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Implicit Maximum a Posteriori Filtering via Adaptive Optimization
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Frames Bayesian filtering as optimization over a time-varying objective, implicitly defining the filter via adaptive optimizer hyperparameters (Implicit MAP Filter), and evaluates it on toy nonlinear, stochastic Lorenz, and Yearbook CNN adaptation tasks.
Problem
Bayesian filtering requires explicit matrix storage, inversion, or particle simulation, which is impractical in high-dimensional state spaces such as neural network weight spaces, and existing filters are sensitive to process noise and dynamics misspecification.
Approach
Reinterpret the filter update as K steps of an adaptive optimizer (SGD, Adam, RMSprop, etc.) on the negative log-likelihood starting from the predicted mean, so the optimizer and its hyperparameters implicitly define the prior/predictive covariance; in linear-Gaussian systems, K steps of gradient descent with a suitable learning-rate matrix recover the Kalman filter mode.
Key results
- On the toy nonlinear system (Q=1, R=2), IMAP with RMSprop reaches RMSE 5.304 vs UKF 4.550 and PF 1.565; EKF diverges with RMSE 24.619.
- On the stochastic Lorenz attractor (RK4, alpha=10), IMAP with gradient descent achieves RMSE 0.701, statistically matching the optimized-Q EKF (0.692).
- On Yearbook test years, Implicit MAP Filter reaches 84.747% accuracy vs 80.449% for direct fit and 79.967% for variational Kalman filter.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use IMAP with Adam instead of the particle filter | Toy nonlinear filtering · Toy nonlinear system, Q=1/3/5 and R=2 | RMSE: 5.699 \pm 0.190 vs 1.565 \pm 0.047; 5.842 \pm 0.231 vs 2.800 \pm 0.108; 7.964 \pm 0.334 vs 4.518 \pm 0.160 | Hurt |
| Use IMAP with gradient descent instead of EKF with optimized process noise | Stochastic Lorenz filtering · Stochastic Lorenz attractor, RK4 and Euler | RMSE: 0.701 \pm 0.018 vs 0.692 \pm 0.014; 0.960 \pm 0.012 vs 0.952 \pm 0.011 | No clear effect |
| Use IMAP with gradient descent instead of EKF with optimized process noise | Stochastic Lorenz filtering · Stochastic Lorenz attractor, GRW | RMSE: 1.561 \pm 0.010 vs 1.561 \pm 0.010 | No clear effect |
| Use Implicit MAP Filter instead of Direct Fit | 4-layer CNN on Yearbook · Yearbook, validation years | accuracy: 94.416 \pm 0.924 -> 94.973 \pm 0.837 | Helped |
| Use Implicit MAP Filter instead of Direct Fit | 4-layer CNN on Yearbook · Yearbook, test years | accuracy: 80.449 \pm 1.845 -> 84.747 \pm 2.030 | Helped |
Surprising findings
- The Implicit MAP Filter can fix measurement noise to identity without loss of generality, since the implied predictive covariance is compensated accordingly.
- Under the toy nonlinear system, the EKF and IEKF tend to diverge, while the implicit filter with momentum-based optimizers outperforms gradient-based EKF.
Limitations
- Framework provides point estimates and loses calibrated uncertainty estimates, limiting use in risk-sensitive settings.
- Requires grid search over optimizer hyperparameters and a validation set to choose settings.
- Interoperability with the explicit filtering equations is lost.
Experimental setup
Datasets: Yearbook simulated toy nonlinear system simulated stochastic Lorenz '63 system
Benchmarks: Yearbook toy nonlinear system stochastic Lorenz attractor
Metrics: RMSE classification accuracy
Baselines: EKF IEKF UKF particle filter variational Kalman filter direct fit static weights
Builds on: Kalman filter extended Kalman filter unscented Kalman filter Adam RMSprop
Closest papers by meaning
- Predictively-Oriented Kalman Filtering (NeurIPS 2026 · similarity 0.90)
- Nonlinear Filtering with Brenier Optimal Transport Maps (ICML 2024 · similarity 0.88)
- Outlier-robust Kalman Filtering through Generalised Bayes (ICML 2024 · similarity 0.88)
- Deep Bayesian Filter for Bayes-Faithful Data Assimilation (ICML 2025 · similarity 0.87)
- Understanding High-Dimensional Bayesian Optimization (ICML 2025 · similarity 0.85)
- Max Entropy Moment Kalman Filter for Polynomial Systems with Arbitrary Noise (NeurIPS 2025 · similarity 0.84)
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