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Learning to be Smooth: An End-to-End Differentiable Particle Smoother
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Proposes the Mixture Density Particle Smoother (MDPS), a fully differentiable two-filter particle smoother with stratified importance-weighted resampling that learns neural dynamics and measurement models end-to-end, evaluated on bearings-only tracking and city-scale global vehicle localization (MGL, KITTI).
Problem
Particle smoothers offer more accurate offline state estimation than filters, but classically require human-engineered dynamics and observation models and cannot be trained end-to-end with deep networks; gradients through resampling are also biased or high-variance.
Approach
Combine forward and backward mixture density particle filters (MDPF) in a two-filter smoothing framework, sampling smoothed particles from a mixture of filtered posteriors and learning a neural posterior weight network, with importance-weighted sample gradients and stratified resampling to enable low-variance end-to-end training.
Key results
- On MGL, MDPS outperforms MDPF and search/retrieval baselines in position and rotation recall, using best of top-3 posterior modes.
- On bearings-only tracking, MDPS substantially outperforms MDPF, TG-PF, SR-PF, and classic FFBS across 11 runs.
- On KITTI, MDPS maintains a more diverse posterior and gains over MDPF when lateral error exceeds 2 meters.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use full-sequence smoothing with forward and backward particle filters instead of forward filtering alone | MDPS vs MDPF · Bearings-Only Tracking | state estimation accuracy | Helped |
| use stratified resampling instead of multinomial resampling | TG-PF and SR-PF · Bearings-Only Tracking | performance | Helped |
| use stratified resampling instead of multinomial resampling | MDPF · Bearings-Only Tracking | performance | Helped |
| use residual resampling instead of stratified resampling | particle filters · Bearings-Only Tracking | performance | Hurt |
| use MDPF instead of SR-PF or TG-PF | particle filters · Bearings-Only Tracking | performance | Helped |
| use end-to-end learned dynamics and measurement models instead of retrieval-based methods | MDPF and MDPS · MGL | position and rotation recall | Helped |
| incorporate future observations through smoothing instead of using past observations alone | MDPS vs MDPF · MGL | position and rotation recall | Helped |
| use a smoother that moves particles through end-to-end learning rather than FFBS reweighting | MDPS vs FFBS · Bearings-Only Tracking | state estimation accuracy | Helped |
| incorporate future observations to retain diverse posterior modes | MDPS vs MDPF · KITTI | Top 3 mode performance for lateral error greater than 2 meters | Helped |
| increase the number of particles at inference time (ablation) | MDPS · MGL | position and angle recall | Mixed |
Surprising findings
- Classic FFBS performs poorly even when given the true observation likelihood, because it only reweights rather than moves particles.
- MDPF with uniformly initialized particles still outperforms SR-PF and TG-PF.
Limitations
- Curse of dimensionality limits particle-based posterior expressiveness as state dimension grows.
- Training cost is high (about 160 GPU hours for MDPS on bearings-only task); OT-PF was not compared due to slowness.
- Custom MGL splits were created, so results are not directly comparable to prior MGL benchmarks.
Experimental setup
Datasets: Mapillary Geo-Localization (MGL) KITTI OpenStreetMap
Benchmarks: Bearings-Only tracking MGL KITTI
Metrics: position recall rotation recall top-3 mode recall lateral and longitudinal error recall
Baselines: TG-PF SR-PF FFBS MDPF Dense Search Retrieval Retrieval (PF)
Builds on: MDPF two-filter smoothing forward-filtering backward-smoothing soft resampling particle filter dense search
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