# Multivariate Latent Recalibration for Conditional Normalizing Flows

arXiv: https://arxiv.org/abs/2505.16636 · HTML: https://mlsanity.com/paper/2505.16636/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 71 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 30 of 49)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [On Fairness and Calibration](https://mlsanity.com/paper/1709.02012/) (2017)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Why do tree-based models still outperform deep learning on tabular data?](https://mlsanity.com/paper/2207.08815/) (2022)
- [Normalizing Flows are Capable Generative Models](https://mlsanity.com/paper/2412.06329/) (2024)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Distribution Calibration for Regression](https://mlsanity.com/paper/1905.06023/) (2019)
- [Conformal Prediction using Conditional Histograms](https://mlsanity.com/paper/2105.08747/) (2021)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [Individual Calibration with Randomized Forecasting](https://mlsanity.com/paper/2006.10288/) (2020)
- [Calibrated Multiple-Output Quantile Regression with Representation Learning](https://mlsanity.com/paper/2110.00816/) (2021)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Distributional Random Forests: Heterogeneity Adjustment and Multivariate Distributional Regression](https://mlsanity.com/paper/2005.14458/) (2020)
- [Calibrated and Sharp Uncertainties in Deep Learning via Density Estimation](https://mlsanity.com/paper/2112.07184/) (2021)

## Built on by (most cited first, 3 of 3)

- [metabeta -- A fast neural model for Bayesian mixed-effects regression](https://mlsanity.com/paper/2510.07473/) (2025)
- [Calibrated Multivariate Distributional Regression with Pre-Rank Regularization](https://mlsanity.com/paper/2601.22895/) (2026)
- [Super-Level-Set Regression: Conditional Quantiles via Volume Minimization](https://mlsanity.com/paper/2605.06210/) (2026)

## Abstract (arXiv)

Reliably characterizing the full conditional distribution of a multivariate response variable given a set of covariates is crucial for trustworthy decision-making. However, misspecified or miscalibrated multivariate models may yield a poor approximation of the joint distribution of the response variables, leading to unreliable predictions and suboptimal decisions. Furthermore, standard recalibration methods are primarily limited to univariate settings, while conformal prediction techniques, despite generating multivariate prediction regions with coverage guarantees, do not provide a full probability density function. We address this gap by first introducing a novel notion of latent calibration, which assesses probabilistic calibration in the latent space of a conditional normalizing flow. Second, we propose latent recalibration (LR), a novel post-hoc model recalibration method that learns a transformation of the latent space with finite-sample bounds on latent calibration. Unlike existing methods, LR produces a recalibrated distribution with an explicit multivariate density function while remaining computationally efficient. Extensive experiments on both tabular and image datasets show that LR consistently improves latent calibration error and the negative log-likelihood of the recalibrated models.