Non-Parametric Mixed-Manifold Products Using Multiscale Kernel Densities

We extend the core operation of non-parametric belief propagation (NBP), also known as multi-scale sequential Gibbs sampling, to approximate products of kernel density estimated beliefs that reside on some manifold. The original algorithm, though multidimensional, implicitly assumes the beliefs to reside on the Euclidean R-d. space only. The proposed extension generalizes to any mixture of Riemannian manifolds, provided the primary operations-addition and subtraction-are defined. Our motivation is primarily focused on state-estimation using non-Gaussian factor graphs for multimodal simultaneous localization and mapping in robotics. The paper presents the method as well as simulation and experimental results for validation. Our implementation is publicly available and allows for expansion with user-defined manifold mixtures.