Representation of Lipschitz Maps and Metric Coordinate Systems

[EN] Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze ¿Pietsch Theorem inspired factorizations" through subspaces of `¿ and L1, which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane¿Whitney formulas, are also given.The first author was supported by a contract of the Programa de Ayudas de Investigacion y Desarrollo (PAID-01-21), Universitat Politecnica de Valencia. The third author was supported by Grant PID2020-112759GB-I00 funded by MCIN/AEI/10.13039/501100011033.Arnau-Notari, AR.; Calabuig, JM.; Sánchez Pérez, EA. (2022). Representation of Lipschitz Maps and Metric Coordinate Systems. Mathematics. 10(20):1-23. https://doi.org/10.3390/math10203867123102

Measure-Based Extension of Continuous Functions and p-Average-Slope-Minimizing Regression

[EN] This work is inspired by some recent developments on the extension of Lipschitz real functions based on the minimization of the maximum value of the slopes of a reference set for this function. We propose a new method in which an integral p-average is optimized instead of its maximum value. We show that this is a particular case of a more general theoretical approach studied here, provided by measure-valued representations of the metric spaces involved, and a duality formula. For p = 2, explicit formulas are proved, which are also shown to be a particular case of a more general class of measure-based extensions, which we call ellipsoidal measure extensions. The Lipschitz-type boundedness properties of such extensions are shown. Examples and concrete applications are also given.The first author was supported by a contract of the Programa de Ayudas de Investigación y Desarrollo (PAID-01-21), Universitat Politècnica de València. The third author was supported by Grant PID2020-112759GB-I00 funded by MCIN/AEI /10.13039/501100011033.Arnau-Notari, AR.; Calabuig, JM.; Sánchez Pérez, EA. (2023). Measure-Based Extension of Continuous Functions and p-Average-Slope-Minimizing Regression. Axioms. 12(4):359-379. https://doi.org/10.3390/axioms1204035935937912

Approximation of Almost Diagonal Non-linear Maps by Lattice Lipschitz Operators

[EN] Lattice Lipschitz operators define a new class of nonlinear Banach-lattice-valued maps that can be written as diagonal functions with respect to a certain basis. In the n-dimensional case, such a map can be represented as a vector of size n of real-valued functions of one variable. In this paper we develop a method to approximate almost diagonal maps by means of lattice Lipschitz operators. The proposed technique is based on the approximation properties and error bounds obtained for these operators, together with a pointwise version of the interpolation of McShane and Whitney extension maps that can be applied to almost diagonal functions. In order to get the desired approximation, it is necessary to previously obtain an approximation to the set of eigenvectors of the original function. We focus on the explicit computation of error formulas and on illustrative examples to present our construction.The first author was supported by a contract of the Programa de Ayudas de Investigacion y Desarrollo (PAID-01-21), Universitat Politecnica de Valencia. This publication is part of the R & D & I project PID2020-112759GB-I00 funded by MCIN/AEI /10.13039/501100011033. This publication is part of the R & D & I project PID2022-138342NB-I00 funded by MCIN/AEI /10.13039/501100011033. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Arnau-Notari, AR.; Calabuig, JM.; Erdogan, E.; Sánchez Pérez, EA. (2024). Approximation of Almost Diagonal Non-linear Maps by Lattice Lipschitz Operators. Bulletin of the Brazilian Mathematical Society New Series. 55(1). https://doi.org/10.1007/s00574-024-00385-955

Extension procedures for lattice Lipschitz operators on Euclidean spaces

[EN] We present a new class of Lipschitz operators on Euclidean lattices that we call lattice Lipschitz maps, and we prove that the associated McShane and Whitney formulas provide the same extension result that holds for the real valued case. Essentially, these maps satisfy a (vector-valued) Lipschitz inequality involving the order of the lattice, with the peculiarity that the usual Lipschitz constant becomes a positive real function. Our main result shows that, in the case of Euclidean space, being lattice Lipschitz is equivalent to having a diagonal representation, in which the coordinate coefficients are real-valued Lipschitz functions. We also show that in the linear case the extension of a diagonalizable operator from the values in their eigenvectors coincide with the operator obtained both from the McShane and the Whitney formulae. Our work on such extension/representation formulas is intended to follow current research on the design of machine learning algorithms based on the extension of Lipschitz functions.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Arnau-Notari, AR.; Calabuig, JM.; Erdogan, E.; Sánchez Pérez, EA. (2023). Extension procedures for lattice Lipschitz operators on Euclidean spaces. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 117(2):1-16. https://doi.org/10.1007/s13398-023-01402-0116117

Enseñanza del aprendizaje por refuerzo con un sencillo ejemplo de minimización de funciones

[ES] En este trabajo se presenta una sesión a modo de taller orientada al estudiantado universitarios para que entiendan los fundamentos del aprendizaje por refuerzo (RL). Esta técnica de inteligencia artificial no es comúnmente estudiada por su dificultad, por ello se expone una simplificación del RL, que se aplica a la resolución de un problema de optimización. Además se analizará la manera de abordar el problema de optmización como un juego, puesto que este es una aplicación natural del RL.[EN] This paper presents a practical session for university students to introduce them on the fundamentals of reinforcement learning (RL). The difficulty of this technique means that it is not studied, so a simplification of RL is presented, which is applied to the solution of an optimization problem. In addition to this technique, we study how to approach the optimization problem as a game, since this is a natural application of RL.Proyectos financiados por la Universitat Politècnica de València (PAID-01-21), el Ministerio de Ciencia e Innovación (PID2020-112759GB-I00) y Polish National Agency for Strategic Partnership (BPI/PST/2021/1/00031/U/00001)Arnau Notari, AR.; García Raffi, LM.; Calabuig Rodriguez, JM.; Sánchez Pérez, EA. (2023). Enseñanza del aprendizaje por refuerzo con un sencillo ejemplo de minimización de funciones. Editorial Universitat Politècnica de València. 133-147. https://doi.org/10.4995/INRED2023.2023.1661713314