Efficient Continuous Manifold Learning for Time Series Modeling

Modeling non-Euclidean data is drawing attention along with the unprecedented successes of deep neural networks in diverse fields. In particular, symmetric positive definite (SPD) matrix is being actively studied in computer vision, signal processing, and medical image analysis, thanks to its ability to learn appropriate statistical representations. However, due to its strong constraints, it remains challenging for optimization problems or inefficient computation costs, especially, within a deep learning framework. In this paper, we propose to exploit a diffeomorphism mapping between Riemannian manifolds and a Cholesky space, by which it becomes feasible not only to efficiently solve optimization problems but also to reduce computation costs greatly. Further, in order for dynamics modeling in time series data, we devise a continuous manifold learning method by integrating a manifold ordinary differential equation and a gated recurrent neural network in a systematic manner. It is noteworthy that because of the nice parameterization of matrices in a Cholesky space, it is straightforward to train our proposed network with Riemannian geometric metrics equipped. We demonstrate through experiments that the proposed model can be efficiently and reliably trained as well as outperform existing manifold methods and state-of-the-art methods in two classification tasks: action recognition and sleep staging classification

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