Supervised dimensionality reduction has emerged as an important theme in the last decade. Despite the plethora of models and formulations, there is a lack of a simple model which aims to project the set of patterns into a space defined by the classes (or categories). To this end, we set up a model in which each class is represented as a 1D subspace of the vector space formed by the features. Assuming the set of classes does not exceed the cardinality of the features, the model results in multi-class supervised learning in which the features of each class are projected into the class subspace. Class discrimination is automatically guaranteed via the imposition of orthogonality of the 1D class sub-spaces. The resulting optimization problem - formulated as the minimization of a sum of quadratic functions on a Stiefel manifold - while being non-convex (due to the constraints), nevertheless has a structure for which we can identify when we have reached a global minimum. After formulating a version with standard inner products, we extend the formulation to reproducing kernel Hilbert spaces in a straightforward manner. The optimization approach also extends in a similar fashion to the kernel version. Results and comparisons with the multi-class Fisher linear (and kernel) discriminants and principal component analysis (linear and kernel) showcase the relative merits of this approach to dimensionality reduction.
From the above, the modest goal of the present work should be clear. We seek to project the input feature vectors to a category space a subspace formed by category basis vectors. The multi-class FLD falls short of this goal since the number of projected dimensions is one less than the number of classes. The multi-class (and more recently multi-label) SVM @cite_24 literature is fragmented due to lack of agreement regarding the core issue of multi-class discrimination. The varieties of supervised PCA do not begin by clearly formulating a criterion for category space projection. Variants such as CCA @cite_19 @cite_15 , PLS @cite_29 and structured SVM's @cite_34 while attempting to add structure to the categories do not go as far as the present work in attempting to fit a category subspace. Kernel variants of the above also do not touch the basic issue addressed in the present work. Nonlinear (and manifold learning-based) dimensionality reduction techniques @cite_18 @cite_5 @cite_22 are unsupervised and therefore do not qualify.
@cite_18 @cite_22 @cite_29 @cite_24 @cite_19 @cite_5 @cite_15 @cite_34
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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. Subspace learning techniques have been extensively used for dimensionality reduction (DR) in many pattern classification problem domains. Recently, methods like Subclass Discriminant Analysis (SDA) and Clustering-based Discriminant Analysis (CDA), which use subclass information for the discrimination between the data classes, have attracted much attention. In parallel, important work has been accomplished on Graph Embedding (GE), which is a general framework unifying several subspace learning techniques. In this paper, GE has been extended in order to integrate subclass discriminant information resulting to the novel Subclass Graph Embedding (SGE) framework, which is the main contribution of our work. It is proven that SGE encapsulates a diversity of both supervised and unsupervised unimodal methods like Locality Preserving Projections (LPP), Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA). The theoretical link of SDA and CDA methods with SGE is also established. Along these lines, it is shown that SGE comprises a generalization of the typical GE framework including subclass DR methods. Moreover, it allows for an easy utilization of kernels for confronting non-linearly separable data. Employing SGE, in this paper a novel DR algorithm, which uses subclass discriminant information, called Subclass Marginal Fisher Analysis (SMFA) has been proposed. Through a series of experiments on various real-world datasets, it is shown that SMFA outperforms in most of the cases the state-of-the-art demonstrating the efficacy and power of SGE as a platform to develop new methods. HighlightsGraph Embedding is extended in order to integrate subclass informationThe novel Subclass Graph Embedding framework is proposed.The kernelized version of the new framework is presentedSubclass Graph Embedding encapsulates various subspace learning methods.A novel Subclass Marginal Fisher Analysis method is proposed. In this paper, we focus on improving the proposal classification stage in the object detection task and present implicit negative sub-categorization and sink diversion to lift the performance by strengthening loss function in this stage. First, based on the observation that the “background” class is generally very diverse and thus challenging to be handled as a single indiscriminative class in existing state-of-the-art methods, we propose to divide the background category into multiple implicit sub-categories to explicitly differentiate diverse patterns within it. Second, since the ground truth class inevitably has low-value probability scores for certain images, we propose to add a “sink” class and divert the probabilities of wrong classes to this class when necessary, such that the ground truth label will still have a higher probability than other wrong classes even though it has low probability output. Additionally, we propose to use dilated convolution, which is widely used in the semantic segmentation task, for efficient and valuable context information extraction. Extensive experiments on PASCAL VOC 2007 and 2012 data sets show that our proposed methods based on faster R-CNN implementation can achieve state-of-the-art mAPs, i.e., 84.1 , 82.6 , respectively, and obtain 2.5 improvement on ILSVRC DET compared with that of ResNet. We derive a least-squares formulation for MDDMp technique.A novel multi-label feature extraction algorithm is proposed.Our algorithm maximizes both feature variance and feature-label dependence.Experiments show that our algorithm is a competitive candidate. Dimensionality reduction is an important pre-processing procedure for multi-label classification to mitigate the possible effect of dimensionality curse, which is divided into feature extraction and selection. Principal component analysis (PCA) and multi-label dimensionality reduction via dependence maximization (MDDM) represent two mainstream feature extraction techniques for unsupervised and supervised paradigms. They produce many small and a few large positive eigenvalues respectively, which could deteriorate the classification performance due to an improper number of projection directions. It has been proved that PCA proposed primarily via maximizing feature variance is associated with a least-squares formulation. In this paper, we prove that MDDM with orthonormal projection directions also falls into the least-squares framework, which originally maximizes Hilbert-Schmidt independence criterion (HSIC). Then we propose a novel multi-label feature extraction method to integrate two least-squares formulae through a linear combination, which maximizes both feature variance and feature-label dependence simultaneously and thus results in a proper number of positive eigenvalues. Experimental results on eight data sets show that our proposed method can achieve a better performance, compared with other seven state-of-the-art multi-label feature extraction algorithms.