Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices --especially of high-dimensional ones-- comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods.
@cite_14 , the Covariance Discriminative Learning (CDL) algorithm was proposed to embed the SPD manifold into a Euclidean space. In contrast to PGA, CDL utilizes class labels to learn a discriminative subspace using Partial Least Squares (PLS) or Linear Discriminant Analysis (LDA). However, CDL relies on mapping the SPD manifold to the space of symmetric matrices via the principal matrix logarithm. While this embedding has some nice properties ( , diffeomorphism), it can also be thought of as embedding the SPD manifold into its tangent space at the identity matrix. Therefore, although supervised, CDL also exploits data potentially distorted by the use of a single tangent space, as PGA.
@cite_14
2144093206 566612420 603167732 2085044339
We propose a novel discriminative learning approach to image set classification by modeling the image set with its natural second-order statistic, i.e. covariance matrix. Since nonsingular covariance matrices, a.k.a. symmetric positive definite (SPD) matrices, lie on a Riemannian manifold, classical learning algorithms cannot be directly utilized to classify points on the manifold. By exploring an efficient metric for the SPD matrices, i.e., Log-Euclidean Distance (LED), we derive a kernel function that explicitly maps the covariance matrix from the Riemannian manifold to a Euclidean space. With this explicit mapping, any learning method devoted to vector space can be exploited in either its linear or kernel formulation. Linear Discriminant Analysis (LDA) and Partial Least Squares (PLS) are considered in this paper for their feasibility for our specific problem. We further investigate the conventional linear subspace based set modeling technique and cast it in a unified framework with our covariance matrix based modeling. The proposed method is evaluated on two tasks: face recognition and object categorization. Extensive experimental results show not only the superiority of our method over state-of-the-art ones in both accuracy and efficiency, but also its stability to two real challenges: noisy set data and varying set size. The manifold of Symmetric Positive Definite (SPD) matrices has been successfully used for data representation in image set classification. By endowing the SPD manifold with Log-Euclidean Metric, existing methods typically work on vector-forms of SPD matrix logarithms. This however not only inevitably distorts the geometrical structure of the space of SPD matrix logarithms but also brings low efficiency especially when the dimensionality of SPD matrix is high. To overcome this limitation, we propose a novel metric learning approach to work directly on logarithms of SPD matrices. Specifically, our method aims to learn a tangent map that can directly transform the matrix logarithms from the original tangent space to a new tangent space of more discriminability. Under the tangent map framework, the novel metric learning can then be formulated as an optimization problem of seeking a Mahalanobis-like matrix, which can take the advantage of traditional metric learning techniques. Extensive evaluations on several image set classification tasks demonstrate the effectiveness of our proposed metric learning method. 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. Linear discriminant analysis (LDA) has been a popular method for dimensionality reduction, which preserves class separability. The projection vectors are commonly obtained by maximizing the between-class covariance and simultaneously minimizing the within-class covariance. LDA can be performed either in the original input space or in the reproducing kernel Hilbert space (RKHS) into which data points are mapped, which leads to kernel discriminant analysis (KDA). When the data are highly nonlinear distributed, KDA can achieve better performance than LDA. However, computing the projective functions in KDA involves eigen-decomposition of kernel matrix, which is very expensive when a large number of training samples exist. In this paper, we present a new algorithm for kernel discriminant analysis, called Spectral Regression Kernel Discriminant Analysis (SRKDA). By using spectral graph analysis, SRKDA casts discriminant analysis into a regression framework, which facilitates both efficient computation and the use of regularization techniques. Specifically, SRKDA only needs to solve a set of regularized regression problems, and there is no eigenvector computation involved, which is a huge save of computational cost. The new formulation makes it very easy to develop incremental version of the algorithm, which can fully utilize the computational results of the existing training samples. Moreover, it is easy to produce sparse projections (Sparse KDA) with a L 1-norm regularizer. Extensive experiments on spoken letter, handwritten digit image and face image data demonstrate the effectiveness and efficiency of the proposed algorithm.