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Gated Inference Network: Inference and Learning State-Space Models
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Proposes the Gated Inference Network (GIN), a GRU-gated deep extended Kalman filter/smoother for nonlinear SSMs learned end-to-end from video, with a spectral-norm constraint to prevent GRU gradient explosion.
Problem
Approximate Bayesian filtering and smoothing for state-space models with nonlinear, unknown dynamics and high-dimensional noisy video observations, where classical EKF is O(n^3) per step and prone to model mismatch.
Approach
Replace the Kalman gain and smoothing gain matrix inversions with compact GRU networks operating on convolutionally pooled covariances, use K learned transition/emission matrix sets mixed by a state-dependent softmax, and constrain the GRU recurrent weight's largest singular value below 2 to guarantee local stability and avoid gradient explosion.
Key results
- On the single pendulum imputation task, GIN (smoothed) reaches log likelihood -11.215 vs -12.395 for LGSSM and -12.495 for CRU.
- On KITTI visual odometry, GIN achieves mean trel 7.03% and rrel 2.75°, versus 7.55%/3.28° for LGSSM and 5.96%/6.12° for DeepVO.
- Gradient-clipping with theta=15 or 20 fails (0-20% success) while the SVD and GCT spectral constraints train successfully in 100% of runs.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GIN smoothing for state estimation instead of the strongest listed baseline, CRU | GIN · Double pendulum | Log Likelihood: 4.269 ± 0.237 -> 4.977 ± 0.168 | Helped |
| Use GIN smoothing for state estimation instead of the strongest listed baseline, CRU | GIN · Single pendulum | Log Likelihood: 6.348 ± 0.19 -> 7.445 ± 0.165 | Helped |
| Use GIN smoothing for state estimation instead of the strongest listed baseline, CRU | GIN · Irregular polygon | Log Likelihood: 3.366 ± 0.039 -> 4.342 ± 0.051 | Helped |
| Use GIN image imputation instead of LGSSM | GIN · Single pendulum | Log Likelihood: -12.395 ± 0.048 -> -11.215 ± 0.027 | Helped |
| Use GIN image imputation instead of CRU | GIN · Double pendulum | Log Likelihood: -13.668 ± 0.038 -> -12.584 ± 0.021 | Helped |
| Use GIN image imputation instead of CRU | GIN · Irregular polygon | Log Likelihood: -15.884 ± 0.297 -> -14.628 ± 0.226 | Helped |
| Use GIN instead of DeepVO | GIN · KITTI seq 03–10 mean | rrel(°): 6.12 -> 2.75 | Helped |
| Use GIN instead of DeepVO | GIN · KITTI seq 03–10 mean | trel(%): 5.96 -> 7.03 | Hurt |
| Increase state dimension from n=2m to n=3m (ablation) | GIN smoother · Single pendulum | Log Likelihood: 7.292 ± 0.173 -> 7.445 ± 0.165 | Helped |
| Use informed rather than uninformed missing-image masks (ablation) | GIN · Single pendulum | Log Likelihood: -11.246 ± 0.029 -> -11.215 ± 0.027 | No clear effect |
| Increase stability threshold from δ=0.1 to δ=1 (ablation) | GIN with SVD stability constraint · Single pendulum | Objective: 7.445 ± 0.165 -> 6.935 ± 0.271 | Hurt |
| Use SVD stability constraint instead of conventional gradient clipping (ablation) | GIN · Double pendulum | Objective: 4.765 ± 0.349 -> 4.977 ± 0.168 | Helped |
Surprising findings
- The GRU-based gain networks degrade smoothing when the constraint pushes the spectral radius toward 2-delta, so larger delta (stronger constraint) lowers likelihood, indicating a stability-accuracy trade-off.
- DeepVO attains lower rotational error (6.12 deg mean in the table is GIN-competitive but DeepVO's rrel is 6.12 vs GIN's 2.75; DeepVO has lower translational error on some sequences than GIN's average on KITTI reporting in the table lines).
Limitations
- Theorem 3 requires shrinking input covariances toward zero over time, which limits posterior inference and is acknowledged as an open problem.
- Gradient explosion mitigation via spectral constraint trades off accuracy: larger delta reduces performance because long time dependencies are damped.
- Training uses K=15 transition matrix sets and a grid-searched learning rate on a single GTX 1050 Ti; scalability to large-scale real data is only shown on KITTI with reduced observation sizes.
Experimental setup
Datasets: simulated single pendulum video simulated double pendulum video irregular polygon bouncing ball simulation KITTI
Benchmarks: KITTI single pendulum double pendulum irregular polygon bouncing ball
Metrics: log likelihood squared error of state estimates translational error trel rotational error rrel objective and success rate under gradient explosion
Baselines: VAE IW-VAE VAE-RNN (LSTM) VAE-RNN (GRU) SVAE KVAE EKVAE MVAE DeepAR RKN CRU LGSSM
Builds on: Kalman filter KalmanNet Kalman VAE RKN Hammerstein-Wiener model
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