Variational Gaussian approximation of the Kushner optimal filter (2310.01859v1)

Abstract: In estimation theory, the Kushner equation provides the evolution of the probability density of the state of a dynamical system given continuous-time observations. Building upon our recent work, we propose a new way to approximate the solution of the Kushner equation through tractable variational Gaussian approximations of two proximal losses associated with the propagation and Bayesian update of the probability density. The first is a proximal loss based on the Wasserstein metric and the second is a proximal loss based on the Fisher metric. The solution to this last proximal loss is given by implicit updates on the mean and covariance that we proposed earlier. These two variational updates can be fused and shown to satisfy a set of stochastic differential equations on the Gaussian's mean and covariance matrix. This Gaussian flow is consistent with the Kalman-Bucy and Riccati flows in the linear case and generalize them in the nonlinear one.

• The paper develops a variational Gaussian framework to approximate the Kushner equation by minimizing proximal losses based on the Wasserstein and Fisher metrics.
• It demonstrates how gradient flows derived from JKO and LMMR schemes reveal geometric and statistical insights for state estimation.
• The approach unifies into a continuous-time variational Kalman filter that generalizes the Kalman-Bucy filter with promising theoretical guarantees for nonlinear systems.

Variational Gaussian Approximation of the Kushner Optimal Filter

The paper "Variational Gaussian approximation of the Kushner optimal filter" addresses a significant problem in estimation theory, focusing on approximating the solution to the Kushner equation, which describes the evolution of the state probability density of a dynamical system given continuous-time observations. This problem is essential for a wide range of applications in fields such as control systems, robotics, and signal processing, where understanding the underlying state of a system from noisy measurements is crucial. The authors propose a novel approach that utilizes variational Gaussian approximations to tackle this challenge.

Overview of the Research

The Kushner equation provides an optimal Bayesian filter for the continuous-time state estimation problem. However, solving it directly is infeasible in most practical scenarios. The paper presents a framework to approximate the solution using variational methods that handle two proximal losses—one based on the Wasserstein metric and the other on the Fisher Information metric. By approaching the problem from a geometric viewpoint, the authors achieve a variational approximation that automatically integrates the propagation and Bayesian update of probability densities.

The research is grounded on a solid mathematical foundation, leveraging the properties of variational calculus and information geometry. Specifically, it draws from recent developments in the Jordan-Kinderlehrer-Otto (JKO) and Laugesen-Mehta-Meyn-Raginsky (LMMR) proximal flow paradigms. In doing so, the authors manage to derive a set of stochastic differential equations (SDEs) that describe the Gaussian flow consistent with known solutions in special cases, such as the linear Kalman-Bucy filter, yet sufficiently general to extend these notions to nonlinear systems.

Key Contributions

1. Gaussian Approximation in Variational Loss Framework: The paper develops a variational Gaussian framework to solve the Kushner equation approximately. By constraining the candidate solutions to a space of Gaussian distributions, the problem becomes tractable, reducing it to minimizing variational losses over Gaussian parameters.
2. Informative Gradient Flows: The authors establish that the gradient flows generated by the proximal schemes offer insightful connections to geometric and statistical interpretations. The JKO scheme is shown to evolve according to the Wasserstein metric, while the LMMR update connects to the natural gradient and Fisher metric, highlighting the interplay between probability distributions and their geometric properties.
3. Continuous-Time Variational Kalman Filter: By uniting the results from both proximal schemes, the paper synthesizes these into a unified continuous-time filter, presenting a generalization of the well-known Riccati equations in the nonlinear scenario. This continuous variational Kalman filter retains consistency with the Kalman-Bucy filter under linear system assumptions.
4. Numerical Results and Theoretical Insight: Although primarily theoretical, the presented framework is equipped with formal derivations and guarantees that could guide practical implementations. The potential for convergence and stability under realistic conditions suggests broader applicability in future computational experiments.

Implications and Future Work

The implications of this research are both practical and theoretical. Practically, the proposed variational Gaussian approximation method could enhance the state estimation processes in complex dynamical systems, allowing for more precise and efficient filtering solutions in nonlinear settings.

Theoretically, the results suggest rich avenues for future research, specifically in exploring further connections between variational inference, optimal transport theory, and information geometry. Investigating these intersections might pave the way for new, more powerful approximation techniques for non-linear filtering problems.

Moreover, while this paper provides the groundwork for a novel approach to approximating the Kushner filter, future work could involve the development and analysis of numerical algorithms that implement these theoretical insights effectively. This would include evaluating performance on real-world datasets and exploring the computational landscape to optimize these approaches for large-scale systems.

In conclusion, the paper presents a coherent and well-constructed approximation technique for dealing with complex filtering problems, offering a substantial contribution to both the theoretical framework and its potential applications in estimation theory and beyond.