Frédéric Pascal

Frédéric Chazal, Laure Ferraris, Pablo Groisman et al.

The Fermat distance has been recently established as a useful tool for machine learning tasks when a natural distance is not directly available to the practitioner or to improve the results given by Euclidean distances by exploding the geometrical and statistical properties of the dataset. This distance depends on a parameter $α$ that greatly impacts the performance of subsequent tasks. Ideally, the value of $α$ should be large enough to navigate the geometric intricacies inherent to the problem. At the same, it should remain restrained enough to sidestep any deleterious ramifications stemming from noise during the process of distance estimation. We study both theoretically and through simulations how to select this parameter.

Imen Ayadi, Florent Bouchard, Frédéric Pascal

This paper deals with Elliptical Wishart distributions - which generalize the Wishart distribution - in the context of signal processing and machine learning. Two algorithms to compute the maximum likelihood estimator (MLE) are proposed: a fixed point algorithm and a Riemannian optimization method based on the derived information geometry of Elliptical Wishart distributions. The existence and uniqueness of the MLE are characterized as well as the convergence of both estimation algorithms. Statistical properties of the MLE are also investigated such as consistency, asymptotic normality and an intrinsic version of Fisher efficiency. On the statistical learning side, novel classification and clustering methods are designed. For the $t$-Wishart distribution, the performance of the MLE and statistical learning algorithms are evaluated on both simulated and real EEG and hyperspectral data, showcasing the interest of our proposed methods.

Florent Bouchard, Ammar Mian, Malik Tiomoko et al.

In this study, we consider the realm of covariance matrices in machine learning, particularly focusing on computing Fréchet means on the manifold of symmetric positive definite matrices, commonly referred to as Karcher or geometric means. Such means are leveraged in numerous machine-learning tasks. Relying on advanced statistical tools, we introduce a random matrix theory-based method that estimates Fréchet means, which is particularly beneficial when dealing with low sample support and a high number of matrices to average. Our experimental evaluation, involving both synthetic and real-world EEG and hyperspectral datasets, shows that we largely outperform state-of-the-art methods.

Jasin Machkour, Arnaud Breloy, Michael Muma et al.

Sparse principal component analysis (PCA) aims at mapping large dimensional data to a linear subspace of lower dimension. By imposing loading vectors to be sparse, it performs the double duty of dimension reduction and variable selection. Sparse PCA algorithms are usually expressed as a trade-off between explained variance and sparsity of the loading vectors (i.e., number of selected variables). As a high explained variance is not necessarily synonymous with relevant information, these methods are prone to select irrelevant variables. To overcome this issue, we propose an alternative formulation of sparse PCA driven by the false discovery rate (FDR). We then leverage the Terminating-Random Experiments (T-Rex) selector to automatically determine an FDR-controlled support of the loading vectors. A major advantage of the resulting T-Rex PCA is that no sparsity parameter tuning is required. Numerical experiments and a stock market data example demonstrate a significant performance improvement.

Florian Mouret, Alexandre Hippert-Ferrer, Frédéric Pascal et al.

This paper tackles the problem of missing data imputation for noisy and non-Gaussian data. A classical imputation method, the Expectation Maximization (EM) algorithm for Gaussian mixture models, has shown interesting properties when compared to other popular approaches such as those based on k-nearest neighbors or on multiple imputations by chained equations. However, Gaussian mixture models are known to be non-robust to heterogeneous data, which can lead to poor estimation performance when the data is contaminated by outliers or follows non-Gaussian distributions. To overcome this issue, a new EM algorithm is investigated for mixtures of elliptical distributions with the property of handling potential missing data. This paper shows that this problem reduces to the estimation of a mixture of Angular Gaussian distributions under generic assumptions (i.e., each sample is drawn from a mixture of elliptical distributions, which is possibly different for one sample to another). In that case, the complete-data likelihood associated with mixtures of elliptical distributions is well adapted to the EM framework with missing data thanks to its conditional distribution, which is shown to be a multivariate $t$-distribution. Experimental results on synthetic data demonstrate that the proposed algorithm is robust to outliers and can be used with non-Gaussian data. Furthermore, experiments conducted on real-world datasets show that this algorithm is very competitive when compared to other classical imputation methods.

Pierre Houdouin, Frédéric Pascal, Matthieu Jonckheere et al.

Linear and Quadratic Discriminant Analysis are well-known classical methods but can heavily suffer from non-Gaussian distributions and/or contaminated datasets, mainly because of the underlying Gaussian assumption that is not robust. To fill this gap, this paper presents a new robust discriminant analysis where each data point is drawn by its own arbitrary Elliptically Symmetrical (ES) distribution and its own arbitrary scale parameter. Such a model allows for possibly very heterogeneous, independent but non-identically distributed samples. After deriving a new decision rule, it is shown that maximum-likelihood parameter estimation and classification are very simple, fast and robust compared to state-of-the-art methods.

Malik Tiomoko, Romain Couillet, Frédéric Pascal

The article proposes and theoretically analyses a \emph{computationally efficient} multi-task learning (MTL) extension of popular principal component analysis (PCA)-based supervised learning schemes \cite{barshan2011supervised,bair2006prediction}. The analysis reveals that (i) by default learning may dramatically fail by suffering from \emph{negative transfer}, but that (ii) simple counter-measures on data labels avert negative transfer and necessarily result in improved performances. Supporting experiments on synthetic and real data benchmarks show that the proposed method achieves comparable performance with state-of-the-art MTL methods but at a \emph{significantly reduced computational cost}.

Alexandre Hippert-Ferrer, Ammar Mian, Florent Bouchard et al.

This paper proposes a strategy to handle missing data for the classification of electroencephalograms using covariance matrices. It relies on the observed-data likelihood within an expectation-maximization algorithm. This approach is compared to two existing state-of-the-art methods: (i) covariance matrices computed with imputed data; (ii) Riemannian averages of partially observed covariance matrix. All approaches are combined with the minimum distance to Riemannian mean classifier and applied to a classification task of two widely known paradigms of brain-computer interfaces. In addition to be applicable for a wider range of missing data scenarios, the proposed strategy generally performs better than other methods on the considered real EEG data.

Violeta Roizman, Matthieu Jonckheere, Frédéric Pascal

Though very popular, it is well known that the EM for GMM algorithm suffers from non-Gaussian distribution shapes, outliers and high-dimensionality. In this paper, we design a new robust clustering algorithm that can efficiently deal with noise and outliers in diverse data sets. As an EM-like algorithm, it is based on both estimations of clusters centers and covariances. In addition, using a semi-parametric paradigm, the method estimates an unknown scale parameter per data-point. This allows the algorithm to accommodate for heavier tails distributions and outliers without significantly loosing efficiency in various classical scenarios. We first derive and analyze the proposed algorithm in the context of elliptical distributions, showing in particular important insensitivity properties to the underlying data distributions. We then study the convergence and accuracy of the algorithm by considering first synthetic data. Then, we show that the proposed algorithm outperforms other classical unsupervised methods of the literature such as k-means, the EM for Gaussian mixture models and its recent modifications or spectral clustering when applied to real data sets as MNIST, NORB, and 20newsgroups.