Douglas-Rachford Algorithm
The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. However, the behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. More than a decade ago, however, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and it is weak cluster points solve a best approximation problem. In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Several examples illustrate the results. …

Distance Correlation
In statistics and in probability theory, distance correlation is a measure of statistical dependence between two random variables or two random vectors of arbitrary, not necessarily equal dimension. An important property is that this measure of dependence is zero if and only if the random variables are statistically independent. This measure is derived from a number of other quantities that are used in its specification, specifically: distance variance, distance standard deviation and distance covariance. These take the same roles as the ordinary moments with corresponding names in the specification of the Pearson product-moment correlation coefficient.
These distance-based measures can be put into an indirect relationship to the ordinary moments by an alternative formulation (described below) using ideas related to Brownian motion, and this has led to the use of names such as Brownian covariance and Brownian distance covariance. …

Neural Reasoner
We propose Neural Reasoner, a framework for neural network-based reasoning over natural language sentences. Given a question, Neural Reasoner can infer over multiple supporting facts and find an answer to the question in specific forms. Neural Reasoner has 1) a specific interaction-pooling mechanism, allowing it to examine multiple facts, and 2) a deep architecture, allowing it to model the complicated logical relations in reasoning tasks. Assuming no particular structure exists in the question and facts, Neural Reasoner is able to accommodate different types of reasoning and different forms of language expressions. Despite the model complexity, Neural Reasoner can still be trained effectively in an end-to-end manner. Our empirical studies show that Neural Reasoner can outperform existing neural reasoning systems with remarkable margins on two difficult artificial tasks (Positional Reasoning and Path Finding) proposed in. For example, it improves the accuracy on Path Finding(10K) from 33.4% to over 98%. …