Understanding Image Representations by Measuring Their Equivariance and Equivalence Despite the importance of image representations such as histograms of oriented gradients and deep Convolutional Neural Networks (CNN), our theoretical understanding of them remains limited. Aimed at filling this gap, we investigate two key mathematical properties of representations: equivariance and equivalence. Equivariance studies how transformations of the input image are encoded by the representation, invariance being a special case where a transformation has no effect. Equivalence studies whether two representations, for example two different parameterizations of a CNN, two different layers, or two different CNN architectures, share the same visual information or not. A number of methods to establish these properties empirically are proposed, including introducing transformation and stitching layers in CNNs. These methods are then applied to popular representations to reveal insightful aspects of their structure, including clarifying at which layers in a CNN certain geometric invariances are achieved and how various CNN architectures differ. We identify several predictors of geometric and architectural compatibility, including the spatial resolution of the representation and the complexity and depth of the models. While the focus of the paper is theoretical, direct applications to structuredoutput regression are demonstrated too.

Unsupervised Representation Learning by Discovering Reliable Image Relations Abstract   Learning robust representations that allow to reliably establish relations between images is of paramount importance for virtually all of computer vision. Annotating the quadratic number of pairwise relations between training images is simply not feasible, while unsupervised inference is prone to noise, thus leaving the vast majority of these relations to be unreliable. To nevertheless find those relations which can be reliably utilized for learning, we follow a divideandconquer strategy: We find reliable similarities by extracting compact groups of images and reliable dissimilarities by partitioning these groups into subsets, converting the complicated overall problem into few reliable local subproblems. For each of the subsets we obtain a representation by learning a mapping to a target feature space so that their reliable relations are kept. Transitivity relations between the subsets are then exploited to consolidate the local solutions into a concerted global representation. While iterating between grouping, partitioning, and learning, we can successively use more and more reliable relations which, in turn, improves our image representation. In experiments, our approach shows stateoftheart performance on unsupervised classification on ImageNet with 46.0% and competes favorably on different transfer learning tasks on PASCAL VOC.