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Multivariate Latent Recalibration for Conditional Normalizing Flows
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Proposes latent recalibration (LR), a post-hoc method that rescales the latent norms of conditional normalizing flows so the PIT of latent norms is uniform, yielding a recalibrated explicit multivariate density with finite-sample latent calibration guarantees.
Problem
Multi-output distributional regression models (normalizing flows, flow matching) are often miscalibrated, and existing multivariate recalibration or conformal methods either give only prediction sets or lack an explicit recalibrated density and are computationally expensive.
Approach
Define latent calibration as uniformity of the PIT of the latent norm in the base latent space of an invertible conditional generative model, then recalibrate by radially transforming the latent vectors via the CDF-quantile map of latent norms, which gives a closed-form Jacobian and an explicit recalibrated PDF.
Key results
- On AFHQ with TarFlow, LR reduces L-ECE from 0.474 to 0.00895 and BPD from 5.477 to 5.465.
- LR consistently reduces L-ECE on tabular datasets and reduces NLL on the majority of datasets, while energy score is largely unchanged.
- In a SLUMP decision task, LR with numerical integration raises average utility from 76.23 (BASE) to 113.31.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply latent recalibration to the TarFlow model | TarFlow · AFHQ | L-ECE: $\text{0.474}_{\text{0.000625}}$ -> $\text{0.00895}_{\text{0.00160 }}$ | Helped |
| Apply latent recalibration to the TarFlow model | TarFlow · AFHQ | BPD: $\text{5.477}_{\text{3.523e-05}}$ -> $\text{5.465}_{\text{1.772e-05}}$ | Helped |
| Use PDF numerical integration instead of sampling to estimate region probabilities | BASE convex potential flow · SLUMP decision-making | Average Utility: 62.53 $\pm$ 11.33 -> 76.23 $\pm$ 11.99 | Helped |
| Apply HDR-R resampling | BASE convex potential flow · SLUMP decision-making | Average Utility: 62.53 $\pm$ 11.33 -> 32.38 $\pm$ 10.81 | Hurt |
| Apply latent recalibration and use the recalibrated PDF for numerical integration | BASE convex potential flow · SLUMP decision-making | Average Utility: 76.23 $\pm$ 11.99 -> \textbf{113.31 $\pm$ 12.91} | Helped |
| Apply latent recalibration | Convex potential flows on tabular datasets · 29 tabular datasets | NLL | Helped |
| Apply latent recalibration | Convex potential flows on tabular datasets · 29 tabular datasets | L-ECE | Helped |
| Apply latent recalibration | Convex potential flows on tabular datasets · Most tabular datasets | HDR-ECE | Helped |
| Apply latent recalibration | Tabular models · Tabular datasets | Energy Score | No clear effect |
| Apply latent recalibration | Convex potential flow · blog_data (BLO) | Energy Score | Hurt |
| Apply latent recalibration | TarFlow · AFHQ | Sample visual quality | No clear effect |
Surprising findings
- Energy score remains largely unchanged after LR despite NLL improvements, attributed to its insensitivity to variance and correlation misspecification.
- On a blog dataset with a discrete output, LR increased energy score, leading the authors to advise against using LR on discrete outputs.
Limitations
- LR only rescales latent norms, not directions, so it cannot correct orientation errors of the latent manifold.
- Requires the latent norm to have a tractable distribution and an invertible model, excluding VAEs and diffusion models.
- On the image experiment, calibration is done on training data, sacrificing finite-sample guarantees.
Experimental setup
Datasets: 29 UCI-style multi-output tabular datasets AFHQ
Benchmarks: 29 multi-output tabular datasets AFHQ
Metrics: L-ECE HDR-ECE NLL energy score BPD
Baselines: HDR-R BASE (uncalibrated model) CONTRA L-CP
Builds on: quantile recalibration HDR recalibration conformal prediction convex potential flows TarFlow
Closest papers by meaning
- Sampling-based Multi-dimensional Recalibration (ICML 2024 · similarity 0.85)
- Linear combinations of latents in generative models: subspaces and beyond (ICLR 2025 · similarity 0.84)
- Generalized Venn and Venn-Abers Calibration with Applications in Conformal Prediction (ICML 2025 · similarity 0.84)
- Not all distributional shifts are equal: Fine-grained robust conformal inference (ICML 2024 · similarity 0.84)
- Reassessing How to Compare and Improve the Calibration of Machine Learning Models (ICLR 2025 · similarity 0.84)
- CONTRA: Conformal Prediction Region via Normalizing Flow Transformation (ICLR 2025 · similarity 0.84)
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