Recently, some works have suggested methods to combine variational probabilistic inference with Monte Carlo sampling. One promising approach is via local optimal transport. In this approach, a gradient steepest descent method based on local optimal transport principles is formulated to transform deterministically point samples from an intermediate density to a posterior density. The local mappings that transform the intermediate densities are embedded in a reproducing kernel Hilbert space (RKHS). This variational mapping method requires the evaluation of the log-posterior density gradient and therefore the adjoint of the observational operator. In this work, we evaluate nonlinear observational mappings in the variational mapping method using two approximations that avoid the adjoint, an ensemble based approximation in which the gradient is approximated by the particle covariances in the state and observational spaces the so-called ensemble space and an RKHS approximation in which the observational mapping is embedded in an RKHS and the gradient is derived there. The approximations are evaluated for highly nonlinear observational operators and in a low-dimensional chaotic dynamical system. The RKHS approximation is shown to be highly successful and superior to the ensemble approximation.
During the last two decades there has been a growing interest in Particle Filtering (PF). However, PF suffers from two long-standing problems that are referred to as sample degeneracy and impoverishment. We are investigating methods that are particularly efficient at Particle Distribution Optimization (PDO) to fight sample degeneracy and impoverishment, with an emphasis on intelligence choices. These methods benefit from such methods as Markov Chain Monte Carlo methods, Mean-shift algorithms, artificial intelligence algorithms (e.g., Particle Swarm Optimization, Genetic Algorithm and Ant Colony Optimization), machine learning approaches (e.g., clustering, splitting and merging) and their hybrids, forming a coherent standpoint to enhance the particle filter. The working mechanism, interrelationship, pros and cons of these approaches are provided. In addition, Approaches that are effective for dealing with high-dimensionality are reviewed. While improving the filter performance in terms of accuracy, robustness and convergence, it is noted that advanced techniques employed in PF often causes additional computational requirement that will in turn sacrifice improvement obtained in real life filtering. This fact, hidden in pure simulations, deserves the attention of the users and designers of new filters.
Stein variational gradient descent (SVGD) was recently proposed as a general purpose nonparametric variational inference algorithm [Liu & Wang, NIPS 2016]: it minimizes the Kullback-Leibler divergence between the target distribution and its approximation by implementing a form of functional gradient descent on a reproducing kernel Hilbert space. In this paper, we accelerate and generalize the SVGD algorithm by including second-order information, thereby approximating a Newton-like iteration in function space. We also show how second-order information can lead to more effective choices of kernel. We observe significant computational gains over the original SVGD algorithm in multiple test cases.
Stein variational gradient descent (SVGD) was recently proposed as a general purpose nonparametric variational inference algorithm: it minimizes the Kullback–Leibler divergence between the target distribution and its approximation by implementing a form of functional gradient descent on a reproducing kernel Hilbert space [Liu & Wang, NIPS 2016]. In this paper, we accelerate and generalize the SVGD algorithm by including second-order information, thereby approximating a Newton-like iteration in function space. We also show how second-order information can lead to more effective choices of kernel. We observe significant computational gains over the original SVGD algorithm in multiple test cases.
Particle-based variational inference offers a flexible way of approximating complex posterior distributions with a set of particles. In this paper we introduce a new particle-based variational inference method based on the theory of semi-discrete optimal transport. Instead of minimizing the KL divergence between the posterior and the variational approximation, we minimize a semi-discrete optimal transport divergence. The solution of the resulting optimal transport problem provides both a particle approximation and a set of optimal transportation densities that map each particle to a segment of the posterior distribution. We approximate these transportation densities by minimizing the KL divergence between a truncated distribution and the optimal transport solution. The resulting algorithm can be interpreted as a form of ensemble variational inference where each particle is associated with a local variational approximation.