# Multivariate Uncertainty Quantification with Tomographic Quantile Forests

arXiv: https://arxiv.org/abs/2512.16383 · HTML: https://mlsanity.com/paper/2512.16383/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 56 of its 131 references that are in the map.
- First posted on arXiv 18 Dec 2025.

## Builds on (most cited first, 30 of 56)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Hindsight Experience Replay](https://mlsanity.com/paper/1707.01495/) (2017)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Revisiting Deep Learning Models for Tabular Data](https://mlsanity.com/paper/2106.11959/) (2021)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Estimation and Inference of Heterogeneous Treatment Effects using Random Forests](https://mlsanity.com/paper/1510.04342/) (2015)
- [Tabular Data: Deep Learning is Not All You Need](https://mlsanity.com/paper/2106.03253/) (2021)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2101.12072/) (2021)
- [Generalized Random Forests](https://mlsanity.com/paper/1610.01271/) (2016)
- [When Do Neural Nets Outperform Boosted Trees on Tabular Data?](https://mlsanity.com/paper/2305.02997/) (2023)
- [Conformal Prediction: a Unified Review of Theory and New Challenges](https://mlsanity.com/paper/2005.07972/) (2020)
- [NGBoost: Natural Gradient Boosting for Probabilistic Prediction](https://mlsanity.com/paper/1910.03225/) (2019)
- [A Survey on Uncertainty Quantification Methods for Deep Learning](https://mlsanity.com/paper/2302.13425/) (2023)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [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)
- [Deeply Uncertain: Comparing Methods of Uncertainty Quantification in Deep Learning Algorithms](https://mlsanity.com/paper/2004.10710/) (2020)
- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Multivariate Uncertainty in Deep Learning](https://mlsanity.com/paper/1910.14215/) (2019)
- [Minimum Volume Conformal Sets for Multivariate Regression](https://mlsanity.com/paper/2503.19068/) (2025)
- [Stochastic Wasserstein Barycenters](https://mlsanity.com/paper/1802.05757/) (2018)
- [A Unified Comparative Study with Generalized Conformity Scores for Multi-Output Conformal Regression](https://mlsanity.com/paper/2501.10533/) (2025)
- [Distributional Regression Forests for Probabilistic Precipitation Forecasting in Complex Terrain](https://mlsanity.com/paper/1804.02921/) (2018)
- [Multivariate Quantile Function Forecaster](https://mlsanity.com/paper/2202.11316/) (2022)

## Built on by (most cited first, 1 of 1)

- [Multivariate quantile regression via Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2609.23906/) (2026)

## Abstract (arXiv)

Quantifying predictive uncertainty is essential for safe and trustworthy real-world AI deployment. Yet, fully nonparametric estimation of conditional distributions remains challenging for multivariate targets. We propose Tomographic Quantile Forests (TQF), a nonparametric, uncertainty-aware, tree-based regression model for multivariate targets. TQF learns conditional quantiles of directional projections ntopy as functions of the input x and the unit direction n. At inference, it aggregates quantiles across many directions and reconstructs the multivariate conditional distribution by minimizing the sliced Wasserstein distance via an efficient alternating scheme with convex subproblems. Unlike classical directional-quantile approaches that typically produce only convex quantile regions and require training separate models for different directions, TQF covers all directions with a single model without imposing convexity restrictions. We evaluate TQF on synthetic and real-world datasets, and release the source code on GitHub.