Busemann functions and barrier functions

We show that Busemann functions on a smooth, non-compact, complete, boundaryless, connected Riemannian manifold are viscosity solutions with respect to the Hamilton-Jacobi equation determined by the Riemannian metric and consequently they are locally semi-concave with linear modulus. We also analysis the structure of singularity sets of Busemann functions. Moreover we study barrier functions, which are analogues to Mather's barrier functions in Mather theory, and provide some fundamental properties. Based on barrier functions, we could define some relations on the set of lines and thus classify them. We also discuss some initial relations with the ideal boundary of the Riemannian manifold.Comment: comments are welcome

Fixed-point Factorized Networks

In recent years, Deep Neural Networks (DNN) based methods have achieved remarkable performance in a wide range of tasks and have been among the most powerful and widely used techniques in computer vision. However, DNN-based methods are both computational-intensive and resource-consuming, which hinders the application of these methods on embedded systems like smart phones. To alleviate this problem, we introduce a novel Fixed-point Factorized Networks (FFN) for pretrained models to reduce the computational complexity as well as the storage requirement of networks. The resulting networks have only weights of -1, 0 and 1, which significantly eliminates the most resource-consuming multiply-accumulate operations (MACs). Extensive experiments on large-scale ImageNet classification task show the proposed FFN only requires one-thousandth of multiply operations with comparable accuracy