A Framework for Improving the Characterization Scope of Stein's Method on Riemannian Manifolds

Stein's method has been widely used to achieve distributional approximations for probability distributions defined in Euclidean spaces. Recently, techniques to extend Stein's method to manifold-valued random variables with distributions defined on the respective manifolds have been reported. However, several of these methods impose strong regularity conditions on the distributions as well as the manifolds and/or consider very special cases. In this paper, we present a novel framework for Stein's method on Riemannian manifolds using the Friedrichs extension technique applied to self-adjoint unbounded operators. This framework is applicable to a variety of conventional and unconventional situations, including but not limited to, intrinsically defined non-smooth distributions, truncated distributions on Riemannian manifolds, distributions on incomplete Riemannian manifolds, etc. Moreover, the stronger the regularity conditions imposed on the manifolds or target distributions, the stronger will be the characterization ability of our novel Stein pair, which facilitates the application of Stein's method to problem domains hitherto uncharted. We present several (non-numeric) examples illustrating the applicability of the presented theory

Horocycle Decision Boundaries for Large Margin Classification in Hyperbolic Space

Hyperbolic spaces have been quite popular in the recent past for representing hierarchically organized data. Further, several classification algorithms for data in these spaces have been proposed in the literature. These algorithms mainly use either hyperplanes or geodesics for decision boundaries in a large margin classifiers setting leading to a non-convex optimization problem. In this paper, we propose a novel large margin classifier based on horocycle (horosphere) decision boundaries that leads to a geodesically convex optimization problem that can be optimized using any Riemannian gradient descent technique guaranteeing a globally optimal solution. We present several experiments depicting the performance of our classifier

Maximizing all margins: Pushing face recognition with Kernel Plurality

We present two theses in this paper: First, performance of most existing face recognition algorithms improves if instead of the whole image, smaller patches are individually classified followed by label aggregation using voting. Second, weighted plurality voting outperforms other popular voting methods if the weights are set such that they maximize the victory margin for the winner with respect to each of the losers. Moreover, this can be done while taking higher order relationships among patches into account using kernels. We call this scheme Kernel Plurality. We verify our proposals with detailed experimental results and show that our framework with Kernel Plurality improves the performance of various face recognition algorithms beyond what has been previously reported in the literature. Furthermore, on five different benchmark datasets - Yale A, CMU PIE, MERL Dome, Extended Yale B and Multi-PIE, we show that Kernel Plurality in conjunction with recent face recognition algorithms can provide state-of-the-art results in terms of face recognition rates.Engineering and Applied Science