Distribution-free Measures of Association based on Optimal Transport

Bodhisattva Sen; Nabarun Deb; Promit Ghosal

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Distribution-free Measures of Association based on Optimal Transport

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Pith's one-line read Replacing observations with optimal-transport multivariate ranks makes a kernel graph-based association measure exactly distribution-free, with 0 exactly under independence and 1 exactly when one variable is a measurable function of the…

desk verdict A clever and mostly correct construction of distribution-free OT-rank-based association measures, but the consistency proof has a real index-set error that should be fixable. read the letter →

arxiv 2411.13080 v1 pith:LVAIAFTD submitted 2024-11-20 math.ST math.PRstat.MEstat.TH

Nabarun Deb , Promit Ghosal , Bodhisattva Sen This is my paper · ORCID

classification math.STmath.PRstat.MEstat.TH MSC 62G1062H2060F0560D05

keywords measureofassociationmaximummeandiscrepancymultivariateranksoptimaltransportdistribution-freetestgeometricgraphsreproducingkernelHilbertspaceuniformcentrallimittheorem

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

This paper proposes a class of association measures between random vectors $X$ and $Y$ that combines reproducing kernel Hilbert spaces, geometric graphs, and optimal-transport multivariate ranks. The key claim is that the sample measure $\hat\eta_n^{\mathrm{rank}}$ is exactly distribution-free when $X$ and $Y$ are independent, so it yields an exact finite-sample test of independence in any dimension. The same statistic consistently estimates a population limit $\eta_K^{\mathrm{rank}}$ that is $0$ exactly under independence and $1$ exactly when $Y$ is a measurable function of $X$. If correct, this gives practitioners a single nonparametric number that both measures dependence strength and tests for independence without permutations or asymptotic calibration.

What carries the argument

The load-bearing object is the empirical multivariate rank map, defined by optimally transporting the sample to a fixed set $H_n$ of $n$ uniform-like points in $[0,1]^d$ using the Brenier-McCann optimal transport map. Because the observations are exchangeable, the rank vectors are uniformly distributed over permutations of $H_n$, which is exactly what makes the statistic pivotal under independence. The kernel $K$ measures similarity of rank-transformed $Y$-values, and the geometric graph on the rank-transformed $X$-values estimates the conditional expectation $E[K(R_Y(Y'), R_Y(\tilde Y'))\mid X']$. The statistic is the rank analogue of Spearman correlation: compute the kernel association on ranks rather than raw data.

What would settle it

Under independence, fix small $n$, the grids, the kernel, and the graph, and enumerate the exact distribution of $\hat\eta_n^{\mathrm{rank}}$ for two very different marginal pairs; Theorem 3.1(a) predicts identical distributions, so any discrepancy would disprove the pivotal claim.

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Extended reading notes

Core claim

The central discovery is that replacing $X_i$ and $Y_i$ by their empirical multivariate ranks, defined as the optimal-transport map to a fixed uniform-like grid, makes the kernel-based geometric-graph estimator $\hat\eta_n$ distribution-free while preserving consistency and interpretability. Under the null $X \perp\!\!\perp Y$, the ranks are a uniform random permutation of fixed points, so $\hat\eta_n^{\mathrm{rank}}$ has a pivotal distribution. The paper proves that $\hat\eta_n^{\mathrm{rank}}$ converges to $\eta_K^{\mathrm{rank}}$, establishes that $\eta_K^{\mathrm{rank}}\in[0,1]$ with equality to $0$ iff independence and to $1$ iff $Y=g(X)$ almost surely, and gives a uniform central limit theorem under the null over a large class of geometric graphs. The authors further claim that, to their knowledge, this is the only class of procedures that simultaneously has all of these properties.

Load-bearing premise

The consistency proof needs a Hölder (power-law smoothness) condition on the conditional kernel inner product after rank transformation; the authors say this condition is used only by their proof technique and expect it can be relaxed.

Editorial extensions

If this is right

For any $d_1,d_2\ge 1$, one can test mutual independence at an exact level without permutations, resampling, or asymptotic critical values, because the null distribution of $\hat\eta_n^{\mathrm{rank}}$ is known once $n$, the grid, the kernel, and the graph are fixed.
The population measure gives a single interpretable number: $0$ means independence, $1$ means $Y$ is a measurable function of $X$, and intermediate values compare strength of association.
Consistency holds for estimators based on $k$-nearest-neighbor graphs and minimum spanning trees, so the method inherits the flexibility of graph-based dependence estimation.
The uniform CLT permits data-dependent graph choices, such as a nearest-neighbor radius $k$ that grows logarithmically with $n$, while retaining Gaussian limiting behavior.
For univariate $Y$ with the kernel $K(y_1,y_2)=|y_1|+|y_2|-|y_1-y_2|$, the population limit reduces to the previously proposed scalar regression-dependence coefficient, placing that measure as a special case of this family.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the pivotal property relies only on permutation uniformity of the ranks, the same construction should yield valid exact tests when $H_n$ is any deterministic low-discrepancy sequence; the paper only details uniform-grid choices, but the mechanism is general.
The uniform CLT opens the door to data-driven tuning of the graph, such as choosing $k$ by a pilot estimate of dependence, without invalidating the test; the paper mentions this possibility but does not develop a concrete procedure.
Symmetrizing by $\max(\hat\eta_n^{\mathrm{rank}}(X,Y), \hat\eta_n^{\mathrm{rank}}(Y,X))$ gives a distribution-free measure of mutual dependence that is $1$ iff either variable is a measurable function of the other; the paper notes this but does not analyze its limit distribution.
One could use other reference measures than uniform on the cube, such as a Gaussian or Student-$t$ distribution, to emphasize tail regions; the paper says proofs carry over for compactly supported references but leaves the unbounded-support case open.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper has a genuinely nice idea and mostly delivers what it claims, but the proof of Theorem 3.1(b) has a concrete gap in the second-moment computation. The gap is repairable, but it is real.

What is new and good: the construction is simple once you see it. Applying OT-based multivariate ranks to the authors' earlier kernel/graph measure gives an estimator that is exactly distribution-free under independence, has a well-defined population limit with the 0/1 independence/functional-dependence characterization, and recovers Chatterjee / Azadkia-Chatterjee at the population level for a particular kernel. The distribution-free property is not trivial because OT ranks are exchangeable; the proof is short and correct. The 0/1 characterization also goes through cleanly by inverting the rank map. The uniform null CLT is valuable, and the variance estimator is explicit and pivotal.

The main issue is in Section 5.1. The decomposition E[Z_n^2] = (I)+(II)+(III) is not an identity. (III) as written sums over all ordered pairs of edges (k,i) and (l,j) without requiring the edges to be disjoint. Same-edge pairs and adjacent-edge pairs are therefore counted in (III) as well as in (I) and (II). With n=2 and a single edge, (I)+(II)+(III) gives 1.5 E[K^2] while E[Z_n^2] = E[K^2]. The limits (5.2)-(5.5) are only justified for the disjoint part, so the proof of part (a) is incomplete as printed. I think the fix is straightforward: split (III) into disjoint and overlapping parts and show the overlapping part is negligible using (3.4)-(3.5). For k-NN and MST graphs the overlapping contributions are O(1/n k^2) or O(1/n). But the authors need to write this out.

Two lesser issues. Condition (S1) is a genuine smoothness restriction on the conditional kernel mean embedding; the authors admit in Remark 3.1 that it is only needed for their proof technique. That is honest, but it narrows the scope of the consistency result. The proof also leans on the authors' unpublished preprint [22] for two key ingredients (Theorem 2.1 and Theorem 4.1). That is acceptable if [22] is public and the results are cited precisely, but it makes independent verification harder.

One thing the paper does not do is any simulation; the authors explicitly defer it. For a new measure of association that is a real omission, though not a correctness issue.

Who this is for: theoretical nonparametrics people working on dependence measures, OT ranks, and graph-based inference. It deserves a serious referee. I would send it out despite the proof gap, with instructions to check the repair of (III).

Referee Report

2 major / 4 minor

Summary. This paper introduces a class of distribution-free measures of association between random vectors X in R^{d1} and Y in R^{d2}, constructed by applying the graph-based kernel estimator of Deb et al. [22] to optimal-transport multivariate ranks of the observations. The empirical measure \hat\eta_n^{rank} is shown to have a pivotal distribution under independence (Theorem 3.1(a)), to converge to a population measure \eta_K^{rank} (Theorem 3.1(b)) that is 0 iff X and Y are independent and 1 iff Y is a measurable function of X (Theorem 3.2), and to satisfy a uniform CLT under the null (Theorem 3.3). For d2=1 and a particular kernel, \eta_K^{rank} is shown to coincide with the Azadkia-Chatterjee/Chatterjee coefficient (Proposition 3.1). The proofs are collected in Section 5.

Significance. The proposed construction is conceptually appealing: OT-based multivariate ranks deliver exact distribution-freeness, while the RKHS/graph framework provides interpretable population limits and consistency. If the results hold, this is, to my knowledge, the first class of multivariate association measures combining all three properties (zero/one characterization, consistent distribution-free estimation, and a null CLT). The connection with Chatterjee's coefficient is a nice sanity check. The paper is honest about limitations, e.g., Condition (S1) being a proof-technique assumption. However, a load-bearing gap in the consistency proof needs to be addressed before the results can be considered established.

major comments (2)

[Section 5.1, decomposition of E[Z_n^2] into (I)+(II)+(III)] The set defining (III) only imposes i≠j and k≠ℓ, so it contains reverse-edge terms (k=j, ℓ=i) and two-edge path terms (i=ℓ or k=j) that are already counted in (I) or (II). For n=2 and a single-edge graph, (I)+(II)+(III)=1.5·E[K(Y1,Y2)^2] while E[Z_n^2]=E[K(Y1,Y2)^2]. The later limits (5.2) and the error bound (5.3) require factorizing conditional expectations, which is valid only when the two edges are disjoint. Since the proof of consistency (Theorem 3.1(b)) and hence the consistency of the independence test rest on this step, the gap is load-bearing. The likely repair is to add the distinctness condition on i,j,k,ℓ in (III) and in (~III), but the corrected proof must be supplied.
[Section 5.1, Eq. (5.4)] The sentence 'the last inequality follows once again from Proposition 5.1' is not justified as written. Proposition 5.1 gives n^{-1}∑_i ||\hat R^X_n(X_i)-R^X(X_i)||^2 → 0 almost surely, but (5.4) requires control of the β-Hölder edge sum ∑_i d_i^{-1}∑_{k∼i} ||R^X(X_i)-R^X(X_k)||^β in terms of the corresponding sum with \hat R^X_n. An additional argument (e.g., triangle inequality plus (a+b+c)^β ≤ a^β+b^β+c^β for 0<β≤1) is needed to show the replacement error is o(1) under (S2).

minor comments (4)

[Section 5.1, definition of (III)] The subscript 'RX(Xl)' should read 'RX(X_\ell)'.
[Section 5.1, after Eq. (5.2)] The symbols ~Y1 and ~Y2 are introduced without definition; please align the notation with the earlier (Y', ~Y').
[Theorem 3.3, proof of (5.13)] The step 'The conclusion follows directly from [22, Theorem 4.1]' should be expanded to state why the uniformity over ~G ∈ Jθ is covered and why the rank-transformed observations satisfy the conditions of that theorem.
[Section 3.1 and Remark 3.4] The theorems should clarify whether the sets H^{d1}_n and H^{d2}_n are fixed or random; several steps (e.g., the determinism of D_n and the ~g_j's) assume fixed design points, while (S3) only requires weak convergence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rank-based measure is defined independently, and the load-bearing prior results are external support.

full rationale

The paper's central object, \hat\eta_n^{rank}, is defined directly as a sample statistic in (3.1), and its population limit \eta_K^{rank} is derived as an asymptotic limit in Theorem 3.1(b), not imposed as an input. The distribution-free property in Theorem 3.1(a) follows from the permutation-uniformity of OT-based empirical ranks, a result imported from Deb and Sen [23, Proposition 2.2]. Although this is a self-citation, the cited result is a published, independently stated property of multivariate ranks and is not equivalent to the paper's target claim; it is external support for the rank construction. Similarly, the population properties (P1)-(P3) in Theorem 3.2 are obtained by applying Theorem 2.1 of Deb et al. [22] to the transformed pair (X, R_Y(Y)). This is a legitimate application of an earlier theorem about the unranked kernel measure \eta_K, not a redefinition of the current measure. The CLT in Theorem 3.3 invokes [22, Theorem 4.1] only as one ingredient, alongside new arguments showing the rank approximation error is negligible. Proposition 3.1 derives the equality \eta_K^{rank} = \xi(\mu) for the specific kernel, rather than defining \eta_K^{rank} to equal Chatterjee's coefficient. The skeptical concern about the index sets in the decomposition of E[Z_n^2] in Section 5.1 is a correctness issue in the proof as printed, not a circularity: even if that step is repaired, the claim does not reduce to its inputs by construction. No fitted parameters are renamed as predictions, and no uniqueness theorem from the authors' prior work is used to forbid alternatives. The derivation chain is therefore not circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard OT/RKHS machinery, plus the stated regularity conditions (S1)-(S3) on the distribution and graphs. No numeric free parameters are fitted; the user chooses a characteristic kernel and a geometric graph, but the 0/1 characterization holds for any characteristic kernel and the distribution-free property holds for any choice of the reference sets H_n. The main burden beyond standard theorems is the Holder condition (S1) and the dependence on the authors' earlier preprint [22].

assumptions (7)

domain assumption µ is absolutely continuous on R^{d1+d2} with absolutely continuous marginals (P_ac(R^d))
Required for existence and uniqueness of OT rank maps in Definition 2.5 and Proposition 2.1; stated at the start of Section 3.
domain assumption K is a characteristic and continuous kernel with µ_Y in M^1_K(Y)
Needed for Theorem 2.1/3.2 properties (P1)-(P3), from [22, Theorem 2.1].
ad hoc to paper Condition (S1): r(x1,x2) uniformly beta-Holder continuous
Assumption (3.3) in Theorem 3.1, used in consistency proof; authors note it is needed for the proof technique but expected to hold more generally.
ad hoc to paper Condition (S2): graph degree balance and edge-length condition (3.4)-(3.5)
Assumptions on geometric graphs in Theorem 3.1, analogous to [22].
domain assumption Condition (S3): empirical distributions on H_n converge weakly to U[0,1]^d
Needed for rank maps to converge; standard quasi-Monte Carlo condition.
standard math Brenier-McCann theorem for existence and uniqueness of OT maps
Proposition 2.1, cited from [47].
standard math Moore-Aronszajn theorem for RKHS embedding
Section 2.1, cited from [5].

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Pith. "Pith review of Distribution-free Measures of Association based on Optimal Transport." pith.science (2026). https://pith.science/paper/LVAIAFTD

@misc{pith2026241113080,
  author       = {Pith},
  title        = {Pith review of: Distribution-free Measures of Association based on Optimal Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LVAIAFTD}},
  note         = {Machine review of arXiv:2411.13080}
}

abstract

In this paper we propose and study a class of nonparametric, yet interpretable measures of association between two random vectors $X$ and $Y$ taking values in $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$ respectively ($d_1, d_2\ge 1$). These nonparametric measures -- defined using the theory of reproducing kernel Hilbert spaces coupled with optimal transport -- capture the strength of dependence between $X$ and $Y$ and have the property that they are 0 if and only if the variables are independent and 1 if and only if one variable is a measurable function of the other. Further, these population measures can be consistently estimated using the general framework of geometric graphs which include $k$-nearest neighbor graphs and minimum spanning trees. Additionally, these measures can also be readily used to construct an exact finite sample distribution-free test of mutual independence between $X$ and $Y$. In fact, as far as we are aware, these are the only procedures that possess all the above mentioned desirable properties. The correlation coefficient proposed in Dette et al. (2013), Chatterjee (2021), Azadkia and Chatterjee (2021), at the population level, can be seen as a special case of this general class of measures.

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