Kinetic Euclidean Distance Matrices Euclidean distance matrices (EDMs) are a major tool for localization from distances, with applications ranging from protein structure determination to global positioning and manifold learning. They are, however, static objects which serve to localize points from a snapshot of distances. If the objects move, one expects to do better by modeling the motion. In this paper, we introduce Kinetic Euclidean Distance Matrices (KEDMs)xe2x80x94a new kind of timedependent distance matrices that incorporate motion. The entries of KEDMs become functions of time, the squared timevarying distances. We study two smooth trajectory modelsxe2x80x94polynomial and bandlimited trajectoriesxe2x80x94and show that these trajectories can be reconstructed from incomplete, noisy distance observations, scattered over multiple time instants. Our main contribution is a semidefinite relaxation, inspired by similar strategies for static EDMs. Similarly to the static case, the relaxation is followed by a spectral factorization step; however, because spectral factorization of polynomial matrices is more challenging than for constant matrices, we propose a new factorization method that uses anchor measurements. Extensive numerical experiments show that KEDMs and the new semidefinite relaxation accurately reconstruct trajectories from noisy, incomplete distance data and that, in fact, motion improves rather than degrades localization if properly modeled. This makes KEDMs a promising tool for problems in geometry of dynamic points sets.

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.