Given labeled data represented by a binary matrix, we consider the task to derive a Boolean matrix factorization which identifies commonalities and specifications among the classes. While existing works focus on rank-one factorizations which are either specific or common to the classes, we derive class-specific alterations from common factorizations as well. Therewith, we broaden the applicability of our new method to datasets whose class-dependencies have a more complex structure. On the basis of synthetic and real-world datasets, we show on the one hand that our method is able to filter structure which corresponds to our model assumption, and on the other hand that our model assumption is justified in real-world application. Our method is parameter-free.
For social media retrieval, introduce Joint Subspace Matrix Factorization (JSMF) @cite_9 . Focusing on the two-class setting, they assume that data points (rows of the data matrix) emerge not only from discriminative but also from common subspaces. JSMF infers for a given nonnegative data matrix and ranks @math and @math a factorization as displayed in Fig. . Multiplicative updates minimize the weighted sum of class-wise computed RSS. In Regularized JSNMF (RJSNMF), a regularization term is used to prevent that shared feature vectors swap into discriminative subspaces and vice versa @cite_2 . The arising optimization problem is solved by the method of Lagrange multipliers. Furthermore, a provisional method to determine the rank automatically is evaluated. However, this involves multiple runs of the algorithm with increasing rank of shared and discriminative subspaces, until the approximation error barely decreases. A pioneering extension to the multi-class case is provided in @cite_13 .
@cite_9 @cite_13 @cite_2
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Abstract This paper mainly focuses on dimensional reduction of fused dataset of holistic and geometrical face features vectors by solving singularity problem of linear discriminant analysis and maximizing the Fisher ratio in nonlinear subspace region with the preservation of local discriminative features. The combinational feature vector space is projected into low dimensional subspace using proposed Kernel Locality Preserving Symmetrical Weighted Fisher Discriminant Analysis (KLSWFDA) method. Matching score level fusion technique has been applied on projected subspace and combinational entire Gabor subspace is framed. Euclidean distance metric (L2) and support vector machine (SVM) classifier has been implemented to recognize and classify the expressions. Performance of proposed approach is evaluated and compared with state of art approaches. Experimental results on JAFFE, YALE and FD expression database demonstrate the effectiveness of the proposed approach. To uncover an appropriate latent subspace for data representation, in this paper we propose a novel Robust Structured Subspace Learning (RSSL) algorithm by integrating image understanding and feature learning into a joint learning framework. The learned subspace is adopted as an intermediate space to reduce the semantic gap between the low-level visual features and the high-level semantics. To guarantee the subspace to be compact and discriminative, the intrinsic geometric structure of data, and the local and global structural consistencies over labels are exploited simultaneously in the proposed algorithm. Besides, we adopt the @math -norm for the formulations of loss function and regularization respectively to make our algorithm robust to the outliers and noise. An efficient algorithm is designed to solve the proposed optimization problem. It is noted that the proposed framework is a general one which can leverage several well-known algorithms as special cases and elucidate their intrinsic relationships. To validate the effectiveness of the proposed method, extensive experiments are conducted on diversity datasets for different image understanding tasks, i.e., image tagging, clustering, and classification, and the more encouraging results are achieved compared with some state-of-the-art approaches. 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 We consider the problem of approximating a set P of η points in Rd by a j-dimensional subspace under the lp measure, in which we wish to minimize the sum of lp distances from each point of P to this subspace. More generally, the Fq (lp)-subspace approximation problem asks for a j-subspace that minimizes the sum of qth powers of lp-distances to this subspace, up to a multiplicative factor of (1 + e). We develop techniques for subspace approximation, regression, and matrix approximation that can be used to deal with massive data sets in high dimensional spaces. In particular, we develop coresets and sketches, i.e. small space representations that approximate the input point set P with respect to the subspace approximation problem. Our results are: • A dimensionality reduction method that can be applied to Fq (lp)-clustering and shape fitting problems, such as those in [8, 15]. • The first strong coreset for F1 (l2)-subspace approximation in high-dimensional spaces, i.e. of size polynomial in the dimension of the space. This coreset approximates the distances to any j-subspace (not just the optimal one). • A (1 + e)-approximation algorithm for the j-dimensional F1 (l2)-subspace approximation problem with running time ηd(j e)O(1) + (η + d)2poly(j e). • A streaming algorithm that maintains a coreset for the F1 (l2)-subspace approximation problem and uses a space of d [EQUATION] (weighted) points. • Streaming algorithms for the above problems with bounded precision in the turnstile model, i.e, when coordinates appear in an arbitrary order and undergo multiple updates. We show that bounded precision can lead to further improvements. We extend results of [7] for approximate linear regression, distances to subspace approximation, and optimal rank-j approximation, to error measures other than the Frobenius norm.