Beyond Gauss: Image-Set Matching on the Riemannian Manifold of PDFs

State-of-the-art image-set matching techniques typically implicitly model each image-set with a Gaussian distribution. Here, we propose to go beyond these representations and model image-sets as probability distribution functions (PDFs) using kernel density estimators. To compare and match image-sets, we exploit Csiszar f-divergences, which bear strong connections to the geodesic distance defined on the space of PDFs, i.e., the statistical manifold. Furthermore, we introduce valid positive definite kernels on the statistical manifolds, which let us make use of more powerful classification schemes to match image-sets. Finally, we introduce a supervised dimensionality reduction technique that learns a latent space where f-divergences reflect the class labels of the data. Our experiments on diverse problems, such as video-based face recognition and dynamic texture classification, evidence the benefits of our approach over the state-of-the-art image-set matching methods

Second-order Democratic Aggregation

Aggregated second-order features extracted from deep convolutional networks have been shown to be effective for texture generation, fine-grained recognition, material classification, and scene understanding. In this paper, we study a class of orderless aggregation functions designed to minimize interference or equalize contributions in the context of second-order features and we show that they can be computed just as efficiently as their first-order counterparts and they have favorable properties over aggregation by summation. Another line of work has shown that matrix power normalization after aggregation can significantly improve the generalization of second-order representations. We show that matrix power normalization implicitly equalizes contributions during aggregation thus establishing a connection between matrix normalization techniques and prior work on minimizing interference. Based on the analysis we present {\gamma}-democratic aggregators that interpolate between sum ({\gamma}=1) and democratic pooling ({\gamma}=0) outperforming both on several classification tasks. Moreover, unlike power normalization, the {\gamma}-democratic aggregations can be computed in a low dimensional space by sketching that allows the use of very high-dimensional second-order features. This results in a state-of-the-art performance on several datasets

Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces

Abstract This paper introduces a novel mathematical and computational framework, namely Log-Hilbert-Schmidt metric between positive definite operators on a Hilbert space. This is a generalization of the Log-Euclidean metric on the Riemannian manifold of positive definite matrices to the infinite-dimensional setting. The general framework is applied in particular to compute distances between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), for which we obtain explicit formulas via the corresponding Gram matrices. Empirically, we apply our formulation to the task of multi-category image classification, where each image is represented by an infinite-dimensional RKHS covariance operator. On several challenging datasets, our method significantly outperforms approaches based on covariance matrices computed directly on the original input features, including those using the Log-Euclidean metric, Stein and Jeffreys divergences, achieving new state of the art results

Bregman Divergences for Infinite Dimensional Covariance Matrices

Abstract We introduce an approach to computing and comparing Covariance Descriptors (CovDs