A Bayesian Approach for Noisy Matrix Completion: Optimal Rate under General Sampling Distribution

Bayesian methods for low-rank matrix completion with noise have been shown to be very efficient computationally. While the behaviour of penalized minimization methods is well understood both from the theoretical and computational points of view in this problem, the theoretical optimality of Bayesian estimators have not been explored yet. In this paper, we propose a Bayesian estimator for matrix completion under general sampling distribution. We also provide an oracle inequality for this estimator. This inequality proves that, whatever the rank of the matrix to be estimated, our estimator reaches the minimax-optimal rate of convergence (up to a logarithmic factor). We end the paper with a short simulation study

A method of integrating correlation structures for a generalized recursive route choice model

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A reduced-rank approach to predicting multiple binary responses through machine learning

This paper investigates the problem of simultaneously predicting multiple binary responses by utilizing a shared set of covariates. Our approach incorporates machine learning techniques for binary classification, without making assumptions about the underlying observations. Instead, our focus lies on a group of predictors, aiming to identify the one that minimizes prediction error. Unlike previous studies that primarily address estimation error, we directly analyze the prediction error of our method using PAC-Bayesian bounds techniques. In this paper, we introduce a pseudo-Bayesian approach capable of handling incomplete response data. Our strategy is efficiently implemented using the Langevin Monte Carlo method. Through simulation studies and a practical application using real data, we demonstrate the effectiveness of our proposed method, producing comparable or sometimes superior results compared to the current state-of-the-art method

Numerical comparisons between Bayesian and frequentist low-rank matrix completion: estimation accuracy and uncertainty quantification

In this paper we perform a numerious numerical studies for the problem of low-rank matrix completion. We compare the Bayesain approaches and a recently introduced de-biased estimator which provides a useful way to build confidence intervals of interest. From a theoretical viewpoint, the de-biased estimator comes with a sharp minimax-optinmal rate of estimation error whereas the Bayesian approach reaches this rate with an additional logarithmic factor. Our simulation studies show originally interesting results that the de-biased estimator is just as good as the Bayesain estimators. Moreover, Bayesian approaches are much more stable and can outperform the de-biased estimator in the case of small samples. However, we also find that the length of the confidence intervals revealed by the de-biased estimator for an entry is absolutely shorter than the length of the considered credible interval. These suggest further theoretical studies on the estimation error and the concentration for Bayesian methods as they are being quite limited up to present

On a low-rank matrix single index model

In this paper, we present a theoretical study of a low-rank matrix single index model. This model is recently introduced in biostatistics however its theoretical properties on estimating together the link function and the coefficient matrix are not yet carried out. Here, we advance on using PAC-Bayesian bounds technique to provide a rigorous theoretical understanding for jointly estimation of the link function and the coefficient matrix

From bilinear regression to inductive matrix completion: a quasi-Bayesian analysis

In this paper we study the problem of bilinear regression and we further address the case when the response matrix contains missing data that referred as the problem of inductive matrix completion. We propose a quasi-Bayesian approach first to the problem of bilinear regression where a quasi-likelihood is employed. Then, we adapt this approach to the context of inductive matrix completion. Under a low-rankness assumption and leveraging PAC-Bayes bound technique, we provide statistical properties for our proposed estimators and for the quasi-posteriors. We propose a Langevin Monte Carlo method to approximately compute the proposed estimators. Some numerical studies are conducted to demonstrated our methods.Comment: arXiv admin note: substantial text overlap with arXiv:2206.0861

Simulation study on the influence of a dielectric constant gradient in the concrete on the direct wave of GPR measurements

Ground Penetrating Radar (GPR) is a non-destructive technique based on the propagation of electromagnetic wave, the propagation characteristics (speed, level of signal attenuation, ...) depend on the electromagnetic properties of the material through which it passes, where the dielectric constant of the material is one of the key parameters. Previous studies mainly based on the analysis of reflected signals to evaluate the transmission medium. Recently, the research to exploit the application of GPR direct wave (wave propagate directly in the material from the transmitting antenna to the receiving antenna) for characterization of the structural materials is becoming a matter of great interest.This paper focuses on studying on the influence of a gradient of dielectric constant in concrete on GPR direct wave, and at the same time evaluates the survey depth of GPR direct wave in concrete materials. The studying was carried out by simulation method based on GPRmax - 2D software fortwo concrete models with two gradients of dielectric constant and simultaneously with two antennas with center frequencies of 1.5 GHz and 2.6 GHz, respectively.Analysis of the obtained simulation results allows to evaluate the influence of the dielectric constant gradient on the GPR direct wave and the survey depth of the GPR direct wave.