Building Deep Networks on Grassmann Manifolds

Learning representations on Grassmann manifolds is popular in quite a few visual recognition tasks. In order to enable deep learning on Grassmann manifolds, this paper proposes a deep network architecture by generalizing the Euclidean network paradigm to Grassmann manifolds. In particular, we design full rank mapping layers to transform input Grassmannian data to more desirable ones, exploit re-orthonormalization layers to normalize the resulting matrices, study projection pooling layers to reduce the model complexity in the Grassmannian context, and devise projection mapping layers to respect Grassmannian geometry and meanwhile achieve Euclidean forms for regular output layers. To train the Grassmann networks, we exploit a stochastic gradient descent setting on manifolds of the connection weights, and study a matrix generalization of backpropagation to update the structured data. The evaluations on three visual recognition tasks show that our Grassmann networks have clear advantages over existing Grassmann learning methods, and achieve results comparable with state-of-the-art approaches.Comment: AAAI'18 pape

Manifold-valued Image Generation with Wasserstein Generative Adversarial Nets

Generative modeling over natural images is one of the most fundamental machine learning problems. However, few modern generative models, including Wasserstein Generative Adversarial Nets (WGANs), are studied on manifold-valued images that are frequently encountered in real-world applications. To fill the gap, this paper first formulates the problem of generating manifold-valued images and exploits three typical instances: hue-saturation-value (HSV) color image generation, chromaticity-brightness (CB) color image generation, and diffusion-tensor (DT) image generation. For the proposed generative modeling problem, we then introduce a theorem of optimal transport to derive a new Wasserstein distance of data distributions on complete manifolds, enabling us to achieve a tractable objective under the WGAN framework. In addition, we recommend three benchmark datasets that are CIFAR-10 HSV/CB color images, ImageNet HSV/CB color images, UCL DT image datasets. On the three datasets, we experimentally demonstrate the proposed manifold-aware WGAN model can generate more plausible manifold-valued images than its competitors.Comment: Accepted by AAAI 201

Wasserstein Divergence for GANs

In many domains of computer vision, generative adversarial networks (GANs) have achieved great success, among which the family of Wasserstein GANs (WGANs) is considered to be state-of-the-art due to the theoretical contributions and competitive qualitative performance. However, it is very challenging to approximate the $k$-Lipschitz constraint required by the Wasserstein-1 metric~(W-met). In this paper, we propose a novel Wasserstein divergence~(W-div), which is a relaxed version of W-met and does not require the $k$-Lipschitz constraint. As a concrete application, we introduce a Wasserstein divergence objective for GANs~(WGAN-div), which can faithfully approximate W-div through optimization. Under various settings, including progressive growing training, we demonstrate the stability of the proposed WGAN-div owing to its theoretical and practical advantages over WGANs. Also, we study the quantitative and visual performance of WGAN-div on standard image synthesis benchmarks of computer vision, showing the superior performance of WGAN-div compared to the state-of-the-art methods.Comment: accepted by eccv_2018, correct minor error

Sliced Wasserstein Generative Models

In generative modeling, the Wasserstein distance (WD) has emerged as a useful metric to measure the discrepancy between generated and real data distributions. Unfortunately, it is challenging to approximate the WD of high-dimensional distributions. In contrast, the sliced Wasserstein distance (SWD) factorizes high-dimensional distributions into their multiple one-dimensional marginal distributions and is thus easier to approximate. In this paper, we introduce novel approximations of the primal and dual SWD. Instead of using a large number of random projections, as it is done by conventional SWD approximation methods, we propose to approximate SWDs with a small number of parameterized orthogonal projections in an end-to-end deep learning fashion. As concrete applications of our SWD approximations, we design two types of differentiable SWD blocks to equip modern generative frameworks---Auto-Encoders (AE) and Generative Adversarial Networks (GAN). In the experiments, we not only show the superiority of the proposed generative models on standard image synthesis benchmarks, but also demonstrate the state-of-the-art performance on challenging high resolution image and video generation in an unsupervised manner.Comment: This paper is accepted by CVPR 2019, accidentally uploaded as a new submission (arXiv:1904.05408, which has been withdrawn). The code is available at this https URL https:// github.com/musikisomorphie/swd.gi

Electromagnetic Wave Theory and Applications

Contains table of contents for Section 3, reports on six research projects and a list of publications and conference papers.Joint Services Electronics Program Contract DAAL03-89-C-0001National Science Foundation Grant ECS 86-20029Schlumberger- Doll ResearchU.S. Army Research Office Contract DAAL03 88-K-0057U.S. Navy - Office of Naval Research Contract N00014-90-J-1002National Aeronautics and Space Administration Grant NAGW-1617U.S. Navy - Office of Naval Research Grant N00014-89-J-1107National Aeronautics and Space Administration Grant NAGW-1272National Aeronautics and Space Administration Agreement 958461U.S. Army - Corps of Engineers Contract DACA39-87-K-0022U.S. Air Force - Electronic Systems Division Contract F19628-88-K-0013U.S. Navy - Office of Naval Research Grant N00014-89-J-1019Digital Equipment CorporationIBM CorporationU.S. Department of Transportation Contract DTRS-57-88-C-00078Defence Advanced Research Projects Agency Contract MDA972-90-C-002

Electromagnetic Wave Theory and Applications

Contains table of contents for Section 3, reports on three research projects and a list of publications.California Institute of Technology/Jet Propulsion Laboratory Contract 959548National Aeronautics and Space Administration Grant NAGW-1617National Aeronautics and Space Administration Grant Contract 958461U.S. Navy - Office of Naval Research Grant N00014-92-J-1616U.S. Navy - Office of Naval Research Grant N00014-92-J-4098Digital Equipment Corporation AGMT DTD 11/16/93Joint Services Electronics Program Contract DAAL03-92-C-0001Joint Services Electronics Program Grant DAAH04-95-1-0038MIT Lincoln Laboratory P.O. No. BX-5424U.S. Navy - Office of Naval Research Grant N00014-90-J-1002U.S. Navy - Office of Naval Research Grant N00014-89-J-1019DEMACO Agreement 11/15/93Federal Aviation Administration Grant 94-G-007U.S. Army Cold Regions Research and Engineering Laboratory Contract DACA89-93-K-000

Electromagnetic Wave Theory and Applications

Contains table of contents for Section 3 and reports on five research projects.U.S. Department of Transportation Contract DTRS-57-88-C-00078TTD13U.S. Department of Transportation Contract DTRS-57-88-C-00078TTD30Defense Advanced Research Projects Agency Contract MDA972-90-C-0021Digital Equipment CorporationIBM CorporationJoint Services Electronics Program Contract DAAL03-89-C-0001Joint Services Electronics Program Contract DAAL03-92-C-0001Schlumberger-Doll ResearchU.S. Navy - Office of Naval Research Grant N00014-90-J-1002U.S. Navy - Office of Naval Research Grant N00014-89-J-1019National Aeronautics and Space Administration Grant NAGW-1617National Aeronautics and Space Administration Grant 958461National Aeronautics and Space Administration Grant NAGW-1272U.S. Army Corp of Engineers Contract DACA39-87-K-0022U.S. Navy - Office of Naval Research Grant N00014-89-J-110