Nonparametric MixedManifold Products using Multiscale Kernel Densities We extend the core operation of nonparametric belief propagation (NBP), also known as multiscale 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 mathbbRd space only. The proposed extension generalizes to any mixture of Riemannian manifolds, provided the primary operationsxe2x80x94addition and subtractionxe2x80x94 are defined. Our motivation is primarily focused on stateestimation using nonGaussian 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 userdefined manifold mixtures.

Information Filter Occupancy Mapping using Decomposable Radial Kernels Building occupancy maps of the environment is a fundamental problem for robot autonomy. A common assumption in early work was that the occupancy states of different map elements are independent. Recently, Gaussian Process (GP) techniques were proposed to capture correlation, which is important not only for improved accuracy but also for uncertainty quantification and autonomous exploration based on the predicted occupancy of nearby unexplored areas. Despite these desirable properties, current GP mapping techniques are limited to small maps and slow inference speeds. This paper proposes an information space formulation of the GP mapping problem. If a decomposable radial kernel is evaluated over a latent grid of pseudoinput points, the resulting kernel matrix has a KroneckerproductofToeplitzmatrices structure that allows very efficient representation of the occupancy distribution. We utilize this structure to design an information filter occupancy mapping algorithm with linear time and memory complexity that still permits continuous space observations and predictions.