Variational Dimension Lifting for Robust Tracking of Nonlinear Stochastic Dynamics

Nonlinear stochastic motion presents significant challenges for Bayesian particle tracking. To address this challenge, we propose a lifting framework that constructs a higher-dimensional linear stochastic representation of nonlinear state-space models. The resulting surrogates enable the use of computationally efficient linear filtering techniques while retaining a direct connection to the underlying nonlinear dynamics. The paper derives the necessary conditions for such transformations using Ito's lemma and variational calculus, and illustrates the method on a bistable cubic motion model, radial Brownian process model, and a logistic model with multiplicative noise. Simulations confirm that the transformed linear systems, when projected back, accurately reconstruct the nonlinear dynamics and, in distinct regimes of stiffness and singularity, yield tracking accuracy competitive with conventional filters, while avoiding their structural instabilities.