46 research outputs found
Rank Equalities Related to Generalized Inverses of Matrices and Their Applications
This paper is divided into two parts. In the first part, we develop a general
method for expressing ranks of matrix expressions that involve Moore-Penrose
inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose
inverses of matrices. Through this method we establish a variety of valuable
rank equalities related to generalized inverses of matrices mentioned above.
Using them, we characterize many matrix equalities in the theory of generalized
inverses of matrices and their applications. In the second part, we consider
maximal and minimal possible ranks of matrix expressions that involve variant
matrices, the fundamental work is concerning extreme ranks of the two linear
matrix expressions and . As applications,
we present a wide range of their consequences and applications in matrix
theory.Comment: 245 pages, LaTe
Balanced truncation for linear switched systems
In this paper, we present a theoretical analysis of the model reduction
algorithm for linear switched systems. This algorithm is a reminiscence of the
balanced truncation method for linear parameter varying systems. Specifically
in this paper, we provide a bound on the approximation error in L2 norm for
continuous-time and l2 norm for discrete-time linear switched systems. We
provide a system theoretic interpretation of grammians and their singular
values. Furthermore, we show that the performance of bal- anced truncation
depends only on the input-output map and not on the choice of the state-space
representation. For a class of stable discrete-time linear switched systems (so
called strongly stable systems), we define nice controllability and nice
observability grammians, which are genuinely related to reachability and
controllability of switched systems. In addition, we show that quadratic
stability and LMI estimates of the L2 and l2 gains depend only on the
input-output map.Comment: We have corrected a number of typos and inconsistencies. In addition,
we added new results in Theorem
-laminations as bases for cluster varieties for surfaces
In this paper we partially settle Fock-Goncharov's duality conjecture for
cluster varieties associated to their moduli spaces of -local systems
on a punctured surface with boundary data, when is a group
of type , namely and . Based on Kuperberg's
-webs, we introduce the notion of -laminations on
defined as certain -webs with integer weights. We
introduce coordinate systems for -laminations, and show that -laminations satisfying a congruence property are geometric realizations
of the tropical integer points of the cluster -moduli space
. Per each such -lamination, we
construct a regular function on the cluster -moduli space
. We show that these functions form a basis
of the ring of all regular functions. For a proof, we develop
quantum and classical trace maps for any triangulated bordered surface with
marked points, and state-sum formulas for them. We construct quantum versions
of the basic regular functions on . The
bases constructed in this paper are built from non-elliptic webs, hence could
be viewed as higher `bangles' bases, and the corresponding `bracelets' versions
can also be considered as direct analogs of Fock-Goncharov's and
Allegretti-Kim's bases for the - case.Comment: ver2: A major change is that the quantization is added. Some
terminology changed, hence the title changed / ver3: Relatively minor
corrections, and bibliography update / ver4: Some errors corrected, gaps
filled, subsection 5.7 added, conjectural section 6 cut down, referee's
feedbacks applied. To appear in Memoirs of the AM
SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group
Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown poses given noisy measurements of a subset of their pairwise relative transforms. Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides an exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so more than an order of magnitude faster than the Gauss-Newton-based approach that forms the basis of current state-of-the-art techniques
SE-Sync: a certifiably correct algorithm for synchronization over the special Euclidean group
Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown group elements (Formula presented.) given noisy measurements of a subset of their pairwise relative transforms (Formula presented.). Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a non-convex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation (MLE) whose minimizer provides an exact maximum-likelihood estimate so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so significantly faster than the Gauss–Newton-based approach that forms the basis of current state-of-the-art techniques
Linearizations of rational matrices
Mención Internacional en el título de doctorThis PhD thesis belongs to the area of Numerical Linear Algebra. Specifically, to
the numerical solution of the Rational Eigenvalue Problem (REP). This is a type
of eigenvalue problem associated with rational matrices, which are matrices whose
entries are rational functions. REPs appear directly from applications or as approx imations to arbitrary Nonlinear Eigenvalue Problems (NLEPs). Rational matrices
also appear in linear systems and control theory, among other applications. Nowa days, a competitive method for solving REPs is via linearization. This is due to the
fact that there exist backward stable and efficient algorithms to solve the linearized
problem, which allows to recover the information of the original rational problem.
In particular, linearizations transform the REP into a generalized eigenvalue pro blem in such a way that the pole and zero information of the corresponding rational
matrix is preserved. To recover the pole and zero information of rational matrices, it
is fundamental the notion of polynomial system matrix, introduced by Rosenbrock
in 1970, and the fact that rational matrices can always be seen as transfer functions
of polynomial system matrices.
This thesis addresses different topics regarding the problem of linearizing REPs.
On the one hand, one of the main objectives has been to develop a theory of li nearizations of rational matrices to study the properties of the linearizations that
have appeared so far in the literature in a general framework. For this purpose,
a definition of local linearization of rational matrix is introduced, by developing as
starting point the extension of Rosenbrock’s minimal polynomial system matrices to
a local scenario. This new theory of local linearizations captures and explains rigor ously the properties of all the different linearizations that have been used from the
1970’s for computing zeros, poles and eigenvalues of rational matrices. In particu lar, this theory has been applied to a number of pencils that have appeared in some
influential papers on solving numerically NLEPs through rational approximation.
On the other hand, the work has focused on the construction of linearizations
of rational matrices taking into account different aspects. In some cases, we focus
on preserving particular structures of the corresponding rational matrix in the li nearization. The structures considered are symmetric (Hermitian), skew-symmetric
(skew-Hermitian), among others. In other cases, we focus on the direct construc tion of the linearizations from the original representation of the rational matrix.
The representations considered are rational matrices expressed as the sum of their
polynomial and strictly proper parts, rational matrices written as general trans fer function matrices, and rational matrices expressed by their Laurent expansion
around the point at infinity. In addition, we describe the recovery rules of the
information of the original rational matrix from the information of the new lineari zations, including in some cases not just the zero and pole information but also the
information about the minimal indices. Finally, in this dissertation we tackle one of the most important open problems
related to linearizations of rational matrices. That is the analysis of the backward
stability for solving REPs by running a backward stable algorithm on a linearization.
On this subject, a global backward error analysis has been developed by considering
the linearizations in the family of “block Kronecker linearizations”. An analysis of
this type had not been developed before in the literature.Este trabajo ha sido desarrollado en el Departamento de
Matemáticas de la Universidad Carlos III de Madrid (UC3M)
bajo la dirección del profesor Froilán Martínez Dopico y codirección de la profesora Silvia Marcaida Bengoechea. Se contó
durante cuatro años con un contrato predoctoral FPI, referencia BES-2016-076744, asociado al proyecto ALGEBRA LINEAL NUMERICA ESTRUCTURADA PARA MATRICES CONSTANTES, POLINOMIALES Y RACIONALES,
referencia MTM2015-65798-P, del Ministerio de Economía
y Competitividad, y cuyo investigador principal fue Froilán
Martínez Dopico. Asociado a este contrato, se contó con
una ayuda para realizar parte de este trabajo durante dos es tancias internacionales de investigación. La primera estancia
de investigación se realizó del 30 de enero de 2019 hasta el
1 de marzo de 2019 en el Department of Mathematical En gineering, Université catholique de Louvain (Bélgica), bajo
la supervisión del profesor Paul Van Dooren. La segunda
estancia de investigación se realizó del 15 de septiembre de
2019 hasta el 19 de noviembre de 2019 en el Department
of Mathematical Sciences, University of Montana (EEUU),
bajo la supervisión del profesor Javier Pérez Alvaro. Dado que la entidad beneficiaria del contrato predoctoral es la
UC3M mientras que el otro codirector de tesis, la profesora
Silvia Marcaida Bengoechea, pertenece al Departamento de
Matemáticas de la Universidad del País Vasco (UPV/EHU),
el trabajo con la profesora Silvia Marcaida se reforzó mediante visitas a la UPV/EHU, financiadas por ayudas de
la RED temática de Excelencia ALAMA (Algebra Lineal, Análisis Matricial y Aplicaciones) asociadas al los proyectos
MTM2015-68805-REDT y MTM2017-90682-REDT.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Ion Zaballa Tejada.- Secretario: Fernando de Terán Vergara.- Vocal: Vanni Noferin