By always mapping data from lower dimensional space into higher or even infinite dimensional space, kernel k-means is able to organize data into groups when data of different clusters are not linearly separable. However, kernel k-means incurs the large scale computation due to the representation theorem, i.e. keeping an extremely large kernel matrix in memory when using popular Gaussianand spatial pyramid matching kernels, which largely limits its use for processing large scale data. Also, existing kernel clustering can be overfitted by outliers as well. In this paper, we introduce an Euler clustering, which can not only maintain the benefit of nonlinear modeling using kernel function but also significantly solve the large scale computational problem in kernel-based clustering. This is realized by incorporating Euler kernel. Euler kernel is relying on a nonlinear and robust cosine metric that is less sensitive to outliers. More important it intrinsically induces an empirical map which maps data onto a complex space of the same dimension. Euler clustering takes these advantages to measure the similarity between data in a robust way without increasing the dimensionality of data, and thus solves the large scale problem in kernel k-means. We evaluate Euler clustering and show its superiority against related methods on five publicly available datasets.
This paper proposes a novel robust sparse representation method, called the two-stage sparse representation (TSR), for robust recognition on a large-scale database. Based on the divide and conquer strategy, TSR divides the procedure of robust recognition into outlier detection stage and recognition stage. In the first stage, a weighted linear regression is used to learn a metric in which noise and outliers in image pixels are detected. In the second stage, based on the learnt metric, the large-scale dataset is firstly filtered into a small set according to the nearest neighbor criterion. Then a sparse representation is computed by the non-negative least squares technique. The sparse solution is unique and can be optimized efficiently. The extensive numerical experiments on several public databases demonstrate that the proposed TSR approach generally obtains better classification accuracy than the state of the art Sparse Representation Classification (SRC). At the same time, by using the TSR, a significant reduction of computational cost is reached by over fifty times in comparison with the SRC, which enables the TSR to be deployed more suitably for large-scale dataset.
Most existing robust principal component analysis (PCA) involve mean estimation for extracting low-dimensional representation. However, they do not get the optimal mean for real data, which include outliers, under the different robust distances metric learning, such as L 1 -norm and L 2,1 -norm. This affects the robustness of algorithms. Motivated by the fact that the variance of data can be characterized by the variation between each pair of data, we propose a novel robust formulation for PCA. It avoids computing the mean of data in the criterion function. Our method employs L 2 ,p-norm as the distance metric to measure the variation in the criterion function and aims to seek the projection matrix that maximizes the sum of variation between each pair of the projected data. Both theoretical analysis and experimental results demonstrate that our methods are efficient and superior to most existing robust methods for data reconstruction.
Li, Yong (Chinese Academy of Sciences) | Liu, Jing (Chinese Academy of Sciences) | Li, Zechao (Nanjing University of Science and Technology) | Zhang, Yangmuzi (University of Maryland, College Park) | Lu, Hanqing (Chinese Academy of Sciences) | Ma, Songde (Chinese Academy of Sciences)
Face recognition has been widely studied due to its importance in various applications. However, the case that both training images and testing images are corrupted is not well addressed. Motivated by the success of low-rank matrix recovery, we propose a novel semi-supervised low-rank matrix recovery algorithm for robust face recognition. The proposed method can learn robust discriminative representations for both training images and testing images simultaneously by exploiting the classwise block-diagonal structure. Specifically, low-rank matrix approximation can handle the possible contamination of data. Moreover, the classwise block-diagonal structure is exploited to promote discrimination of representations for robust recognition. The above issues are formulated into a unified objective function and we design an efficient optimization procedure based on augmented Lagrange multiplier method to solve it. Extensive experiments on three public databases are performed to validate the effectiveness of our approach. The strong identification capability of representations with block-diagonal structure is verified.
Sparse representation based classification (SRC) has gained great success in image recognition. Motivated by the fact that kernel trick can capture the nonlinear similarity of features, which may help improve the separability and margin between nearby data points, we propose Euler SRC for image classification, which is essentially the SRC with Euler sparse representation. To be specific, it first maps the images into the complex space by Euler representation, which has a negligible effect for outliers and illumination, and then performs complex SRC with Euler representation. The major advantage of our method is that Euler representation is explicit with no increase of the image space dimensionality, thereby enabling this technique to be easily deployed in real applications. To solve Euler SRC, we present an efficient algorithm, which is fast and has good convergence. Extensive experimental results illustrate that Euler SRC outperforms traditional SRC and achieves better performance for image classification.