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Ilaria Giulini

Publications and source records attributed to Ilaria Giulini.

8 recordsLinked to original sources
arXiv2026

Optimal minimization of an unknown function in a nonparametric multivariate regression model thanks to a dimension reduction approach

In this paper, we propose a novel approach for estimating the minimum of a smooth function and its location from observations corresponding to a multivariate regression function depending a priori on d variables but actually only on r < d active variables and corrupted by some additional noise. Our method consists of two steps: The rst one is a variable selection approach which is used for identifying the r active variables on which f depends and the second one consists in estimating the minimum of the function and its location. The estimation of the minimizers is obtained by using a projected gradient descent where the gradient is estimated using a local polynomial approximation of the regression function limited to its active variables obtained in the rst step. The estimation of the minimum is obtained by evaluating the estimator of the regression function using a local polynomial approach at the estimator of one of the minimizers previously obtained. We establish non asymptotic upper bounds for the quadratic risk of the estimators of the minimizers and of the minimum and prove that they reach the optimal rate that could be expected as if the active variables were known beforehand up to a factor smaller than a power of a logarithmic term.

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arXiv2017

Dimension-free PAC-Bayesian bounds for matrices, vectors, and linear least squares regression

This paper is focused on dimension-free PAC-Bayesian bounds, under weak polynomial moment assumptions, allowing for heavy tailed sample distributions. It covers the estimation of the mean of a vector or a matrix, with applications to least squares linear regression. Special efforts are devoted to the estimation of Gram matrices, due to their prominent role in high-dimension data analysis.

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arXiv2017

Robust dimension-free Gram operator estimates

In this paper we investigate the question of estimating the Gram operator by a robust estimator from an i.i.d. sample in a separable Hilbert space and we present uniform bounds that hold under weak moment assumptions. The approach consists in first obtaining non-asymptotic dimension-free bounds in finite-dimensional spaces using some PAC-Bayesian inequalities related to Gaussian perturbations of the parameter and then in generalizing the results in a separable Hilbert space. We show both from a theoretical point of view and with the help of some simulations that such a robust estimator improves the behavior of the classical empirical one in the case of heavy tail data distributions.

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arXiv2017

Robust Principal Component Analysis in Hilbert spaces

We propose a stable version of Principal Component Analysis (PCA) in the general framework of a separable Hilbert space. It consists in interpreting the projection on the first eigenvectors as a step function applied to the spectrum of the covariance operator and in replacing it with a smooth cut-off of the eigenvalues. We study the problem from a statistical point of view, so that we assume that we do not have direct access to the covariance operator but we have to estimate it from an i.i.d. sample. We provide some results on the quality of the approximation of our spectral cut-off in terms of the quality of the approximation of the eigenvalues of the covariance operator.

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arXiv2016

Data driven estimation of Laplace-Beltrami operator

Approximations of Laplace-Beltrami operators on manifolds through graph Lapla-cians have become popular tools in data analysis and machine learning. These discretized operators usually depend on bandwidth parameters whose tuning remains a theoretical and practical problem. In this paper, we address this problem for the unnormalized graph Laplacian by establishing an oracle inequality that opens the door to a well-founded data-driven procedure for the bandwidth selection. Our approach relies on recent results by Lacour and Massart [LM15] on the so-called Lepski's method.

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arXiv2016

Kernel Spectral Clustering

We investigate the question of studying spectral clustering in a Hilbert space where the set of points to cluster are drawn i.i.d. according to an unknown probability distribution whose support is a union of compact connected components. We modify the algorithm proposed by Ng, Jordan and Weiss in order to propose a new algorithm that automatically estimates the number of clusters and we characterize the convergence of this new algorithm in terms of convergence of Gram operators. We also give a hint of how this approach may lead to learn transformation-invariant representations in the context of image classification.

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arXiv2015

PAC-Bayesian bounds for Principal Component Analysis in Hilbert spaces

Based on some new robust estimators of the covariance matrix, we propose stable versions of Principal Component Analysis (PCA) and we qualify it independently of the dimension of the ambient space. We first provide a robust estimator of the orthogonal projector on the largest eigenvectors of the covariance matrix. The behavior of such an estimator is related to the size of the gap in the spectrum of the covariance matrix and in particular a large gap is needed in order to get a good approximation. To avoid the assumption of a large eigengap in the spectrum of the covariance matrix we propose a robust version of PCA that consists in performing a smooth cut-off of the spectrum via a Lipschitz function. We provide bounds on the approximation error in terms of the operator norm and of the Frobenius norm.

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