arxiv: 2605.10406 · v1 · submitted 2026-05-11 · 📊 stat.ME · stat.AP· stat.ML

Recognition: 2 theorem links

· Lean Theorem

Multi-Fidelity Quantile Regression

Yao Zhang, Yixiang Liu

Pith reviewed 2026-05-12 05:09 UTC · model grok-4.3

classification 📊 stat.ME stat.APstat.ML

keywords quantile regressionmulti-fidelitylevel functionhigh-fidelity datalow-fidelity dataconditional quantilesconformal prediction

0 comments

The pith

The high-fidelity quantile equals the low-fidelity quantile evaluated at a covariate-dependent level.

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

This paper develops a two-stage method for estimating conditional quantiles when high-fidelity observations are scarce by borrowing strength from cheaper low-fidelity data. It represents the target high-fidelity quantile as the low-fidelity quantile computed at a level that changes with the covariate. When the conditional distributions at the two fidelity levels have similar shapes, this level function is smoother than the quantile surface and therefore easier to estimate from limited high-fidelity samples. The authors give convergence theory that identifies when the resulting estimator improves on direct high-fidelity quantile regression and add a correction step for cases in which the shape similarity is weaker. Experiments on both synthetic and real data confirm lower estimation error and narrower conformal prediction intervals.

Core claim

The central claim is that a local quantile link exists under which the high-fidelity conditional quantile equals the low-fidelity conditional quantile evaluated at a covariate-dependent level function. This reformulation converts multi-fidelity quantile regression into the simpler task of estimating the level function, which converges faster than the original quantile when the low- and high-fidelity conditional distributions share similar shapes; a correction step restores robustness when that similarity weakens.

What carries the argument

The local quantile link, which expresses each high-fidelity quantile as the low-fidelity quantile at a covariate-dependent level.

If this is right

When the level function is smoother, the estimator converges faster than standard quantile regression that uses only high-fidelity data.
The correction step improves accuracy in regimes where distributional shapes differ more strongly.
Quantile estimates obtained this way are more accurate on both synthetic and real datasets.
Conformal prediction intervals constructed from the estimates are tighter while preserving valid coverage.

Where Pith is reading between the lines

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

This link representation could be applied to other conditional functionals such as means or tail probabilities if comparable fidelity relations hold.
Chaining the link across a hierarchy of fidelity levels might produce cumulative savings in data collection cost.
Performance will depend on having low-fidelity coverage over the full covariate domain, suggesting tests in settings with sparse low-fidelity observations.
The approach is model-agnostic, so it can be paired with any base quantile estimator.

Load-bearing premise

The low-fidelity and high-fidelity conditional distributions have similar shapes, so the level function varies more smoothly than the target high-fidelity quantile.

What would settle it

On synthetic data with deliberately dissimilar low- and high-fidelity conditional shapes, the multi-fidelity estimator shows no reduction in error or no faster convergence rate compared with direct high-fidelity quantile regression.

Figures

Figures reproduced from arXiv: 2605.10406 by Yao Zhang, Yixiang Liu.

**Figure 2.** Figure 2: Conformal prediction intervals based on quantile estimates from [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

**Figure 3.** Figure 3: MFQR results in the informative LF regime. The level functions [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

**Figure 4.** Figure 4: Illustration of LF and HF data in the non-informative LF regime; [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗

**Figure 5.** Figure 5: Conformal prediction intervals in the non-informative LF regime. [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗

**Figure 6.** Figure 6: Illustration of LF and HF data in the misinformative LF regime; [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗

**Figure 7.** Figure 7: Prediction intervals of HF-Only in the misinformative LF regime. [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

**Figure 8.** Figure 8: MFQR results in the misinformative LF regime. One-step and [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗

**Figure 9.** Figure 9: Coverage and interval width on the QeMFi datasets. [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗

**Figure 10.** Figure 10: Coverage and interval width on the Burgers and F-Energy datasets. [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗

**Figure 11.** Figure 11: Coverage and interval width on the QeMFi datasets under the [PITH_FULL_IMAGE:figures/full_fig_p041_11.png] view at source ↗

**Figure 12.** Figure 12: Coverage and interval width on the Burgers and F-Energy datasets [PITH_FULL_IMAGE:figures/full_fig_p042_12.png] view at source ↗

read the original abstract

High-fidelity (HF) data are often expensive to collect and therefore scarce, making conditional quantiles difficult to estimate accurately. We propose a two-stage, model-agnostic method for multi-fidelity quantile regression. The central idea is a local quantile link: at each covariate value, the HF quantile is represented as a low-fidelity (LF) quantile evaluated at a covariate-dependent level. This reformulation reduces the problem to estimating the level function, which can be smoother than the HF quantile itself when the LF and HF conditional distributions have similar shapes. We also study the complementary regime in which this advantage weakens and introduce a correction step to improve robustness. Our theory characterizes when the proposed estimator converges faster than direct quantile regression using HF data alone and when the correction step provides further improvement. Experiments on synthetic and real data show that our method yields more accurate quantile estimates and tighter conformal prediction intervals.

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

The paper gives a practical two-stage reformulation for multi-fidelity quantile regression that can speed up estimation when low- and high-fidelity conditional distributions are similar, but the gains hinge on an assumption that still needs sharper quantitative conditions.

read the letter

The central contribution is the local quantile link: the high-fidelity quantile at x is expressed as the low-fidelity quantile evaluated at a covariate-dependent level α(x). This reduces the task to estimating α(x), which can be smoother than the target quantile surface itself under the similar-shapes regime. They handle the estimation in two stages, keep the method model-agnostic, and add a correction step for the complementary case where the shapes diverge more. The theory is said to characterize the faster convergence rates and when the correction helps, and the experiments on synthetic and real data report better accuracy plus tighter conformal intervals than plain high-fidelity quantile regression.

Referee Report

1 major / 2 minor

Summary. The paper proposes a two-stage, model-agnostic multi-fidelity quantile regression method. The central idea is a local quantile link reformulation in which the high-fidelity (HF) conditional quantile at covariate x is expressed as the low-fidelity (LF) quantile evaluated at a covariate-dependent level α(x). This reduces the problem to estimating the level function α(x), which is claimed to be smoother than the direct HF quantile surface when the LF and HF conditional distributions have similar shapes. The work provides theoretical results characterizing convergence rates under this setup and in complementary regimes, introduces a correction step for robustness, and reports improved accuracy and tighter conformal prediction intervals on synthetic and real data.

Significance. If the similarity assumption holds with sufficient strength, the method offers a practical way to improve quantile estimation accuracy when HF data are scarce but LF data are plentiful. The model-agnostic two-stage structure and the correction step for robustness are clear strengths. The potential for tighter conformal intervals adds applied value. The absence of quantitative conditions on the similarity regime, however, limits the strength of the theoretical guarantees.

major comments (1)

[Theory section] The theory section characterizing faster convergence: the claimed rate improvement requires that α(x) be smoother than the HF quantile surface, which occurs 'when the LF and HF conditional distributions have similar shapes.' No explicit quantitative bound is supplied on the deviation between the conditional distributions (e.g., a bound on sup_x |F_HF(x,·) - F_LF(x,·)| or on the difference in conditional densities) that would guarantee the smoothness ordering or the rate gain. Without such a condition the advantage is not assured even inside the regime the method targets.

minor comments (2)

[Experiments] The experimental section would benefit from explicit reporting of the HF and LF sample sizes used in each synthetic example and from a quantitative metric (or diagnostic plot) confirming that the 'similar shapes' condition holds in the cases where improvement is observed.
[Introduction / Methods] The definition of the level function α(x) and its estimation procedure should be stated more explicitly in the introduction or early methods section to improve accessibility for readers new to the reformulation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below and have revised the manuscript to incorporate a quantitative condition strengthening the theoretical guarantees.

read point-by-point responses

Referee: [Theory section] The theory section characterizing faster convergence: the claimed rate improvement requires that α(x) be smoother than the HF quantile surface, which occurs 'when the LF and HF conditional distributions have similar shapes.' No explicit quantitative bound is supplied on the deviation between the conditional distributions (e.g., a bound on sup_x |F_HF(x,·) - F_LF(x,·)| or on the difference in conditional densities) that would guarantee the smoothness ordering or the rate gain. Without such a condition the advantage is not assured even inside the regime the method targets.

Authors: We appreciate the referee's observation. The main theoretical results are formulated directly in terms of the relative smoothness of the level function α(·) and the HF quantile surface, with the similarity of conditional distribution shapes provided as motivation for when the rate improvement is expected. We agree, however, that an explicit quantitative link between the deviation of the conditional distributions and the smoothness ordering would make the conditions more precise. In the revised version we have added a lemma (now Lemma 3.3) that supplies such a bound: under the assumption that the conditional densities are bounded away from zero and Lipschitz continuous, if sup_x ||F_HF(x,·) − F_LF(x,·)||_∞ ≤ δ, then the Hölder exponent of α exceeds that of the HF quantile by an amount controlled by δ. This yields an explicit regime (δ sufficiently small relative to the sample sizes) in which the faster convergence rate is guaranteed. We have also expanded the discussion of the complementary regime to clarify when the advantage does not hold. revision: yes

Circularity Check

0 steps flagged

No circularity: reformulation is an independent modeling step with separate estimation

full rationale

The paper's central step is a modeling reformulation that represents the HF quantile as an LF quantile evaluated at a covariate-dependent level α(x), reducing the task to estimating this level function. This is presented as a choice that can yield smoother targets under the assumption of similar conditional distribution shapes, followed by a two-stage estimation procedure and theoretical characterization of rates. No equations reduce a claimed prediction or result back to fitted inputs by construction, no self-citations are load-bearing for the core claim, and the level-function estimation is treated as a distinct, model-agnostic step rather than a tautology. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the level function is smoother when conditional distributions are similar in shape; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)

domain assumption LF and HF conditional distributions have similar shapes making the level function smoother than the HF quantile
Invoked in the abstract as the condition under which the reformulation yields faster convergence.

pith-pipeline@v0.9.0 · 5442 in / 1144 out tokens · 49878 ms · 2026-05-12T05:09:22.553356+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Assumption 1 (Local Quantile Link). For a fixed target level τ and any covariate value x, the HF and LF quantile functions satisfy Q_H(τ|x) = Q_L(u_τ(x)|x) for some covariate-dependent level function u_τ(x) ∈ (0,1).
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear
Proposition 2. Under Assumptions 3–4, q_τ ∈ H(min{β_μ, β_σ}, C_q), r_τ ∈ H(β_σ, …), u_τ ∈ H(β_ρ, C_τ C_ρ).

Reference graph

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