Ricci Curvature on Alexandrov spaces and Rigidity Theorems

In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.Comment: final versio

Discussion of "EQUI-energy sampler" by Kou, Zhou and Wong

Discussion of ``EQUI-energy sampler'' by Kou, Zhou and Wong [math.ST/0507080]Comment: Published at http://dx.doi.org/10.1214/009053606000000506 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Lipschitz continuity of harmonic maps between Alexandrov spaces

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally H\"older continuous. In [39], F. H. Lin proposed a challenge problem: Can the H\"older continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.Comment: We remove the assumption in the previous version that the domain space has nonnegative generalized Ricci curvature. This solves Lin's conjecture completely. To appear in Invent. Mat

The Structure of Factor Content Predictions

The last decade witnessed an explosion of research into the impact of international technology differences on the factor content of trade. Yet the literature has failed to confront two pivotal issues. First, with international technology differences and traded intermediate inputs there does not exist a Vanek-consistent definition of the factor content of trade. Restated, we do not know what we are trying to explain! We fill this gap by providing the correct definition. Second, as Helpman and Krugman (1985) showed, many models beyond Heckscher-Ohlin imply the Vanek prediction. So what model is being tested? We completely characterize the class of models being tested by providing a familiar `consumption similarity' condition that is necessary and sufficient for the Vanek prediction. We illustrate with a unique dataset containing input-output tables for 41 rich and poor countries. We find modest support for the strong version of the Vanek prediction and impressive support for weaker versions of the prediction.

CERN: Confidence-Energy Recurrent Network for Group Activity Recognition

This work is about recognizing human activities occurring in videos at distinct semantic levels, including individual actions, interactions, and group activities. The recognition is realized using a two-level hierarchy of Long Short-Term Memory (LSTM) networks, forming a feed-forward deep architecture, which can be trained end-to-end. In comparison with existing architectures of LSTMs, we make two key contributions giving the name to our approach as Confidence-Energy Recurrent Network -- CERN. First, instead of using the common softmax layer for prediction, we specify a novel energy layer (EL) for estimating the energy of our predictions. Second, rather than finding the common minimum-energy class assignment, which may be numerically unstable under uncertainty, we specify that the EL additionally computes the p-values of the solutions, and in this way estimates the most confident energy minimum. The evaluation on the Collective Activity and Volleyball datasets demonstrates: (i) advantages of our two contributions relative to the common softmax and energy-minimization formulations and (ii) a superior performance relative to the state-of-the-art approaches.Comment: Accepted to IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 201