Pith. sign in

REVIEW 3 minor 63 references

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

T0 review · 0 major / 3 minor · reviewed 2026-06-27 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read An adapted optimal transport convergence mode restores continuity to dependence measures defined on conditional distributions.

desk verdict The paper sets up an adapted OT convergence mode on conditionals to restore continuity for nonlinear dependence measures and transfers O(N^{-1/3}) rates to their plug-in estimators. read the letter →

arxiv 2606.07354 v1 pith:CNBBFDWE submitted 2026-06-05 math.ST stat.TH

Jonathan Ansari , Johannes Wiesel This is my paper · ORCID
classification math.STstat.TH
keywords dependencemeasuresoptimaltransportadaptedWassersteindistanceplug-inestimatorsconvergenceratesconditionaldistributionscopulasrankcorrelation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

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.

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.

Watch

Extended reading notes

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.

Load-bearing premise

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

Editorial extensions

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.

Reading between the lines

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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 0 assumptions · 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.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dependence Measures via Adapted Optimal Transport: Stability and Rates of Convergence." pith.science (2026). https://pith.science/paper/CNBBFDWE

@misc{pith2026260607354,
  author       = {Pith},
  title        = {Pith review of: Dependence Measures via Adapted Optimal Transport: Stability and Rates of Convergence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CNBBFDWE}},
  note         = {Machine review of arXiv:2606.07354}
}
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.

Figures

Figures reproduced from arXiv: 2606.07354 by the authors.

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. 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. 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 (CˆN ̸= Cˆ# … view at source ↗
Figures from the paper (3 more)
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 …
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 s…
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 ) …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

63 extracted references · 7 canonical work pages

  1. [1]

    Acciaio, B., Backhoff-Veraguas, J., and Carmona, R. (2019). Extended mean field control problems: stochastic maximum principle and transport perspective.SIAM journal on Control and Optimization, 57(6):3666–3693

    2019

  2. [2]

    Acciaio, B., Hou, S., and Pammer, G. (2025). Estimating causal distances with non-causal ones. arXiv preprint arXiv:2506.22421

    work page arXiv 2025

  3. [3]

    Aldous, D. J. (1981). Weak Convergence and General Theory of Processes. Unpublished incomplete draft of monograph; Department of Statistics, University of California, Berkeley, CA 94720

    1981

  4. [4]

    (2005).Gradient flows: in metric spaces and in the space of probability measures

    Ambrosio, L., Gigli, N., and Savar´ e, G. (2005).Gradient flows: in metric spaces and in the space of probability measures. Springer

    2005

  5. [5]

    Ansari, J. (2026). On a copula product linking Wasserstein correlations and rearranged dependence measures. Preprint, arXiv:2604.09950 [math.ST] (2026)

    work page Pith review arXiv 2026

  6. [6]

    and Fuchs, S

    Ansari, J. and Fuchs, S. (2023+). A direct extension of Azadkia & Chatterjee’s rank correlation to multi-response vectors.Available atarXiv: 2212. 01621

    2023

  7. [7]

    and Fuchs, S

    Ansari, J. and Fuchs, S. (2026). An ordering for the strength of functional dependence. Preprint, arXiv:2511.06498 [math.ST] (2026)

    work page arXiv 2026

  8. [8]

    B., Fuchs, S., and Trutschnig, W

    Ansari, J., Langthaler, P. B., Fuchs, S., and Trutschnig, W. (2026). Quantifying and estimating dependence via sensitivity of conditional distributions.Bernoulli, 32(1):179–204

    2026

Show all 63 references
  1. [9]

    and R¨ uschendorf, L

    Ansari, J. and R¨ uschendorf, L. (2021). Sklar’s theorem, copula products, and ordering results in factor models.Depend. Model., 9:267–306

    2021

  2. [10]

    and Chatterjee, S

    Azadkia, M. and Chatterjee, S. (2021). A simple measure of conditional dependence.Ann. Stat., 49(6):3070–3102

    2021

  3. [11]

    and Roudaki, P

    Azadkia, M. and Roudaki, P. (2025+). A new measure of dependence: IntegratedR 2.Available at https: // arxiv. org/ abs/ 2505. 18146

    2025

  4. [12]

    Backhoff, J., Bartl, D., Beiglb¨ ock, M., and Wiesel, J. (2022). Estimating processes in adapted Wasserstein distance.Ann. Appl. Probab., 32(1):529–550

    2022

  5. [13]

    Backhoff-Veraguas, J., Bartl, D., Beiglb¨ ock, M., and Eder, M. (2020). All adapted topologies are equal.Probability Theory and Related Fields, 178:1125–1172

    2020

  6. [14]

    Beiglb¨ ock, M., Pammer, G., and Posch, A. (2023). The Knothe-Rosenblatt distance and its induced topology. Preprint, arXiv:2312.16515 [math.PR] (2023)

    2023

  7. [15]

    Blanchet, J., Larsson, M., Park, J., and Wiesel, J. (2024). Bounding adapted wasserstein metrics. arXiv preprint arXiv:2407.21492

    2024

  8. [16]

    Bonnier, P., Liu, C., and Oberhauser, H. (2023). Adapted topologies and higher rank signatures. The Annals of Applied Probability, 33(3):2136–2175

    2023

  9. [17]

    Borgonovo, E., Figalli, A., Ghosal, P., Plischke, E., and Savar´ e, G. (2025). Convexity and measures of statistical association.J. R. Stat. Soc. Ser. B Stat. Method., 87:1281–1304

    2025

  10. [18]

    and Dette, H

    B¨ ucher, A. and Dette, H. (2024). On the lack of weak continuity of Chatterjee’s correlation coefficient. Preprint, arXiv:2410.11418 [math.ST] (2024)

    2024

  11. [19]

    and Lavenant, H

    Catalano, M. and Lavenant, H. (2025). Measures of dependence based on Wasserstein distances. Preprint, arXiv:2510.06034 [math.ST] (2025)

    2025

  12. [20]

    Chatterjee, S. (2020). A new coefficient of correlation.J. Amer. Statist. Ass., 116(536):2009–2022

    2020

  13. [21]

    Chatterjee, S. (2024). A survey of some recent developments in measures of association. In Athreya, S., Bhatt, A. G., and Rao, B. V., editors,Probability and Stochastic Processes. Indian Statistical Institute Series.Springer, Singapore

    2024

  14. [22]

    Chen, S. X. and Huang, T.-M. (2007). Nonparametric estimation of copula functions for dependence modelling.Can. J. Stat., 35(2):265–282. 34 JONATHAN ANSARI AND JOHANNES WIESEL

    2007

  15. [23]

    F., Nguyen, B., and Olsen, T

    Darsow, W. F., Nguyen, B., and Olsen, T. (1992). Copulas and Markov processes.Illinois J. Math., 36:600–642

    1992

  16. [24]

    Deb, N., Ghosal, P., and Sen, B. (2020). Measuring association on topological spaces using kernels and geometric graphs.Available athttp: // 128. 84. 4. 18/ abs/ 2010. 01768

    2020

  17. [25]

    and Kroll, M

    Dette, H. and Kroll, M. (2025). A simple bootstrap for Chatterjee’s rank correlation.Biometrika, 112(1):15. Id/No asae045

    2025

  18. [26]

    F., and Stoimenov, P

    Dette, H., Siburg, K. F., and Stoimenov, P. A. (2013). A copula-based non-parametric measure of regression dependence.Scand. J. Statist., 40(1):21–41

    2013

  19. [27]

    and Sempi, C

    Durante, F. and Sempi, C. (2016).Principles of Copula Theory. CRC Press, Boca Raton FL

    2016

  20. [28]

    Fermanian, J.-D., Radulovi´ c, D., and Wegkamp, M. (2004). Weak convergence of empirical copula processes.Bernoulli, 10(5):847–860

    2004

  21. [29]

    Gamboa, F., Gremaud, P., Klein, T., and Lagnoux, A. (2022). Global sensitivity analysis: A novel generation of mighty estimators based on rank statistics.Bernoulli, 28(4):2345–2374

    2022

  22. [30]

    G., and R´ emillard, B

    Genest, C., Neˇ slehov´ a, J. G., and R´ emillard, B. (2017). Asymptotic behavior of the empirical multilinear copula process under broad conditions.J. Multivariate Anal., 159:82–110

    2017

  23. [31]

    R., and Trutschnig, W

    Griessenberger, F., Junker, R. R., and Trutschnig, W. (2022). On a multivariate copula-based dependence measure and its estimation.Electron. J. Stat., 16(1):2206–2251

    2022

  24. [32]

    Han, F. (2021). On extensions of rank correlation coefficients to multivariate spaces.Bernoulli, 28(2):7–11

    2021

  25. [33]

    Han, R., Rush, C., and Wiesel, J. (2026). Max-sliced Wasserstein concentration and uniform ratio bounds of empirical measures on RKHS.Bernoulli, 32(3):2123–2148

    2026

  26. [34]

    Hellwig, M. F. (1996). Sequential decisions under uncertainty and the maximum theorem.Journal of Mathematical Economics, 25(4):443–464

    1996

  27. [35]

    Hoover, D. N. and Keisler, H. J. (1984). Adapted probability distributions.Transactions of the American Mathematical Society, 286(1):159–201

    1984

  28. [36]

    Huang, Z., Deb, N., and Sen, B. (2022). Kernel partial correlation coefficient—a measure of condi- tional dependence.J. Mach. Learn. Res., 23(216):1–58

    2022

  29. [37]

    and Sampson, A

    Kimeldorf, G. and Sampson, A. R. (1978). Monotone dependence.Ann. Stat., 6:895–903

    1978

  30. [38]

    Lassalle, R. (2018). Causal transport plans and their Monge–Kantorovich problems.Stochastic Analysis and Applications, 36(3):452–484

    2018

  31. [39]

    Li, X., Mikusi´ nski, P., and Taylor, M. D. (1998). Strong approximation of copulas.J. Math. Anal. Appl., 225(2):608–623

    1998

  32. [40]

    and Han, F

    Lin, Z. and Han, F. (2022). Limit theorems of Chatterjee’s rank correlation.Available athttps: // arxiv. org/ abs/ 2204. 08031v2

    2022

  33. [41]

    and Han, F

    Lin, Z. and Han, F. (2023). On boosting the power of Chatterjee’s rank correlation.Biometrika, 110(2):283–299

    2023

  34. [42]

    Mikusinski, P., Sherwood, H., and Taylor, M. D. (1992). Shuffles of Min.Stochastica, 13(1):61–74

    1992

  35. [43]

    and Taylor, M

    Mikusi´ nski, P. and Taylor, M. D. (2009). Markov operators andn-copulas.Ann. Pol. Math., 96(1):75–95

    2009

  36. [44]

    and Taylor, M

    Mikusi´ nski, P. and Taylor, M. D. (2010). Some approximations ofn-copulas.Metrika, 72(3):385–414

    2010

  37. [45]

    and Stoyan, D

    M¨ uller, A. and Stoyan, D. (2002).Comparison Methods for Stochastic Models and Risks. Wiley, Chichester

    2002

  38. [46]

    Nelsen, R. B. (2006).An Introduction to Copulas.Springer, New York, 2nd edition

    2006

  39. [47]

    Omelka, M., Gijbels, I., and Veraverbeke, N. (2009). Improved kernel estimation of copulas: weak convergence and goodness-of-fit testing.Ann. Stat., 37(5B):3023–3058

    2009

  40. [48]

    Rachev, S. T. and R¨ uschendorf, L. (1998).Mass Transportation Problems. Springer, New York. DEPENDENCE MEASURES VIA ADAPTED OPTIMAL TRANSPORT 35

    1998

  41. [49]

    R¨ uschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics.Ann. Stat., 4:912–923

    1976

  42. [50]

    R¨ uschendorf, L. (1985). The Wasserstein distance and approximation theorems.Probability Theory and Related Fields, 70(1):117–129

    1985

  43. [51]

    (2013).Mathematical Risk Analysis.Springer, Berlin

    R¨ uschendorf, L. (2013).Mathematical Risk Analysis.Springer, Berlin

    2013

  44. [52]

    (2015).Optimal Transport for Applied Mathematicians

    Santambrogio, F. (2015).Optimal Transport for Applied Mathematicians. Birkh¨ auser Cham

    2015

  45. [53]

    and Wolff, E

    Schweizer, B. and Wolff, E. F. (1981). On nonparametric measures of dependence for random variables.Ann. Stat., 9:879–885

    1981

  46. [54]

    Shi, H., Drton, M., and Han, F. (2022). On the power of Chatterjee’s rank correlation.Biometrika, 109(2):317–333

    2022

  47. [55]

    Shi, H., Drton, M., and Han, F. (2024). On Azadkia–Chatterjee’s conditional dependence coefficient. Bernoulli, 30(2):851–877

    2024

  48. [56]

    Siburg, K. F. and Stoimenov, P. A. (2008). Gluing copulas.Commun. Stat., Theory Methods, 37(19):3124–3134

    2008

  49. [57]

    Siburg, K. F. and Strothmann, C. (2021). Stochastic monotonicity and the Markov product for copulas.J. Math. Anal. Appl., 503(2):14

    2021

  50. [58]

    and H¨ ormann, S

    Strenger, D. and H¨ ormann, S. (2025). Azadkia–Chatterjee’s dependence coefficient for infinite di- mensional data.Bernoulli, page to appear

    2025

  51. [59]

    Strothmann, C., Dette, H., and Siburg, K. (2024). Rearranged dependence measures.Bernoulli, 30(2):1055–1078

    2024

  52. [60]

    Trutschnig, W. (2011). On a strong metric on the space of copulas and its induced dependence measure.J. Math. Anal. Appl., 384(2):690–705

    2011

  53. [61]

    (2003).Topics in optimal transportation, volume 58 ofGrad

    Villani, C. (2003).Topics in optimal transportation, volume 58 ofGrad. Stud. Math.Providence, RI: American Mathematical Society (AMS)

    2003

  54. [62]

    (2009).Optimal Transport

    Villani, C. (2009).Optimal Transport. Springer, Berlin

    2009

  55. [63]

    Wiesel, J. (2022). Measuring association with Wasserstein distances.Bernoulli, 28:2816–2832. AppendixA.Auxiliary results Lemma A.1([27, Corollary 4.2.3]).Letπ n, π∈ P(R k ×R l)have cdfs denoted byH n, H. If(π n) converges weakly toπandHis continuous, then∥H n −H∥ ∞ →0. Further...

    2022

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.