Differentiable and Stable Long-Range Tracking of Multiple Posterior Modes

TL;DR

This work tackles the difficulty of learning discriminative particle filters when resampling is non-differentiable by introducing a mixture-density particle filter (MDPF) framework. It identifies severe instability in implicit reparameterization gradients for mixture models and proposes an unbiased, low-variance gradient estimator based on importance weighting (IWSG), enabling end-to-end training. By incorporating KDE-based resampling and decoupled resampling/posterior mixtures (A-MDPF), the approach robustly represents multimodal latent state uncertainty and learns effective dynamics and measurement models from data. Across bearings-only tracking, DeepMind maze localization, and House3D navigation, the proposed methods significantly improve accuracy and training stability over prior discriminative PFs, highlighting the practical potential for vision- and robotics-oriented state estimation in complex environments.

Abstract

Particle filters flexibly represent multiple posterior modes nonparametrically, via a collection of weighted samples, but have classically been applied to tracking problems with known dynamics and observation likelihoods. Such generative models may be inaccurate or unavailable for high-dimensional observations like images. We instead leverage training data to discriminatively learn particle-based representations of uncertainty in latent object states, conditioned on arbitrary observations via deep neural network encoders. While prior discriminative particle filters have used heuristic relaxations of discrete particle resampling, or biased learning by truncating gradients at resampling steps, we achieve unbiased and low-variance gradient estimates by representing posteriors as continuous mixture densities. Our theory and experiments expose dramatic failures of existing reparameterization-based estimators for mixture gradients, an issue we address via an importance-sampling gradient estimator. Unlike standard recurrent neural networks, our mixture density particle filter represents multimodal uncertainty in continuous latent states, improving accuracy and robustness. On a range of challenging tracking and robot localization problems, our approach achieves dramatic improvements in accuracy, while also showing much greater stability across multiple training runs.

Figures (33)

• Figure 1: Sequential state estimation may be formulated via either generative (left) or discriminative (right) graphical models. In either case, dynamics of states $x_t$ are influenced by actions $a_t$, and estimated via data $y_t$.
• Figure 2: Left: Our MDPF method showing the various sub-components. Middle: Our A-MDPF method with decoupled measurement models and bandwidths. Right: Dynamics and measurement model structures used in MDPF and A-MDPF. The dynamics and measurement models are composed of several neural networks as well as some fixed transforms, which convert the angular state dimensions of particles into a vector representation.
• Figure 3: Left: Example changes in samples from a 1D mixture model with two Epanechnikov Epanechnikov components. Under IRG, changes in the mixture are shown by dramatic sample shifting, where some samples must change modes to account for the changes in the relative weights of each mixture mode. Explicitly plotting the particle transformation induced by IRG reveals a discontinuity (bottom). In contrast, IWSG smoothly reweights samples. Right: Example changes in samples from a 2D mixture of two Gaussians. IRG again induces large shifts as particles change modes, as demonstrated by the vector field (bottom).
• Figure 4: A simple temporal prediction problem where IRG has highly unstable gradient estimates. We use $N=25$ particles when training IRG-PF, IWSG-PF, and truncated gradients (TG-PF). We show the learned values for parameters $B$, $w_2$, $C_1$ and their gradients during training. IRG produces unstable gradients which prevent parameters from converging at all, but IWSG allows for smooth convergence of all parameters. IWSG is faster than biased TG, and nearly as effective as (expensive, and for general models intractable) mixture KF.
• Figure 5: Box plots showing median (red line), inter-quartile (colored box) and range (whiskers) over several training runs on the Bearings-Only (11 runs) and Deepmind-Maze (5 runs) tracking tasks. MDPF and A-MDPF consistently perform well on both the NLL and RMSE metrics. Our IWSG estimator is critical: IRG-MDPF performs very poorly due to unstable gradients, TG-MDPF is inconsistent and sometimes becomes trapped in local optima, and baselines with biased gradient estimators typically have inferior performance. Note that IRG-MDPF does not support Epanechnikov kernels, which induce discontinuous CDFs. LSTM achieves low RMSE in the Deepmind-Maze task, but we find it simply propagates the noisy actions blindly. DIS-PF performs well for Maze 3, but has larger variability and sometimes performs worse than our more stable A-MDPF.