FLUX: Progressive State Estimation Based on Zakai-type Distributed Ordinary Differential Equations
For researchers in nonlinear filtering and Bayesian inference, FLUX offers a new computational approach that avoids common bottlenecks like optimization or PDE solving, though its current 1D limitation makes it incremental for now.
FLUX proposes a homotopy continuation method for approximating probability densities by solving distributed ODEs that morph an initial density into a target density. Applied to state estimation in nonlinear systems, it is fast and avoids optimization or PDEs, but is currently limited to 1D with fixed parameters.
We propose a homotopy continuation method called FLUX for approximating complicated probability density functions. It is based on progressive processing for smoothly morphing a given density into the desired one. Distributed ordinary differential equations (DODEs) with an artificial time $γ\in [0,1]$ are derived for describing the evolution from the initial density to the desired final density. For a finite-dimensional parametrization, the DODEs are converted to a system of ordinary differential equations (SODEs), which are solved for $γ\in [0,1]$ and return the desired result for $γ=1$. This includes parametric representations such as Gaussians or Gaussian mixtures and nonparametric setups such as sample sets. In the latter case, we obtain a particle flow between the two densities along the artificial time. FLUX is applied to state estimation in stochastic nonlinear dynamic systems by gradual inclusion of measurement information. The proposed approximation method (1) is fast, (2) can be applied to arbitrary nonlinear systems and is not limited to additive noise, (3) allows for target densities that are only known at certain points, (4) does not require optimization, (5) does not require the solution of partial differential equations, and (6) works with standard procedures for solving SODEs. This manuscript is limited to the one-dimensional case and a fixed number of parameters during the progression. Future extensions will include consideration of higher dimensions and on the fly adaption of the number of parameters.