arXiv subjects

Asma Ben Saber

Publications and source records attributed to Asma Ben Saber.

2 recordsLinked to original sources

arXiv2023

A Distribution Free Truncated Kernel Ridge Regression Estimator and Related Spectral Analyses

It is well known that kernel ridge regression (KRR) is a popular nonparametric regression estimator. Nonetheless, in the presence of a large data set with size $n\gg 1,$ the KRR estimator has the drawback to require an intensive computational load. Recently, scalable KRR approaches have been proposed with the aims to reduce the computational complexity of the KRR, while maintaining its superb convergence rate. In this work, we study a new scalable KRR based approach for nonparametric regression. Our truncated kernel ridge regression (TKRR) approach is simple. It is based on substituting the full $n\times n$ random kernel or Gram matrix $B_n,$ associated with a Mercer's kernel $\mathbb K,$ by its main $n\times N$ sub-matrix $A_N,$ where usually $N \ll n.$ Also, we show that the TKRR works with $d-$dimensional random sampling data following an unknown probability law. To do so, we give a spectral analysis for the compact kernel integral operator, associated with a probability measure, different from its usual probability measure. This decay estimate is then extended to the decay of the tail of the trace of the associated random Gram matrix. A special interest is devoted to develop rules for the optimal choices of the involved truncation order $N$ and the value for regularization parameter $λ>0.$ The proposed rules are based on the behavior and the decay rate of the spectrum of the positive integral operator, associated with the kernel $\mathbb K.$ These optimal values of the parameters ensure that in terms of the empirical risk error, the TKRR and the full KRR estimators have the same optimal convergence rate. Finally, we provide the reader with some numerical simulations that illustrate the performance of our proposed TKRR estimator.

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arXiv2021

Random matrices based schemes for stable and robust nonparametric and functional regression estimators

In the first part of this work, we develop a novel scheme for solving nonparametric regression problems. That is the approximation of possibly low regular and noised functions from the knowledge of their approximate values given at some random points. Our proposed scheme is based on the use of the pseudo-inverse of a random projection matrix, combined with some specific properties of the Jacobi polynomials system, as well as some properties of positive definite random matrices. This scheme has the advantages to be stable, robust, accurate and fairly fast in terms of execution time. In particular, we provide an $L_2$ as well as an $L_2-$risk errors of our proposed nonparametric regression estimator. Moreover and unlike most of the existing nonparametric regression estimators, no extra regularization step is required by our proposed estimator. Although, this estimator is initially designed to work with random sampling set of uni-variate i.i.d. random variables following a Beta distribution, we show that it is still works for a wide range of sampling distribution laws. Moreover, we briefly describe how our estimator can be adapted in order to handle the multivariate case of random sampling sets. In the second part of this work, we extend the random pseudo-inverse scheme technique to build a stable and accurate estimator for solving linear functional regression (LFR) problems. A dyadic decomposition approach is used to construct this last stable estimator for the LFR problem. Alaso, we give an $L_2-$risk error of our proposed LFR estimator. Finally, the performance of the two proposed estimators are illustrated by various numerical simulations. In particular, a real dataset is used to illustrate the performance of our nonparametric regression estimator.

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