Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples

We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family of normal mixtures. We compare this method to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and we compare the L2 mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We prove that for a simple choice of the mixture manifold the L2 mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence does not hold and the L2 mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples.Comment: Updated and expanded version published in the Journal reference below. Preprint updates: January 2016 (v3) added projection of Zakai Equation and difference with projection of Kushner-Stratonovich (section 4.1). August 2014 (v2) added Galerkin equivalence proof (Section 5) to the March 2013 (v1) versio

Langevin and Hamiltonian based Sequential MCMC for Efficient Bayesian Filtering in High-dimensional Spaces

Nonlinear non-Gaussian state-space models arise in numerous applications in statistics and signal processing. In this context, one of the most successful and popular approximation techniques is the Sequential Monte Carlo (SMC) algorithm, also known as particle filtering. Nevertheless, this method tends to be inefficient when applied to high dimensional problems. In this paper, we focus on another class of sequential inference methods, namely the Sequential Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising alternative to SMC methods. After providing a unifying framework for the class of SMCMC approaches, we propose novel efficient strategies based on the principle of Langevin diffusion and Hamiltonian dynamics in order to cope with the increasing number of high-dimensional applications. Simulation results show that the proposed algorithms achieve significantly better performance compared to existing algorithms

A Tractable State-Space Model for Symmetric Positive-Definite Matrices

Bayesian analysis of state-space models includes computing the posterior distribution of the system's parameters as well as filtering, smoothing, and predicting the system's latent states. When the latent states wander around $\mathbb{R}^n$ there are several well-known modeling components and computational tools that may be profitably combined to achieve these tasks. However, there are scenarios, like tracking an object in a video or tracking a covariance matrix of financial assets returns, when the latent states are restricted to a curve within $\mathbb{R}^n$ and these models and tools do not immediately apply. Within this constrained setting, most work has focused on filtering and less attention has been paid to the other aspects of Bayesian state-space inference, which tend to be more challenging. To that end, we present a state-space model whose latent states take values on the manifold of symmetric positive-definite matrices and for which one may easily compute the posterior distribution of the latent states and the system's parameters, in addition to filtered distributions and one-step ahead predictions. Deploying the model within the context of finance, we show how one can use realized covariance matrices as data to predict latent time-varying covariance matrices. This approach out-performs factor stochastic volatility.Comment: 22 pages: 16 pages main manuscript, 4 pages appendix, 2 pages reference

Bags of Affine Subspaces for Robust Object Tracking

We propose an adaptive tracking algorithm where the object is modelled as a continuously updated bag of affine subspaces, with each subspace constructed from the object's appearance over several consecutive frames. In contrast to linear subspaces, affine subspaces explicitly model the origin of subspaces. Furthermore, instead of using a brittle point-to-subspace distance during the search for the object in a new frame, we propose to use a subspace-to-subspace distance by representing candidate image areas also as affine subspaces. Distances between subspaces are then obtained by exploiting the non-Euclidean geometry of Grassmann manifolds. Experiments on challenging videos (containing object occlusions, deformations, as well as variations in pose and illumination) indicate that the proposed method achieves higher tracking accuracy than several recent discriminative trackers.Comment: in International Conference on Digital Image Computing: Techniques and Applications, 201