arxiv: 2512.16383 · v2 · submitted 2025-12-18 · 💻 cs.LG · stat.ML

Multivariate Uncertainty Quantification with Tomographic Quantile Forests

Takuya Kanazawa This is my paper

Pith reviewed 2026-05-16 21:39 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords multivariate uncertainty quantificationquantile forestssliced Wasserstein distancedirectional quantilesnonparametric regressionconditional distributiontree-based modelstomographic reconstruction

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The pith

A single tree model estimates full multivariate conditional distributions from directional quantiles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Tomographic Quantile Forests as a nonparametric method for estimating conditional distributions when the target variable is multivariate. It works by training a tree model to predict quantiles of projections of the target onto arbitrary directions, then combining those predictions across many directions. Reconstruction of the full distribution happens by solving an optimization problem that minimizes the sliced Wasserstein distance between the predicted and observed distributions. A reader would care because this enables uncertainty quantification in settings like multi-output regression without assuming simple shapes for the distribution or needing separate models per direction. This supports safer AI systems that can handle complex dependencies in their predictions.

Core claim

Tomographic Quantile Forests learn conditional quantiles of directional projections as functions of the input and direction using a single tree model. At inference time, these quantiles are aggregated over many directions and the multivariate conditional distribution is reconstructed by minimizing the sliced Wasserstein distance through an alternating optimization scheme whose subproblems are convex. This single-model approach covers all directions and avoids the convexity restrictions typical of classical directional quantile methods.

What carries the argument

Tomographic Quantile Forests that learn directional conditional quantiles and reconstruct the joint distribution by sliced Wasserstein minimization.

If this is right

Supports nonparametric estimation of arbitrary multivariate conditional distributions.
Uses a single model for all projection directions instead of training separate models.
Enables reconstruction without imposing convexity on the quantile regions.
Provides an efficient inference procedure based on alternating convex optimizations.
Validated on both synthetic data and real-world datasets with released code.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be combined with other tree-based or ensemble techniques to further improve calibration in high-stakes applications.
Extensions to streaming or online learning scenarios might follow naturally from the tree structure.
Applications in fields like autonomous driving or medical diagnosis could benefit from full distributional predictions for better risk assessment.
Scalability to very high-dimensional outputs remains an open question that future work could address through dimension reduction techniques.

Load-bearing premise

That minimizing the sliced Wasserstein distance over aggregated directional quantiles from a single tree model will recover the true multivariate conditional distribution without large approximation errors.

What would settle it

A test on synthetic data where the true conditional distribution has a known non-convex shape, checking whether the TQF reconstruction matches the ground truth quantiles or moments more closely than convex-restricted baselines.

Figures

Figures reproduced from arXiv: 2512.16383 by Takuya Kanazawa.

**Figure 1.** Figure 1: Toy datasets in R 2 containing 300 points with identical marginal distributions. For regression with a univariate target y ∈ R, predictive uncertainty is commonly summarized using confidence intervals or quantiles, or modeled using parametric distributions. In contrast, uncertainty quantification for a multivariate target y ∈ R d is considerably more challenging. A naïve approach is to model each compone… view at source ↗

**Figure 2.** Figure 2: Illustration of the predicted quantiles from QRF (left panels) and QRF++ (right panels) on four [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Target importance of QRF++ on the synthetic datasets; error bars show one standard deviation across [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Numerical experiment of the QMEM algorithm. (a) The “two moons” dataset. (b) Best fit of 9 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 6.** Figure 6: Reconstructed point cloud with K = M = 5. After pruning, the population size is reduced from 3,000 to 1,051 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 8.** Figure 8: Left: letter “A” dataset comprising 2,029 points. Right: Distribution reconstructed from 25 projections with QMEM [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

**Figure 10.** Figure 10: Illustration of the function a(x) for x ∈ [−2, 2]2 . In this subsection we evaluate TQF on a synthetic dataset. Dataset. We draw x = (x (ℓ) )ℓ ∈ R p from a uniform distribution over [−2, 2]p , compute the coordinate average xavg := 1 p Pp ℓ=1 x (ℓ) , and set a(x) := σ(xavg) ∈ (0, 1), where σ denotes the sigmoid function. The function a is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

**Figure 11.** Figure 11: Conditional distribution p(y | x) for varying values of a(x) in our synthetic benchmark dataset. Each panel shows 1,500 points. 1 0 1 1 0 1 x = 1.46 0.57 0.89 1.12 1.33 1.53 1.80 2.08 2.70 1 0 1 1 0 1 x = 0.56 0.24 0.43 0.56 0.66 0.75 0.83 0.92 1.06 1 0 1 1 0 1 x = 0.00 0.20 0.39 0.53 0.63 0.70 0.77 0.84 0.96 1 0 1 1 0 1 x = 0.56 0.22 0.40 0.53 0.63 0.71 0.77 0.84 1.01 1 0 1 1 0 1 x = 1.46 0.34 0.62 0.85 … view at source ↗

**Figure 12.** Figure 12: Predictive distributions p(y | x) generated by TQF at x = (x, x) with x shown in the title of each panel, corresponding to a(x) = 0.1, 0.3, 0.5, 0.7 and 0.9, respectively, from left to right. Each panel shows KDE of N points with N = 1475, 2000, 1995, 1910, and 1695. 1.0 0.5 0.0 0.5 1.0 x 1.5 1.0 0.5 0.0 0.5 1.0 1.5 P r oje c tio n o f y = 2 /3 0 4 2 3 4 1 0 1 P r oje c tio n o f y x = 1.50 [PITH_FULL_IM… view at source ↗

**Figure 13.** Figure 13: Quantile predictions for n ⊤y obtained by TQF for levels {0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8} denoted by thin yellow lines. The seven thick colored curves denote quantile predictions by KNN regression: {0.2, 0.3, 0.4} (blue), 0.5 (green), and {0.6, 0.7, 0.8} (red), respectively. Left: predictions at x = (x, x) and n = (cos 2 3 π,sin 2 3 π). Right: predictions at x = (−1.5, −1.5) and n = (cos θ,sin θ). targ… view at source ↗

**Figure 14.** Figure 14: Illustration of the estimated conditional distribution [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗

**Figure 15.** Figure 15: Left: Distribution p(y|x) = p [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗

**Figure 16.** Figure 16: Predictive distributions p(y|x) generated by TQF at x = (x, · · · , x) ∈ R 5 with x in the title of each panel, corresponding to a(x) = 0.1, 0.3, 0.5, 0.7 and 0.9 from top left to bottom right, respectively. Each panel shows a KDE based on N weighted points, with N = 1740, 1605, 1895, 1805 and 1745, respectively. Red contours indicate the boundary of the true support. 1.0 0.5 0.0 0.5 1.0 x 1.5 1.0 0.5 0.0… view at source ↗

**Figure 17.** Figure 17: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p018_17.png] view at source ↗

**Figure 18.** Figure 18: Same as Figure [PITH_FULL_IMAGE:figures/full_fig_p019_18.png] view at source ↗

**Figure 19.** Figure 19: Target-value distribution for the sliding-disk dataset. Colors indicate the input value x ∈ R. Although 4,000 points are shown for illustration, only 30 points are used to train the models. Model. To tune the hyperparameters of TQF and DRF, we generated 30 independent pairs of a training set (30 samples) and a validation set (1,000 samples). We then averaged ES in (13) over the 30 runs to assess perform… view at source ↗

**Figure 20.** Figure 20: Numerical results for the sliding-disk data. (a) Boxplots for the scores of TQF and DRF for 300 [PITH_FULL_IMAGE:figures/full_fig_p020_20.png] view at source ↗

**Figure 21.** Figure 21: Spatial distribution of records in the Califor [PITH_FULL_IMAGE:figures/full_fig_p020_21.png] view at source ↗

**Figure 22.** Figure 22: Feature importances (global absolute SHAP [PITH_FULL_IMAGE:figures/full_fig_p021_22.png] view at source ↗

**Figure 23.** Figure 23: Predictive distributions for the same test sample, provided by KNN, GP, NGBoost, and TQF from [PITH_FULL_IMAGE:figures/full_fig_p022_23.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

TQF uses one tree model for directional quantiles across all n and reconstructs via sliced Wasserstein minimization, which is a clean practical step but leaves the approximation error from finite directions and single-model generalization unquantified.

read the letter

The core contribution is a single quantile forest that predicts conditional quantiles for n^T y for any direction n, then at inference aggregates those and inverts them by minimizing sliced Wasserstein distance with an alternating convex scheme. This lets the method drop the convexity restriction common in earlier directional quantile work and avoids training a separate model per direction. The implementation is released, which is useful for anyone who wants to try it on vector-valued targets. The alternating optimization looks tractable on paper and the abstract reports results on both synthetic and real data, so the method is at least runnable and grounded in external datasets rather than pure self-reference. The main limitation is that the reconstruction step relies on finite directional sampling and the sliced-Wasserstein proxy; the note correctly flags that this can leave noticeable bias for multimodal or heavy-tailed conditionals, and the single forest must generalize over both x and the sphere of n without any consistency proof or error bound shown in the abstract. Experiments would need to demonstrate that the gap stays small in practice, with clear baselines and ablations on the number of directions. This is aimed at practitioners building uncertainty-aware models for multivariate outputs in robotics, finance, or similar areas. A reader looking for a nonparametric tree-based option that produces full conditionals rather than just intervals would find the code and the directional trick worth examining. It is coherent enough on its own terms to merit a serious referee, even if the theory and empirical validation need tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes Tomographic Quantile Forests (TQF), a nonparametric tree-based model for multivariate uncertainty quantification. TQF learns conditional quantiles of directional projections n^T y as functions of input x and unit direction n using a single forest. At inference, quantiles are aggregated over many directions and the full multivariate conditional distribution of y|x is reconstructed by minimizing the sliced Wasserstein distance via an efficient alternating optimization scheme whose subproblems are convex. The method avoids convexity restrictions on quantile regions that affect classical directional approaches and is evaluated on synthetic and real-world datasets, with source code released.

Significance. If the reconstruction step accurately recovers the target conditional law, TQF would offer a scalable, fully nonparametric alternative for multivariate predictive uncertainty that combines the flexibility of quantile forests with tomographic ideas. The single-model coverage of all directions and the convex subproblems in the alternating scheme are practical advantages. Public code release supports reproducibility. Significance is tempered by the need to confirm that finite directional sampling and forest-based quantile estimates do not introduce substantial bias for non-convex or multimodal conditionals.

major comments (2)

[Method (reconstruction and alternating optimization)] The central claim that the sliced-Wasserstein minimization recovers the true multivariate conditional distribution from finite directional quantiles is load-bearing, yet the manuscript provides no statistical consistency result or explicit error bound on the bias arising from (i) finite directional sampling (sliced Wasserstein converges to Wasserstein only in the limit) and (ii) the single forest's generalization over both x and the continuous sphere of n. This gap is especially relevant for multimodal or heavy-tailed targets; see the description of the alternating scheme and the reconstruction procedure.
[Experiments] In the experimental evaluation, the synthetic and real-data comparisons should report quantitative distribution-recovery metrics (e.g., empirical sliced or full Wasserstein distances, or proper scoring rules for the reconstructed law) rather than relying primarily on visual or qualitative assessment; without these, it is difficult to judge whether the approximation error remains negligible as claimed.

minor comments (2)

[Abstract and Method] Clarify the precise number of directions used in the aggregation step and the convergence tolerance of the alternating scheme; these implementation details affect reproducibility.
[Introduction] Notation for the unit vector n should be introduced consistently in the introduction to distinguish it from other vector quantities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the two major comments below and will revise the manuscript accordingly to strengthen both the theoretical discussion and the experimental evaluation.

read point-by-point responses

Referee: The central claim that the sliced-Wasserstein minimization recovers the true multivariate conditional distribution from finite directional quantiles is load-bearing, yet the manuscript provides no statistical consistency result or explicit error bound on the bias arising from (i) finite directional sampling (sliced Wasserstein converges to Wasserstein only in the limit) and (ii) the single forest's generalization over both x and the continuous sphere of n. This gap is especially relevant for multimodal or heavy-tailed targets.

Authors: We agree that a formal consistency analysis would strengthen the paper. While the sliced Wasserstein distance is known to converge to the Wasserstein distance as the number of projections tends to infinity (with explicit rates available in the literature), the manuscript does not derive finite-sample bounds that also account for the quantile forest estimation error. In the revision we will add a dedicated limitations subsection that (a) cites the relevant sliced-Wasserstein convergence results, (b) discusses the additional bias introduced by finite directional sampling and by the single forest’s generalization over the sphere, and (c) highlights the empirical behavior on the multimodal synthetic examples already present in the paper. We do not claim a new theoretical guarantee at this stage. revision: yes
Referee: In the experimental evaluation, the synthetic and real-data comparisons should report quantitative distribution-recovery metrics (e.g., empirical sliced or full Wasserstein distances, or proper scoring rules for the reconstructed law) rather than relying primarily on visual or qualitative assessment.

Authors: We accept this point. The current experiments emphasize visual comparisons and downstream task performance. In the revised version we will augment the experimental section with quantitative tables that report (i) empirical sliced Wasserstein distances between the reconstructed conditional distributions and the ground-truth distributions on the synthetic benchmarks, and (ii) proper scoring rules (energy score and variogram score) on the real-world datasets. These metrics will be computed for TQF as well as the competing methods to allow direct numerical comparison of distribution-recovery quality. revision: yes

Circularity Check

0 steps flagged

No circularity: directional quantile learning and sliced-Wasserstein reconstruction are independently defined and externally evaluated

full rationale

The paper defines TQF as a single tree model that learns conditional quantiles of n^Ty for input x and direction n, then aggregates and inverts via sliced-Wasserstein minimization with an alternating convex scheme. No equation or claim reduces the reconstruction to a quantity defined by the fit itself, nor does any load-bearing step rely on a self-citation chain or imported uniqueness theorem. The method is evaluated on synthetic and real-world datasets with released code, providing external grounding. This matches the default non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities; the method is described as building on existing quantile forest and optimal transport concepts.

pith-pipeline@v0.9.0 · 5442 in / 1035 out tokens · 30480 ms · 2026-05-16T21:39:31.436276+00:00 · methodology

discussion (0)

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