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arxiv: 2606.07354 · v1 · pith:CNBBFDWEnew · submitted 2026-06-05 · 🧮 math.ST · stat.TH

Dependence Measures via Adapted Optimal Transport: Stability and Rates of Convergence

Jonathan Ansari , Johannes Wiesel This is my paper

Pith reviewed 2026-06-27 20:27 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords dependence measuresoptimal transportadapted Wasserstein distanceplug-in estimatorsconvergence ratesconditional distributionscopulasrank correlation
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The pith

An adapted optimal transport convergence mode restores continuity to dependence measures defined on conditional distributions.

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

Dependence measures such as Chatterjee's rank correlation detect nonlinear dependencies but fail to be continuous under ordinary weak convergence because they rely on conditional laws rather than the joint distribution alone. The paper introduces a mode of convergence based on adapted optimal transport that tracks weak convergence of these conditional distributions and thereby makes a broad class of such measures continuous. Using this, the authors obtain O(N^{-1/3}) convergence rates for plug-in estimators of rank-based and rearranged dependence measures. They also compare a new copula estimator built from the adapted empirical measure against the classical checkerboard estimator.

Core claim

The paper establishes that an optimal transport-based mode of convergence, related to the adapted Wasserstein distance, captures weak convergence of conditional distributions. This restores continuity for dependence measures that characterize both independence and perfect functional dependence, allowing the derivation of O(N^{-1/3}) rates for their plug-in estimators based on the adapted empirical measure.

What carries the argument

The adapted empirical measure, which induces a convergence mode on conditional distributions via optimal transport that controls the continuity modulus of dependence measures.

If this is right

  • Plug-in estimators of rank-based dependence measures achieve O(N^{-1/3}) rates under the new convergence.
  • Rearranged dependence measures inherit the same convergence rates.
  • The copula estimator based on the adapted empirical measure converges at O(N^{-1/3}) with respect to metrics capturing conditional weak continuity.
  • The new convergence mode is related to the adapted Wasserstein distance, Knothe-Rosenblatt distance, and d1-metric on copulas.

Where Pith is reading between the lines

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

  • This approach may allow consistent estimation in settings where conditional distributions vary, such as regression or time series.
  • The framework could extend to other functionals of conditional distributions beyond dependence measures.
  • Rates might improve under additional smoothness assumptions on the underlying distributions.

Load-bearing premise

The adapted empirical measure together with the chosen metrics on conditional distributions control the continuity modulus of the target dependence measures.

What would settle it

A counterexample showing a dependence measure that remains discontinuous even when conditional distributions converge weakly in the adapted optimal transport sense, or an empirical rate slower than N to the minus one third for the plug-in estimator.

Figures

Figures reproduced from arXiv: 2606.07354 by Johannes Wiesel, Jonathan Ansari.

Figure 1
Figure 1. Figure 1: Topologies on Π(U, U) [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: summarizes these equivalences. W(π n , π) → 0 AW(π n , π) → 0 CW(π n , π) → 0 KR(π n , π) → 0 d1(Cπn , Cπ) → 0 ∥Cπn − Cπ∥∞ → 0 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Representation of estimated distributions (top row) and their associated checkerboard copulas (bottom row) for Example 4.9. The left column corresponds to the adapted empirical copula estimator, the right column to the checkerboard estimator. By convention, each row/column of the doubly stochastic matrices An and A# n sums up to 1, whereas the densities of ˆπ N c and ˆπ N,# c integrate to 1. Example 4.9 (C… view at source ↗
Figure 4
Figure 4. Figure 4: Average rate of convergence of the adapted copula-based estimator Rˆ ϱ = ϱ(Cˆ↑ N ) (dark blue) and the checkerboard copula-based estimator Rˆ# ϱ = ϱ((Cˆ# N ) ↑ ) (light blue) to Rϱ(π) = 6/π arcsin(r/2) for samples of size N ∈ {3 4 , 3 5 , . . . , 3 13} from a Gaussian copula Cπ with parameter r ∈ {0.01, 0.1, 0.4, 0.7, 0.9, 0.99}. Errors are based on 500 runs. Corollary 6.6 (Spearman’s rho). Assume that π h… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the adapted copula-based estimator Rˆ ϱ = ϱ(Cˆ↑ N ) and the checkerboard copula-based estimator Rˆ# ϱ = ϱ((Cˆ# N ) ↑ ) for Rϱ(π) in the Gaussian copula setting (i.e. Cπ is a Gaussian copula) with parameter r ∈ {0.01, 0.1, 0.4, 0.7, 0.9, 0.99} for sample sizes N ∈ {30, 100, 300, 1000, 3000}. Each boxplot is based on 500 runs. very similarly, exhibiting convergence rates of at least O(N −1/3 ) … view at source ↗
Figure 6
Figure 6. Figure 6: Average rate of convergence of the adapted copula-based estimator Rˆ ϱ = ϱ(Cˆ↑ N ) (dark blue) and the checkerboard copula-based estimator Rˆ# ϱ = ϱ((Cˆ# N ) ↑ ) (light blue) for Rϱ(π) for samples of size N ∈ {3 4 , 3 5 , . . . , 3 11} from the copula Cπ where (X, Y ) ∼ π with Y = X2 + rε has U-shape: X is uniform on [−1, 1] and independent of standard normal ε, and r ∈ {5, 2, 0.7, 0.3, 0.1, 0.03}. The app… view at source ↗
read the original abstract

Recently studied dependence measures, such as Chatterjee's rank correlation, that characterize both independence and perfect functional dependence, provide a powerful framework for detecting nonlinear dependencies. However, these measures cannot be weakly continuous, which limits the applicability of classical plug-in estimators based on empirical distributions. This obstruction is natural, as such measures are defined via conditional distributions and not through their joint law alone. In this paper, we introduce an optimal transport-based mode of convergence that captures weak convergence of conditional distributions and restores continuity for a broad class of dependence measures. We relate this mode of convergence to the adapted Wasserstein distance, the Knothe-Rosenblatt distance and the d1-metric on copulas. Building on this perspective, we propose a copula estimator based on the adapted empirical measure and compare it with the classical rank-based checkerboard estimator. For both estimators, we derive O(N^{-1/3})-rates of convergence with respect to metrics that capture conditional weak continuity. As a consequence, we obtain the same rates for plug-in estimators of several classes of dependence measures, including rank-based and rearranged dependence measures.

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.

Referee Report

0 major / 3 minor

Summary. The paper introduces an optimal transport-based mode of convergence that metrizes weak convergence of conditional distributions, thereby restoring continuity for a broad class of dependence measures (including Chatterjee's rank correlation) that are defined via conditional laws rather than the joint distribution alone. It relates this mode to the adapted Wasserstein distance, the Knothe-Rosenblatt distance, and the d1-metric on copulas; proposes a copula estimator based on the adapted empirical measure and compares it to the classical rank-based checkerboard estimator; derives O(N^{-1/3}) rates of convergence for both estimators with respect to metrics capturing conditional weak continuity; and transfers these rates to plug-in estimators of rank-based and rearranged dependence measures.

Significance. If the central claims hold, the work provides a principled way to obtain explicit convergence rates for plug-in estimators of nonlinear dependence measures that previously lacked weak continuity, which is a meaningful advance for statistical inference on dependence. The explicit connections to adapted OT, Knothe-Rosenblatt, and copula metrics, together with the comparison of the adapted empirical estimator to the checkerboard estimator, strengthen the contribution. The use of standard empirical-process arguments adapted to the conditional setting to obtain the O(N^{-1/3}) rate is a positive technical feature.

minor comments (3)
  1. [§2.2] §2.2: the precise definition of the adapted empirical measure and the metric on conditional distributions should include an explicit statement of the constants or normalizations used, to make the O(N^{-1/3}) rate derivation fully transparent without reference to external results.
  2. [§4] The comparison between the adapted estimator and the checkerboard estimator in §4 would benefit from a short table summarizing the constants in the O(N^{-1/3}) bounds for each, to clarify whether one dominates the other uniformly.
  3. [Introduction / §3] Notation for the dependence measures (e.g., the functional form of Chatterjee's measure) is introduced in the abstract and introduction but should be restated once in the main technical section before the continuity argument is applied.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and the recommendation of minor revision. No specific major comments were raised in the report, so we have no points to address point-by-point. We will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from standard optimal transport objects (adapted Wasserstein, Knothe-Rosenblatt, copula metrics) and defines a new mode of convergence that metrizes weak convergence of conditional distributions. Continuity restoration for dependence measures and the O(N^{-1/3}) rates for plug-in estimators follow from the metric properties and standard empirical-process arguments applied to the adapted empirical measure; neither step reduces to a self-definition, a fitted parameter renamed as prediction, nor a load-bearing self-citation. The construction is externally grounded in existing OT theory and does not invoke uniqueness theorems or ansatzes from the authors' prior work to force the result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the new convergence mode is presented as a definition rather than a derived object.

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