Marie Chavent

Mariette Dupuy, Marie Chavent, Remi Dubois

This paper introduces a methodology based on Denoising AutoEncoder (DAE) for missing data imputation. The proposed methodology, called mDAE hereafter, results from a modification of the loss function and a straightforward procedure for choosing the hyper-parameters. An ablation study shows on several UCI Machine Learning Repository datasets, the benefit of using this modified loss function and an overcomplete structure, in terms of Root Mean Squared Error (RMSE) of reconstruction. This numerical study is completed by comparing the mDAE methodology with eight other methods (four standard and four more recent). A criterion called Mean Distance to Best (MDB) is proposed to measure how a method performs globally well on all datasets. This criterion is defined as the mean (over the datasets) of the distances between the RMSE of the considered method and the RMSE of the best method. According to this criterion, the mDAE methodology was consistently ranked among the top methods (along with SoftImput and missForest), while the four more recent methods were systematically ranked last. The Python code of the numerical study will be available on GitHub so that results can be reproduced or generalized with other datasets and methods.

Marie Chavent, Guy Chavent

Block Principal Component Analysis (Block PCA) of a data matrix A, where loadings Z are determined by maximization of AZ 2 over unit norm orthogonal loadings, is difficult to use for the design of sparse PCA by 1 regularization, due to the difficulty of taking care of both the orthogonality constraint on loadings and the non differentiable 1 penalty. Our objective in this paper is to relax the orthogonality constraint on loadings by introducing new objective functions expvar(Y) which measure the part of the variance of the data matrix A explained by correlated components Y = AZ. So we propose first a comprehensive study of mathematical and numerical properties of expvar(Y) for two existing definitions Zou et al. [2006], Shen and Huang [2008] and four new definitions. Then we show that only two of these explained variance are fit to use as objective function in block PCA formulations for A rid of orthogonality constraints.

Marie Chavent, Vanessa Kuentz-Simonet, Amaury Labenne et al.

In this paper, we propose a Ward-like hierarchical clustering algorithm including spatial/geographical constraints. Two dissimilarity matrices $D_0$ and $D_1$ are inputted, along with a mixing parameter $α\in [0,1]$. The dissimilarities can be non-Euclidean and the weights of the observations can be non-uniform. The first matrix gives the dissimilarities in the "feature space" and the second matrix gives the dissimilarities in the "constraint space". The criterion minimized at each stage is a convex combination of the homogeneity criterion calculated with $D_0$ and the homogeneity criterion calculated with $D_1$. The idea is then to determine a value of $α$ which increases the spatial contiguity without deteriorating too much the quality of the solution based on the variables of interest i.e. those of the feature space. This procedure is illustrated on a real dataset using the R package ClustGeo.

Marie Chavent, Guy Chavent

We address the problem of defining a group sparse formulation for Principal Components Analysis (PCA) - or its equivalent formulations as Low Rank approximation or Dictionary Learning problems - which achieves a compromise between maximizing the variance explained by the components and promoting sparsity of the loadings. So we propose first a new definition of the variance explained by non necessarily orthogonal components, which is optimal in some aspect and compatible with the principal components situation. Then we use a specific regularization of this variance by the group-$\ell_{1}$ norm to define a Group Sparse Maximum Variance (GSMV) formulation of PCA. The GSMV formulation achieves our objective by construction, and has the nice property that the inner non smooth optimization problem can be solved analytically, thus reducing GSMV to the maximization of a smooth and convex function under unit norm and orthogonality constraints, which generalizes Journee et al. (2010) to group sparsity. Numerical comparison with deflation on synthetic data shows that GSMV produces steadily slightly better and more robust results for the retrieval of hidden sparse structures, and is about three times faster on these examples. Application to real data shows the interest of group sparsity for variables selection in PCA of mixed data (categorical/numerical) .