Research map / Probabilistic and scientific modeling / Variational and generative sampling
Enhanced Importance Sampling Through Latent Space Exploration in Normalizing Flows
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Proposes running importance sampling for rare failure events in the latent space of a pre-trained normalizing flow, using Löwner-John ellipsoids as the cost function, and shows gains over target-space IS on three simulated robotics/aerospace tasks.
Problem
Estimating failure probabilities of autonomous systems via importance sampling is inefficient when the proposal distribution poorly covers a non-isotropic, multimodal target distribution of simulator outcomes.
Approach
Perform importance sampling in the latent space of a pre-trained normalizing flow, where the distribution is closer to isotropic Gaussian, and define the failure cost via minimum-volume Löwner-John ellipsoids fit to latent-mapped failure points; samples are mapped back to target space through the invertible flow.
Key results
- Latent-CE achieves the lowest sample count across all datasets (e.g., 4560 vs 11040 for Target-CE on the robot; 300000 vs 895000 on F-16).
- Latent methods achieve lower avg NLL than target methods (e.g., F-16 Latent-CE -32.455 vs Target-CE 22.893), and higher coverage (0.586 vs 0.023 on F-16).
- On F-16, Target-CE and Target-SIS underestimate P_F (relative error about -1.0) while latent methods reach about -0.61 relative error.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| perform cross-entropy importance sampling in latent space instead of target space | Cross-entropy importance sampling · Nonholonomic Robot (3D) | coverage: 0.378 ± 0.222 -> 0.791 ± 0.049 | Helped |
| perform cross-entropy importance sampling in latent space instead of target space | Cross-entropy importance sampling · Nonholonomic Robot (3D) | samples until convergence: 11040 -> 4560 | Helped |
| perform sequential importance sampling in latent space instead of target space | Sequential importance sampling · Nonholonomic Robot (3D) | coverage: 0.565 ± 0.043 -> 0.707 ± 0.031 | Helped |
| perform cross-entropy importance sampling in latent space instead of target space | Cross-entropy importance sampling · Cornering Racecar (8D) | coverage: 0.555 ± 0.139 -> 0.813 ± 0.010 | Helped |
| perform sequential importance sampling in latent space instead of target space | Sequential importance sampling · Cornering Racecar (8D) | coverage: 0.706 ± 0.032 -> 0.791 ± 0.012 | Helped |
| perform cross-entropy importance sampling in latent space instead of target space | Cross-entropy importance sampling · Cornering Racecar (8D) | samples until convergence: 34200 -> 29900 | Helped |
| perform cross-entropy importance sampling in latent space instead of target space | Cross-entropy importance sampling · F-16 Ground Collision Avoidance (12D) | coverage: 0.023 ± 0.016 -> 0.586 ± 0.008 | Helped |
| perform sequential importance sampling in latent space instead of target space | Sequential importance sampling · F-16 Ground Collision Avoidance (12D) | coverage: 0.095 ± 0.011 -> 0.575 ± 0.006 | Helped |
| perform cross-entropy importance sampling in latent space instead of target space | Cross-entropy importance sampling · F-16 Ground Collision Avoidance (12D) | samples until convergence: 895000 -> 300000 | Helped |
Surprising findings
- Target-space IS on the F-16 task yields near-zero coverage (0.023 for Target-CE), i.e., it misses nearly all failure events.
- On the nonholonomic robot, Target-SIS has a smaller relative P_F error (0.058) than Latent-SIS (0.479) despite worse coverage and density.
Limitations
- Latent-space relative error of P_F remains large on F-16 (about -0.61) and the Löwner-John ellipsoid is conservative/overapproximating.
- Requires failure points X from training data or user specification; sparse data may leave failure domain under-covered.
- No theoretical analysis of benefits of latent IS is provided.
Experimental setup
Datasets: simulated nonholonomic robot trials simulated cornering racecar trials (MPOPIS setup) simulated F-16 GCAS trials (jax-f16)
Benchmarks: nonholonomic robot (3D) cornering racecar (8D) F-16 ground collision avoidance (12D)
Metrics: relative error of P_F estimate average negative log likelihood coverage density average samples until convergence
Baselines: Target-CE Target-SIS
Builds on: cross-entropy method sequential importance sampling Löwner-John ellipsoids rational quadratic spline flows coupling flows
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