dynamic Generalized Factor Analysis Models

This paper deals with the estimation of the hidden factor in Dynamic Generalized Factor Analysis via a generalization of Kalman filtering. Asymptotic consistency is discussed and it is shown that the Kalman one-step predictor is not the right tool while the pure filter yields a consistent estimate.

math.ST↗

arXiv2022

Mean-square consistency of the $f$-truncated $\text{M}^2$-periodogram

The paper deals with the problem of estimating the M$^2$ (i.e. multivariate and multidimensional) spectral density function of a stationary random process or random field. We propose the $f$-truncated periodogram, i.e. a truncated periodogram where the truncation point is a suitable function $f$ of the sample size. We discuss the asymptotic consistency of the estimator and we provide three concrete problems that can be solved using the proposed approach. Simulation results show the effectiveness of the procedure.

math.ST↗

arXiv2021

A Robust Approach to ARMA Factor Modeling

This paper deals with the dynamic factor analysis problem for an ARMA process. To robustly estimate the number of factors, we construct a confidence region centered in a finite sample estimate of the underlying model which contains the true model with a prescribed probability. In this confidence region, the problem, formulated as a rank minimization of a suitable spectral density, is efficiently approximated via a trace norm convex relaxation. The latter is addressed by resorting to the Lagrange duality theory, which allows to prove the existence of solutions. Finally, a numerical algorithm to solve the dual problem is presented. The effectiveness of the proposed estimator is assessed through simulation studies both with synthetic and real data.

stat.ME↗

arXiv2020

Learning AR factor models

We face the factor analysis problem using a particular class of auto-regressive processes. We propose an approximate moment matching approach to estimate the number of factors as well as the parameters of the model. This algorithm alternates a step of factor analysis and a step of AR dynamics estimation. Some simulation studies show the effectiveness of the proposed estimator.

math.OC↗

arXiv2020

M$^2$-Spectral Estimation: A Flexible Approach Ensuring Rational Solutions

This paper concerns a spectral estimation problem for multivariate (i.e., vector-valued) signals defined on a multidimensional domain, abbreviated as M$^2$. The problem is posed as solving a finite number of trigonometric moment equations for a nonnegative matricial measure, which is well known as the \emph{covariance extension problem} in the literature of systems and control. This inverse problem and its various generalizations have been extensively studied in the past three decades, and they find applications in diverse fields such as modeling and system identification, signal and image processing, robust control, circuit theory, etc. In this paper, we address the challenging M$^2$ version of the problem, and elaborate on a solution technique via convex optimization with the $\tau$-divergence family. As a major contribution of this work, we show that by properly choosing the parameter of the divergence index, the optimal spectrum is a rational function, that is, the solution is a spectral density which can be represented by a finite-dimensional system, as desired in many practical applications.

math.OC↗

arXiv2020

Link Prediction: A Graphical Model Approach

We consider the problem of link prediction in networks whose edge structure may vary (sufficiently slowly) over time. This problem, with applications in many important areas including social networks, has two main variants: the first, known as positive link prediction or PLP consists in estimating the appearance of a link in the network. The second, known as negative link prediction or NLP consists in estimating the disappearance of a link in the network. We propose a data-driven approach to estimate the appearance/disappearance of edges. Our solution is based on a regularized optimization problem for which we prove existence and uniqueness of the optimal solution.

math.OC↗

arXiv2020

Data-driven Link Prediction over Graphical Models

The positive link prediction (PLP) problem is formulated in a system identification framework: we consider dynamic graphical models for auto-regressive moving-average (ARMA) Gaussian random processes. For the identification of the parameters, we model our network on two different time scales: a quicker one, over which we assume that the process representing the dynamics of the agents can be considered to be stationary, and a slower one in which the model parameters may vary. The latter accounts for the possible appearance of new edges. The identification problem is cast into an optimization framework which can be seen as a generalization of the existing methods for the identification of ARMA graphical models. We prove the existence and uniqueness of the solution of such an optimization problem and we propose a procedure to compute numerically this solution. Simulations testing the performances of our method are provided.

math.OC↗

arXiv2019

A study on Vandermonde-like polynomial matrices

A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.

math.CO↗

arXiv2019

New geometric results in eigenstructure assignment

The focus of this paper is the connection between two foundational areas of LTI systems theory: geometric control and eigenstructure assignment. In particular, we study the properties of the null-spaces of the reachability matrix pencil and of the Rosenbrock system matrix, which have been extensively used as two computational building blocks for the calculation of pole placing state feedback matrices and pole placing friends of output-nulling subspaces. Our objective is to show that the subspaces in the chains of kernels obtained in the construction of these feedback matrices interact with each other in ways that are entirely independent from the choice of eigenvalues. So far, these chains of subspaces have only been studied in the case of stationarity. In this case, it is known that these chains converge to the classic Kalman reachable subspace for the reachability matrix pencil and to the largest reachability subspace in the case of the Rosenbrock matrix, respectively. Here we are interested in showing that even before stationarity has been reached, the partial chains are linked to structural properties of the system, and are therefore independent of the closed-loop eigenvalues that we wish to assign. We further characterize these subspaces by investigating the notion of largest subspace on which it is possible to assign the closed-loop spectrum (possibly maintaining the output at zero) without resorting to non-trivial Jordan forms.

math.OC↗

arXiv2019

M$^2$-Spectral Estimation: A Relative Entropy Approach

This paper deals with M$^2$-signals, namely multivariate (or vector-valued) signals defined over a multidimensional domain. In particular, we propose an optimization technique to solve the covariance extension problem for stationary random vector fields. The multidimensional Itakura-Saito distance is employed as an optimization criterion to select the solution among the spectra satisfying a finite number of moment constraints. In order to avoid technicalities that may happen on the boundary of the feasible set, we deal with the discrete version of the problem where the multidimensional integrals are approximated by Riemann sums. The spectrum solution is also discrete, which occurs naturally when the underlying random field is periodic. We show that a solution to the discrete problem exists, is unique and depends smoothly on the problem data. Therefore, we have a well-posed problem whose solution can be tuned in a smooth manner. Finally, we have applied our theory to the target parameter estimation problem in an integrated system of automotive modules. Simulation results show that our spectral estimator has promising performance.

math.OC↗

arXiv2019

Fusion of Sensors Data in Automotive Radar Systems: A Spectral Estimation Approach

To accurately estimate locations and velocities of surrounding targets (cars) is crucial for advanced driver assistance systems based on radar sensors. In this paper we derive methods for fusing data from multiple radar sensors in order to improve the accuracy and robustness of such estimates. First we pose the target estimation problem as a multivariate multidimensional spectral estimation problem. The problem is multivariate since each radar sensor gives rise to a measurement channel. Then we investigate how the use of the cross-spectra affects target estimates. We see that the use of the magnitude of the cross-spectrum significantly improves the accuracy of the target estimates, whereas an attempt to compensate the phase lag of the cross-spectrum only gives marginal improvement. This paper may be viewed as a first step towards applying high-resolution methods that builds on multidimensional multivariate spectral estimation for sensor fusion.

eess.SP↗

arXiv2019

Robust Identification of "Sparse Plus Low-rank" Graphical Models: An Optimization Approach

Motivated by graphical models, we consider the "Sparse Plus Low-rank" decomposition of a positive definite concentration matrix -- the inverse of the covariance matrix. This is a classical problem for which a rich theory and numerical algorithms have been developed. It appears, however, that the results rapidly degrade when, as it happens in practice, the covariance matrix must be estimated from the observed data and is therefore affected by a certain degree of uncertainty. We discuss this problem and propose an alternative optimization approach that appears to be suitable to deal with robustness issues in the "Sparse Plus Low-rank" decomposition problem.The variational analysis of this optimization problem is carried over and discussed.

math.OC↗