Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis

The geometric median covariation matrix is a robust multivariate indicator of dispersion which can be extended without any difficulty to functional data. We define estimators, based on recursive algorithms, that can be simply updated at each new observation and are able to deal rapidly with large samples of high dimensional data without being obliged to store all the data in memory. Asymptotic convergence properties of the recursive algorithms are studied under weak conditions. The computation of the principal components can also be performed online and this approach can be useful for online outlier detection. A simulation study clearly shows that this robust indicator is a competitive alternative to minimum covariance determinant when the dimension of the data is small and robust principal components analysis based on projection pursuit and spherical projections for high dimension data. An illustration on a large sample and high dimensional dataset consisting of individual TV audiences measured at a minute scale over a period of 24 hours confirms the interest of considering the robust principal components analysis based on the median covariation matrix. All studied algorithms are available in the R package Gmedian on CRAN

Online learning and fusion of orientation appearance models for robust rigid object tracking

We introduce a robust framework for learning and fusing of orientation appearance models based on both texture and depth information for rigid object tracking. Our framework fuses data obtained from a standard visual camera and dense depth maps obtained by low-cost consumer depth cameras such as the Kinect. To combine these two completely different modalities, we propose to use features that do not depend on the data representation: angles. More specifically, our framework combines image gradient orientations as extracted from intensity images with the directions of surface normals computed from dense depth fields. We propose to capture the correlations between the obtained orientation appearance models using a fusion approach motivated by the original Active Appearance Models (AAMs). To incorporate these features in a learning framework, we use a robust kernel based on the Euler representation of angles which does not require off-line training, and can be efficiently implemented online. The robustness of learning from orientation appearance models is presented both theoretically and experimentally in this work. This kernel enables us to cope with gross measurement errors, missing data as well as other typical problems such as illumination changes and occlusions. By combining the proposed models with a particle filter, the proposed framework was used for performing 2D plus 3D rigid object tracking, achieving robust performance in very difficult tracking scenarios including extreme pose variations. © 2014 Elsevier B.V. All rights reserved

he geometry of statistical efficiency

We will place certain parts of the theory of statistical efficiency into the author’s operator trigonometry (1967), thereby providing new geometrical understanding of statistical efficiency. Important earlier results of Bloomfield and Watson, Durbin and Kendall, Rao and Rao, will be so interpreted. For example, worse case relative least squares efficiency corresponds to and is achieved by the maximal turning antieigenvectors of the covariance matrix. Some little-known historical perspectives will also be exposed. The overall view will be emphasized

GrassmannOptim: An R Package for Grassmann Manifold Optimization

The optimization of a real-valued objective function f(U), where U is a p X d,p > d, semi-orthogonal matrix such that UTU=Id, and f is invariant under right orthogonal transformation of U, is often referred to as a Grassmann manifold optimization. Manifold optimization appears in a wide variety of computational problems in the applied sciences. In this article, we present GrassmannOptim, an R package for Grassmann manifold optimization. The implementation uses gradient-based algorithms and embeds a stochastic gradient method for global search. We describe the algorithms, provide some illustrative examples on the relevance of manifold optimization and finally, show some practical usages of the package