Multivariate Uncertainty Quantification with Tomographic Quantile Forests

TL;DR

The paper introduces Tomographic Quantile Forests, a nonparametric tree-based model that estimates multivariate conditional distributions by aggregating directional quantiles, enabling uncertainty quantification without convexity constraints.

Contribution

It presents a novel, single-model approach for multivariate uncertainty quantification using directional quantiles and sliced Wasserstein distance, improving over classical methods.

Findings

TQF accurately estimates multivariate conditional distributions.

The method outperforms classical directional-quantile approaches.

Source code is publicly available on GitHub.

Abstract

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 $\mathbf{n}^{\top}\mathbf{y}$ as functions of the input $\mathbf{x}$ and the unit direction $\mathbf{n}$. 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.