In this paper, we aim to provide a systematic study of the relationship between Chernoff information and topological, as well as algebraic properties of the corresponding Gaussian tree graphs for the underlying graphical testing problems. We first show the relationship between Chernoff information and generalized eigenvalues of the associated covariance matrices. It is then proved that Chernoff information between two Gaussian trees sharing certain local subtree structures can be transformed into that of two smaller trees. Under our proposed grafting operations, bottleneck Gaussian trees, namely, Gaussian trees connected by one such operation, can thus be simplified into two 3-node Gaussian trees, whose topologies and edge weights are subject to the specifics of the operation. Thereafter, we provide a thorough study about how Chernoff information changes when small differences are accumulated into bigger ones via concatenated grafting operations. It is shown that the two Gaussian trees connected by more than one grafting operation may not have bigger Chernoff information than that of one grafting operation unless these grafting operations are separate and independent. At the end, we propose an optimal linear dimensional reduction method related to generalized eigenvalues.
In this paper, we are particularly interested in finding out the determining factors of Chernoff information in distinguishing Gaussian trees with the same amount of randomness, i.e. sharing the same determinant of their covariance matrices and normalized variances. The assumptions are for the ease of analysis and providing insights, as shown later in our results. Generalized eigenvalues of covariance matrices @math and @math are often considered when judging the difference between two Gaussian graphs @cite_13 @cite_15 . We tie the Chernoff information to generalized eigenvalues, which are further shown to be determined by local structure and topological differences between two graphs.
@cite_15 @cite_13
2144093206 2070773367
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. This paper presents region moments, a class of appearance descriptors based on image moments applied to a pool of image features. A careful design of the moments and the image features, makes the descriptors scale and rotation invariant, and therefore suitable for vehicle detection from aerial video, where targets appear at different scales and orientations. Region moments are linearly related to the image features. Thus, comparing descriptors by computing costly geodesic distances and non-linear classifiers can be avoided, because Euclidean geometry and linear classifiers are still effective. The descriptor computation is made efficient by designing a fast procedure based on the integral representation. An extensive comparison between region moments and the region covariance descriptors, reports theoretical, qualitative, and quantitative differences among them, with a clear advantage of the region moments, when used for detecting small image structures, such as vehicles in aerial video. The proposed descriptors hold the promise to become an effective building block in other applications.