Article
Mathematics, Applied
Soumia Aici, Abdelkader Frakis, Fuad Kittaneh
Summary: We introduce a new norm on C-2 x C-2, the Hilbert-Schmidt class, and study its basic properties and inequalities. As a result, new bounds for the Hilbert-Schmidt numerical radii of 2 x 2 operator matrices are derived. We also establish a connection with the classical Hilbert-Schmidt numerical radius of a single operator, and refine some existing bounds.
ACTA APPLICANDAE MATHEMATICAE
(2023)
Article
Computer Science, Information Systems
Francis Bach
Summary: Through the associated covariance operators in reproducing kernel Hilbert spaces, we analyze probability distributions. We find that the von Neumann entropy and relative entropy of these operators are closely related to the usual notions of Shannon entropy and relative entropy, and have many similar properties. They can be used together with efficient estimation algorithms from various oracles on the probability distributions. We also consider product spaces and define notions of mutual information and joint entropies for tensor product kernels, which perfectly characterize independence, but only partially conditional independence. Lastly, we show how these new notions of relative entropy lead to new upper-bounds on log partition functions, which can be utilized in conjunction with convex optimization in variational inference methods, providing a new family of probabilistic inference methods.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2023)
Article
Mathematics
David E. E. Edmunds, Jan Lang
Summary: The paper investigates the possibility of obtaining a series representation for a compact linear map T acting between Banach spaces. It is shown that this representation is possible under certain conditions on T, using the concepts of j-j-eigenfunctions and j-j-eigenvalues. Various specific cases are discussed, including T being factorized through a Hilbert space or having rapidly decaying s-numbers. The concept of p-compactness is found to be useful in this context, and examples of maps possessing this property are provided.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Computer Science, Artificial Intelligence
Tianhao Liu, Daniel Andres Diaz-Pachon, J. Sunil Rao, Jean-Eudes Dazard
Summary: Principal components analysis has long been used for dimensionality reduction. This paper demonstrates the greater importance of components with the smallest variance in mode detection. It proves that implementing petty component analysis leads to boxes of optimal volume in multivariate normal or Laplace distribution, with minimal volume compared to all possible boxes with the same dimensions and fixed probability. Experimental results show that petty components outperform their competitors in a simulation and in finding modal patterns of hand-written numbers using the MNIST database. In fact, the modes obtained with petty components produce better written digits for MNIST than those obtained with principal components.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
(2023)
Article
Computer Science, Artificial Intelligence
Zhu Li, Adrian Perez-Suay, Gustau Camps-Valls, Dino Sejdinovic
Summary: The current use of machine learning in industrial, societal, and economical activities has raised concerns about the fairness, equity, and ethics of automated decisions. This study presents a regularization approach that balances predictive accuracy with fairness in terms of statistical parity, aiming to address biases in machine learning models.
PATTERN RECOGNITION
(2022)
Article
Mathematics
Heng Chen, Di-Rong Chen, Yao Zhao
Summary: The paper introduces the Constrained Covariance (COCO) for measuring dependence between random vectors, and focuses on kernel cross-covariance operators in reproducing kernel Hilbert spaces as a method to extract nonlinear dependence, establishing learning rates for associated estimators. It bounds the squared estimation errors of empirical singular functions in kernel cross-covariance operators and provides a new bound for perturbation of singular functions of Hilbert-Schmidt operators, which is tighter than classical results. A new estimator and learning rate is proposed for normalized cross-covariance operators.
JOURNAL OF APPROXIMATION THEORY
(2021)
Article
Mathematics
Jose Luis Romero, Jordy Timo van Velthoven, Felix Voigtlaender
Summary: In this study, it is demonstrated that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the size profile of a function is fully reflected by the size profile of its sampled values. The methods utilized include a local holomorphic calculus for convolution-dominated operators, applicable to groups with possibly non-polynomial growth. When applied to the matrix coefficients of a group representation, these methods improve upon classical results on atomic decompositions and bridge the gap between abstract and concrete methods.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Automation & Control Systems
Tianhong Sheng, Bharath K. Sriperumbudur
Summary: This work explores the connection between distance measures and kernel functions in measuring conditional dependence, and finds that in certain cases, distance-based measures and kernel-based measures are equivalent.
JOURNAL OF MACHINE LEARNING RESEARCH
(2023)
Article
Automation & Control Systems
Georgios Kissas, Jacob H. Seidman, Leonardo Ferreira Guilhoto, Victor M. Preciado, George J. Pappas, Paris Perdikaris
Summary: In this paper, a novel operator learning method called LOCA is proposed, which can approximate nonlinear operators even with a small number of output function measurements in the training set by coupling attention weights and integral transforms. Empirical results show that LOCA achieves state-of-the-art accuracy and robustness in various operator learning scenarios.
JOURNAL OF MACHINE LEARNING RESEARCH
(2022)
Article
Engineering, Multidisciplinary
Nourhane Attia, Ali Akgul, Djamila Seba, Abdelkader Nour, Jihad Asad
Summary: We employ the reproducing kernel Hilbert space method to construct numerical solutions for fractional ordinary differential equations with fractal fractional derivative. The results demonstrate the effectiveness and superior performance of this method.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Mathematics, Applied
Almudena P. Marquez, Francisco Javier Garcia-Pacheco, Miriam Mengibar-Rodriguez, Alberto Sanchez-Alzola
Summary: This manuscript introduces the concepts of bounded linear operators and supporting vectors, and explores the relationship between the principal components of a matrix and its supporting vectors.
Article
Mathematics, Applied
Mihaly Kovacs, Annika Lang, Andreas Petersson
Summary: The paper derives regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain, which has important implications for stochastic partial differential equations on bounded domains and their numerical approximations.
STOCHASTIC ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Chaojun Yang, Minghua Xu
Summary: In this article, new upper and lower bounds of numerical radius and Hilbert-Schmidt numerical radius inequalities for Hilbert space operators are presented. Specifically, it is shown that if X is an element of C-2 with the Cartesian decomposition X = A+ iB, then 1/4||X||(2) +||X*||(2)parallel(2) <= 1/root 2 ω(2) ([(B2) (0) (0) (A2)]) <= ω(2)(2) (X). This result is an analog of Kittaneh in [Studia Math. 168 (2005): 73-80].
JOURNAL OF MATHEMATICAL INEQUALITIES
(2023)
Article
Mathematics, Applied
Fatemeh Shakeri, Rahim Alizadeh
Summary: In this paper, it is proven that for every Hilbert-Schmidt operator T in a Hilbert space, the inequality r(T) <= root(2 - root 2)(||T||(2) (HS) rho(T)(2)) holds, where rho(middot), r(middot), and || middot ||HS denote the spectral radius, numerical radius, and Hilbert-Schmidt norm, respectively.
COMPLEX ANALYSIS AND OPERATOR THEORY
(2023)
Article
Mathematics, Applied
Masahiro Ikeda, Isao Ishikawa, Yoshihiro Sawano
Summary: In this paper, we investigate the functions that induce bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function. Our results show that only affine transforms can achieve this in a certain class of RKHSs, which includes more general RKHSs. We establish a connection between the behavior of composition operators and the asymptotic properties of the greatest zeros of orthogonal polynomials. Additionally, we examine the compactness of composition operators and prove that bounded composition operators cannot be compact in our situation.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Statistics & Probability
Agustin Alvarez, Graciela Boente, Nadia Kudraszow
JOURNAL OF MULTIVARIATE ANALYSIS
(2019)
Article
Statistics & Probability
Ana M. Bianco, Graciela Boente, Wenceslao Gonzalez-Manteiga, Ana Perez-Gonzalez
Article
Chemistry, Multidisciplinary
Mehrnoosh Arrar, Fernando Martin Boubeta, Maria Eugenia Szretter, Mariela Sued, Leonardo Boechi, Daniela Rodriguez
JOURNAL OF COMPUTATIONAL CHEMISTRY
(2019)
Article
Statistics & Probability
Liliana Forzani, Daniela Rodriguez, Ezequiel Smucler, Mariela Sued
JOURNAL OF MULTIVARIATE ANALYSIS
(2019)
Article
Statistics & Probability
Graciela Boente, Daniela Rodriguez, Mariela Sued
JOURNAL OF MULTIVARIATE ANALYSIS
(2019)
Article
Statistics & Probability
Graciela Boente, Daniela Rodriguez, Pablo Vena
Article
Statistics & Probability
Daniela Rodriguez, Marina Valdora, Pablo Vena
Summary: Partially linear models are important tools in statistical modeling, which combine the flexibility of non-parametric models and the simple interpretation of linear models. Monotonicity constraints naturally appear in certain problems, but the current estimation methods become unreliable when atypical observations are present in the sample. This paper proposes a robust estimation method, which is applied and compared to existing methods in real data sets and Monte Carlo simulations.
COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION
(2022)
Article
Mathematics, Interdisciplinary Applications
Florencia Leonardi, Matias Lopez-Rosenfeld, Daniela Rodriguez, Magno T. F. Severino, Mariela Sued
Summary: In this work, a model selection criterion is proposed to estimate the points of independence of a random vector, decomposing the vector distribution function into independent blocks. The method, based on a general estimator of the distribution function, is applicable for various scenarios, and an efficient algorithm is proposed to approximate the estimator with good performance on simulated data.
JOURNAL OF TIME SERIES ANALYSIS
(2021)
Article
Water Resources
Melanie Meis, Maria Paula Llano, Daniela Rodriguez
Summary: This study quantifies the relationship between extreme events in river discharges in the La Plata Basin and the NINO 3.4 seasonal climate index, developing a simple yet effective model for monitoring discharges. The research aligns series and uses the copula method to fit a joint distribution, enabling the quantification of extreme event probabilities and return periods. By splitting the series into training and test datasets, predictions are generated and the model is validated.
HYDROLOGICAL SCIENCES JOURNAL-JOURNAL DES SCIENCES HYDROLOGIQUES
(2021)
Article
Statistics & Probability
Graciela Boente, Nadia L. Kudraszow
Summary: This paper provides robust estimators for the first canonical correlation and directions of random elements on Hilbert separable spaces using robust association and scale measures, combined with basis expansions and/or penalizations as a regularization tool. The resulting estimators are consistent under regularity conditions. The simulation study shows that the proposed robust method outperforms the existing classical procedure when the data are contaminated. A real data example is also presented.
Article
Statistics & Probability
Ana M. Bianco, Graciela Boente, Gonzalo Chebi
Summary: This paper focuses on robust and penalized estimation for logistic regression parameters, introducing a class of stable penalized estimators for handling atypical data. The study includes analysis of convergence rates, variable selection capability, and asymptotic distribution of estimators under different penalties, as well as comparison of classical and robust estimators' performance in the presence of outliers through numerical simulations.
Article
Water Resources
Melanie Meis, Maria Paula Llano, Daniela Rodriguez
Summary: This study used copula methods to model the distribution of the NINO 3.4 index and river streamflow pair and added the forecast of the streamflow 95% percentile as an exogenous variable in a SARIMAX model. The results showed that the SARIMAX model was more accurate in predicting extreme events compared to the SARIMA model during El Nino events.
INTERNATIONAL JOURNAL OF RIVER BASIN MANAGEMENT
(2023)
Article
Water Resources
Melanie Meis, Manuel Benjamin, Daniela Rodriguez
Summary: The relation between discharge and social economy matters is of constant interest for decision makers, as unexpected fluctuations in discharge can easily impact various sectors such as agriculture and tourism. This study proposes the novel application of the one-sided dynamical principal components (ODPC) technique to hydrology, allowing for improved modeling and prediction of daily streamflow variability. A comparison between ODPC and traditional models showed that ODPC provides some improvement in the treatment and forecasting of discharge variability.
HYDROLOGICAL SCIENCES JOURNAL
(2022)
Article
Computer Science, Interdisciplinary Applications
Graciela Boente, Alejandra Mercedes Martinez
Summary: Partially linear additive models generalize linear regression models by assuming a linear relationship on some covariates while incorporating other covariates through univariate smooth functions. Providing reliable estimators when dealing with additive components, especially in the presence of atypical data, is of practical importance motivating the need for robust procedures. A family of robust estimators for partially linear additive models combining B-splines with robust linear MM-regression estimators is proposed, showing benefits over classical approaches in Monte Carlo studies and real data analysis.
COMPUTATIONAL STATISTICS & DATA ANALYSIS
(2023)
Article
Statistics & Probability
Graciela Boente, Matias Salibian-Barrera
Summary: This paper introduces a new robust method for functional principal component analysis (FPCA) that can handle longitudinal data with limited observations per trajectory. By using local regression to estimate the covariance function, it achieves robust performance against atypical observations. Simulation results show that the proposed method outperforms existing alternatives in the presence of contaminated data, while also comparing favorably in non-outlier scenarios.
METRON-INTERNATIONAL JOURNAL OF STATISTICS
(2021)
Article
Statistics & Probability
Tsung- Lin, Wan-Lun Wang
Summary: This paper derives explicit expressions for the moments of truncated multivariate normal/independent distributions with supports confined within a hyper-rectangle. A Monte Carlo experiment is conducted to validate the proposed formulae for five selected members of the distributions.
JOURNAL OF MULTIVARIATE ANALYSIS
(2024)
Article
Statistics & Probability
Tao Qiu, Qintong Zhang, Yuanyuan Fang, Wangli Xu
Summary: This article introduces a method for testing the homogeneity of two random vectors. The method involves selecting two subspaces and projecting them onto one-dimensional spaces, using the Cramer-von Mises distance to construct the test statistic. The performance is enhanced by repeating this procedure and the effectiveness is demonstrated through numerical simulations.
JOURNAL OF MULTIVARIATE ANALYSIS
(2024)
Article
Statistics & Probability
Alfredo Alegria, Xavier Emery
Summary: This study contributes to covariance modeling by proposing new parametric families of isotropic matrix-valued functions that exhibit non-monotonic behaviors, such as hole effects and cross-dimples. The benefit of these models is demonstrated on a bivariate dataset of airborne particulate matter concentrations.
JOURNAL OF MULTIVARIATE ANALYSIS
(2024)
Article
Statistics & Probability
Kento Egashira, Kazuyoshi Yata, Makoto Aoshima
Summary: This study investigates the asymptotic properties of hierarchical clustering in different settings, including high-dimensional, low-sample-size scenarios. The results show that hierarchical clustering exhibits good asymptotic properties under practical settings for high-dimensional data. The study also extends the analysis to consider scenarios where both the dimension and sample size approach infinity, and generalizes the concept of populations in multiclass HDLSS settings.
JOURNAL OF MULTIVARIATE ANALYSIS
(2024)
Article
Statistics & Probability
Marlene Baumeister, Marc Ditzhaus, Markus Pauly
Summary: This paper introduces a more robust multivariate analysis method by using general quantiles, particularly the median, instead of the traditional mean, and applies and validates this method on various factorial designs. The effectiveness of this method is demonstrated through theoretical and simulation studies on small and moderate sample sizes.
JOURNAL OF MULTIVARIATE ANALYSIS
(2024)
Article
Statistics & Probability
Chuancun Yin, Narayanaswamy Balakrishnan
Summary: The family of multivariate skew-normal distributions has interesting properties, which also hold for a general class of skew-elliptical distributions.
JOURNAL OF MULTIVARIATE ANALYSIS
(2024)
Article
Statistics & Probability
Gaspard Bernard, Thomas Verdebout
Summary: In this paper, we address the problem of testing the relationship between the eigenvalues of a scatter matrix in an elliptical distribution. Using the Le Cam asymptotic theory, we show that the non-specification of nuisance parameters has an asymptotic cost for testing the relationship. We also propose a distribution-free signed-rank test for this problem.
JOURNAL OF MULTIVARIATE ANALYSIS
(2024)
