Variational Dimension Lifting for Robust Tracking of Nonlinear Stochastic Dynamics

TL;DR

nonlinear stochastic dynamics pose challenges for Bayesian tracking; the paper introduces a variational, $It\ô$-consistent dimension lifting that maps nonlinear SDEs into a finite-dimensional linear-Gaussian surrogate with invertibility. The lifted model enables standard linear-Gaussian inference in the lifted space and is weighted by the stationary distribution to prioritize accuracy where the dynamics concentrate. Applied to cubic bistable, radial Bessel, and Wright-Fisher diffusion, the method yields accurate trajectory reconstruction and robust tracking, often outperforming traditional filters in stiff or singular regimes while requiring lower online computation than particle filters. This framework provides a practical middle ground between nonlinear filters and operator-theoretic liftings, enabling efficient, stable tracking of complex stochastic dynamics.

Abstract

Nonlinear stochastic motion presents significant challenges for Bayesian particle tracking. To address this challenge, this paper proposes a framework to construct an invertible transformation that maps the nonlinear state-space model (SSM) into a higher-dimensional linear Gaussian SSM. This approach allows the application of standard linear-Gaussian inference techniques while maintaining a connection to the dynamics of the original system. 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.

Figures (4)

• Figure 1: Optimization diagnostics for the cubic bistable diffusion ($M=4$). Top: The stationary density $\rho(x)$ (black dashed line) and the spatial distribution of the weighted residual error (red solid line). Bottom: Simulated dynamics of the original versus lifted systems (with $dt=0.001$, $x(0)=0.5$, $\mu_{\text{stab}}=1$). We used 5 trajectories of each for the comparison. The lifted dynamics closely reproduces the bistable behavior of the original SDE.
• Figure 2: Comparison of the optimized objective density (left) and the simulated dynamics of the original versus lifted systems (with $dt=0.001$, $x(0)=0.5$). We used 5 trajectories of each for the comparison. The original dynamic grow at an order of square root of $t$ as expected. The lifted dynamics reproduces the growth behavior of the original SDE for our time frame.
• Figure 3: Comparison of the optimized objective density (left) and the simulated dynamics of the original versus lifted systems (with $dt=0.001$, $x(0)=0.5$). We used 5 trajectories of each for the comparison. The original dynamic grow at an order of square root of $t$ as expected. The lifted dynamics reproduces the growth behavior of the original SDE for our time frame.
• Figure 4: RMSE Time-Evolution for the Wright-Fisher case. Average tracking error over time ($T=20$) across 100 trials. The Lifted-KF (blue) maintains stable performance comparable to the computationally expensive Particle Filter (Red). The nonlinear baselines (EKF/UKF) exhibit higher error fluctuations and the regular KF exhibits the most error.