Convolutional Neural Networks (CNNs) have been successfully applied to many computer vision tasks, such as image classification. By performing linear combinations and element-wise nonlinear operations, these networks can be thought of as extracting solely first-order information from an input image. In the past, however, second-order statistics computed from handcrafted features, e.g., covariances, have proven highly effective in diverse recognition tasks. In this paper, we introduce a novel class of CNNs that exploit second-order statistics. To this end, we design a series of new layers that (i) extract a covariance matrix from convolutional activations, (ii) compute a parametric, second-order transformation of a matrix, and (iii) perform a parametric vectorization of a matrix. These operations can be assembled to form a Covariance Descriptor Unit (CDU), which replaces the fully-connected layers of standard CNNs. Our experiments demonstrate the benefits of our new architecture, which outperform the first-order CNNs, while relying on up to 90 fewer parameters.
Covariance Descriptors and Learning. To the best of our knowledge, @cite_28 , and its log-Euclidean metric learning extension @cite_33 , can be thought of as the first attempts to learn RCDs. This, however, was achieved by reducing the dimensionality of input RCDs, and thus has limited learning power. In a work concurrent to ours @cite_22 , the framework of @cite_28 was extended to learning multiple transformations of input RCDs. This approach, however, still relied on RCDs as input. By contrast, here, we introduce an end-to-end learning strategy. As discussed later, this requires special care to transition from the convolutional activations to the covariance matrix, and, as evidenced by our experiments, significantly outperforms the approach of @cite_22 .
@cite_28 @cite_22 @cite_33
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Metric learning has been shown to be highly effective to improve the performance of nearest neighbor classification. In this paper, we address the problem of metric learning for Symmetric Positive Definite (SPD) matrices such as covariance matrices, which arise in many real-world applications. Naively using standard Mahalanobis metric learning methods under the Euclidean geometry for SPD matrices is not appropriate, because the difference of SPD matrices can be a non-SPD matrix and thus the obtained solution can be uninterpretable. To cope with this problem, we propose to use a properly parameterized LogEuclidean distance and optimize the metric with respect to kernel-target alignment, which is a supervised criterion for kernel learning. Then the resulting non-trivial optimization problem is solved by utilizing the Riemannian geometry. Finally, we experimentally demonstrate the usefulness of our LogEuclidean metric learning algorithm on real-world classification tasks for EEG signals and texture patches. By stacking layers of convolution and nonlinearity, convolutional networks (ConvNets) effectively learn from lowlevel to high-level features and discriminative representations. Since the end goal of large-scale recognition is to delineate complex boundaries of thousands of classes, adequate exploration of feature distributions is important for realizing full potentials of ConvNets. However, state-of-theart works concentrate only on deeper or wider architecture design, while rarely exploring feature statistics higher than first-order. We take a step towards addressing this problem. Our method consists in covariance pooling, instead of the most commonly used first-order pooling, of highlevel convolutional features. The main challenges involved are robust covariance estimation given a small sample of large-dimensional features and usage of the manifold structure of covariance matrices. To address these challenges, we present a Matrix Power Normalized Covariance (MPNCOV) method. We develop forward and backward propagation formulas regarding the nonlinear matrix functions such that MPN-COV can be trained end-to-end. In addition, we analyze both qualitatively and quantitatively its advantage over the well-known Log-Euclidean metric. On the ImageNet 2012 validation set, by combining MPN-COV we achieve over 4 , 3 and 2.5 gains for AlexNet, VGG-M and VGG-16, respectively; integration of MPN-COV into 50-layer ResNet outperforms ResNet-101 and is comparable to ResNet-152. The source code will be available on the project page: http: www.peihuali.org MPN-COV. Many learning algorithms use a metric defined over the input space as a principal tool, and their performance critically depends on the quality of this metric. We address the problem of learning metrics using side-information in the form of equivalence constraints. Unlike labels, we demonstrate that this type of side-information can sometimes be automatically obtained without the need of human intervention. We show how such side-information can be used to modify the representation of the data, leading to improved clustering and classification.Specifically, we present the Relevant Component Analysis (RCA) algorithm, which is a simple and efficient algorithm for learning a Mahalanobis metric. We show that RCA is the solution of an interesting optimization problem, founded on an information theoretic basis. If dimensionality reduction is allowed within RCA, we show that it is optimally accomplished by a version of Fisher's linear discriminant that uses constraints. Moreover, under certain Gaussian assumptions, RCA can be viewed as a Maximum Likelihood estimation of the within class covariance matrix. We conclude with extensive empirical evaluations of RCA, showing its advantage over alternative methods. In this paper, we propose a new approach called Deep LogCORAL for unsupervised visual domain adaptation. Our work builds on the recently proposed Deep CORAL method, which aims to train a convolutional neural network and simultaneously minimize the Euclidean distance of convariance matrices between the source and target domains. By observing that the second order statistical information (i.e. the covariance matrix) lies on a Riemannian manifold, we propose to use the Riemannian distance, approximated by Log-Euclidean distance, to replace the naive Euclidean distance in Deep CORAL. We also consider first-order information, and minimize the distance of mean vectors between two domains. We build an end-to-end model, in which we minimize both the classification loss, and the domain difference based on the first-order and second-order information between two domains. Our experiments on the benchmark Office dataset demonstrates the improvements of our newly proposed Deep LogCORAL approach over the Deep CORAL method, as well as the further improvement when optimizing both orders of information.