Minimization of the @math norm, which can be viewed as approximately solving the non-convex least median estimation problem, is a powerful method for outlier removal and hence robust regression. However, current techniques for solving the problem at the heart of @math norm minimization are slow, and therefore cannot scale to large problems. A new method for the minimization of the @math norm is presented here, which provides a speedup of multiple orders of magnitude for data with high dimension. This method, termed Fast @math Minimization, allows robust regression to be applied to a class of problems which were previously inaccessible. It is shown how the @math norm minimization problem can be broken up into smaller sub-problems, which can then be solved extremely efficiently. Experimental results demonstrate the radical reduction in computation time, along with robustness against large numbers of outliers in a few model-fitting problems.
Robust statistical techniques, including the aforementioned robust regression and outlier removal methods, can significantly improve the performance of their classic counterparts. However, they have rarely been applied in image analysis field, to problems such as visual recognition, due to their computational expense. The M-estimator method is utilized in @cite_24 for face recognition and achieved high accuracy even when illumination change and pixel corruption were present. In @cite_40 , the authors propose a theoretical framework combining reconstructive and discriminative subspace methods for robust classification and regression. This framework acts on subsets of pixels in images to detect outliers.
@cite_24 @cite_40
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Linear subspace methods that provide sufficient reconstruction of the data, such as PCA, offer an efficient way of dealing with missing pixels, outliers, and occlusions that often appear in the visual data. Discriminative methods, such as LDA, which, on the other hand, are better suited for classification tasks, are highly sensitive to corrupted data. We present a theoretical framework for achieving the best of both types of methods: an approach that combines the discrimination power of discriminative methods with the reconstruction property of reconstructive methods which enables one to work on subsets of pixels in images to efficiently detect and reject the outliers. The proposed approach is therefore capable of robust classification with a high-breakdown point. We also show that subspace methods, such as CCA, which are used for solving regression tasks, can be treated in a similar manner. The theoretical results are demonstrated on several computer vision tasks showing that the proposed approach significantly outperforms the standard discriminative methods in the case of missing pixels and images containing occlusions and outliers. In this paper, two supervised methods for enhancing the classification accuracy of the Nonnegative Matrix Factorization (NMF) algorithm are presented. The idea is to extend the NMF algorithm in order to extract features that enforce not only the spatial locality, but also the separability between classes in a discriminant manner. The first method employs discriminant analysis in the features derived from NMF. In this way, a two-phase discriminant feature extraction procedure is implemented, namely NMF plus Linear Discriminant Analysis (LDA). The second method incorporates the discriminant constraints inside the NMF decomposition. Thus, a decomposition of a face to its discriminant parts is obtained and new update rules for both the weights and the basis images are derived. The introduced methods have been applied to the problem of frontal face verification using the well-known XM2VTS database. Both methods greatly enhance the performance of NMF for frontal face verification Principal Component Analysis and Fisher Linear Discriminant methods have demonstrated their success in face detection, recognition, and tracking. The representation in these subspace methods is based on second order statistics of the image set, and does not address higher order statistical dependencies such as the relationships among three or more pixels. Recently Higher Order Statistics and Independent Component Analysis (ICA) have been used as informative low dimensional representations for visual recognition. In this paper, we investigate the use of Kernel Principal Component Analysis and Kernel Fisher Linear Discriminant for learning low dimensional representations for face recognition, which we call Kernel Eigenface and Kernel Fisherface methods. While Eigenface and Fisherface methods aim to find projection directions based on the second order correlation of samples, Kernel Eigenface and Kernel Fisherface methods provide generalizations which take higher order correlations into account. We compare the performance of kernel methods with Eigenface, Fisherface and ICA-based methods for face recognition with variation in pose, scale, lighting and expression. Experimental results show that kernel methods provide better representations and achieve lower error rates for face recognition. Convolutional Neural Networks (ConvNets) have successfully contributed to improve the accuracy of regression-based methods for computer vision tasks such as human pose estimation, landmark localization, and object detection. The network optimization has been usually performed with L2 loss and without considering the impact of outliers on the training process, where an outlier in this context is defined by a sample estimation that lies at an abnormal distance from the other training sample estimations in the objective space. In this work, we propose a regression model with ConvNets that achieves robustness to such outliers by minimizing Tukey's biweight function, an M-estimator robust to outliers, as the loss function for the ConvNet. In addition to the robust loss, we introduce a coarse-to-fine model, which processes input images of progressively higher resolutions for improving the accuracy of the regressed values. In our experiments, we demonstrate faster convergence and better generalization of our robust loss function for the tasks of human pose estimation and age estimation from face images. We also show that the combination of the robust loss function with the coarse-to-fine model produces comparable or better results than current state-of-the-art approaches in four publicly available human pose estimation datasets.