Research map / Probabilistic and scientific modeling / Variational and generative sampling
Nonlinear Filtering with Brenier Optimal Transport Maps
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes a likelihood-free nonlinear filter that learns a block-triangular Brenier optimal transport map from prior to posterior via a max-min neural formulation, comparing against SIR particle filter and EnKF.
Problem
Nonlinear filtering (computing posterior of a hidden state given noisy partial observations) fails for SIR particle filters under degenerate likelihoods and high-dimensional states due to weight degeneracy.
Approach
Approximate the posterior by pushing prior particles through a conditional Brenier OT map T(x,y) learned via a max-min stochastic optimization over neural networks (ResNets for f and T, with an EnKF layer), requiring only samples from the likelihood model rather than its analytic form.
Key results
- On the bimodal static example with noise 0.04, SIR collapses to a single mode while the OT method captures both modes.
- On the bimodal dynamic example, OT yields lower MMD to the exact posterior (approximated by SIR with 1e5 particles) than EnKF and SIR, with EnKF plateauing as N grows.
- On Lorenz 63, OT and EnKF converge faster than SIR; OT performs well on multi-modal MNIST in-painting where SIR is overconfident.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use OT filtering instead of SIR | Nonlinear filtering · Bimodal static example, λw=0.04 | posterior mode capture | Helped |
| Use OT filtering instead of EnKF | Nonlinear filtering · Bimodal static example | posterior mode capture | Helped |
| Use OT filtering instead of SIR | Nonlinear filtering · Bimodal dynamic example | MMD to exact posterior | Helped |
| Use OT filtering instead of SIR | Nonlinear filtering · Lorenz 63 | mean squared error | Helped |
| Add the EnKF layer to the OT architecture (ablation) | OT filtering · Lorenz 63 | mean squared error | Helped |
| Remove the EnKF layer from the OT architecture (ablation) | OT filtering · Lorenz 63 | numerical stability | Helped |
| Use OT filtering instead of SIR | Nonlinear filtering · Lorenz 96 | filtering performance | Helped |
| Use OT filtering instead of EnKF | Nonlinear filtering · Lorenz 96 | filtering performance | No clear effect |
| Use OT filtering instead of SIR | Nonlinear filtering · Static MNIST image in-painting | posterior uncertainty capture | Helped |
| Use OT filtering instead of SIR and EnKF | Nonlinear filtering · Dynamic MNIST image in-painting | posterior uncertainty capture | Helped |
| Increase the OT training budget (ablation) | OT filtering · Nonlinear filtering | accuracy | Helped |
| Use EnKF instead of OT | Nonlinear filtering · Unimodal posterior, including Lorenz 96 | computational efficiency | Helped |
Surprising findings
- The learned map is not exactly monotone and the f function is not exactly convex, though violations occur over only a small probability set (about 0.12% and 3.3%).
- Without training, the OT method reduces to the EnKF, and the EnKF layer improves accuracy but reduces numerical stability on Lorenz 63.
Limitations
- Raw computational time of the OT approach is higher than EnKF and SIR.
- Convexity constraint on f is not enforced, so the theoretical error bound applies only if the learned f is c-concave.
- Full bias-variance characterization of the optimality gap is left to future work.
Experimental setup
Datasets: MNIST
Benchmarks: Lorenz 63 Lorenz 96 MNIST in-painting
Metrics: MMD MSE monotonicity measure digit histogram
Baselines: sequential importance resampling particle filter ensemble Kalman filter feedback particle filter ensemble transform particle filter
Builds on: Brenier theorem Kantorovich duality Taghvaei and Mehta variational Bayes OT (TH) ensemble Kalman filter feedback particle filter
Closest papers by meaning
- Implicit Maximum a Posteriori Filtering via Adaptive Optimization (ICLR 2024 · similarity 0.88)
- Estimation of Stochastic Optimal Transport Maps (NeurIPS 2025 · similarity 0.86)
- Predictively-Oriented Kalman Filtering (NeurIPS 2026 · similarity 0.86)
- Outlier-robust Kalman Filtering through Generalised Bayes (ICML 2024 · similarity 0.85)
- Deep Bayesian Filter for Bayes-Faithful Data Assimilation (ICML 2025 · similarity 0.85)
- Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior (ICLR 2025 · similarity 0.84)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.