Sparsity on Statistical Simplexes and Diversity in Social Ranking

We study sparsity on a statistical simplex consisting of all categorical distributions. This is different from the case in < because such a simplex is a Riemannian manifold, a curved space. A learner with sparse constraints would be likely to fall into its low-dimensional boundaries. We present a novel analysis on the statistical simplex as a manifold with boundary. We investigate the learning dynamics in between high-dimensional models in the interior of the simplex and low-dimensional models on its boundaries. We study the differentiability of the cost function and its natural gradient with respect to the Riemannian structure. We apply the proposed technique to social network analysis. Given a directed graph, the task is to rank a subset of influencer nodes. Here, sparsity means that the top-ranked nodes should present diversity in the sense of minimizing influence overlap. We present a ranking algorithm based on the natural gradient. It can scale up to graph datasets with millions of nodes. On real large networks, its top-ranked nodes are the most influential among several commonly-used techniques.