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REVIEW 3 major objections 5 minor 97 references

Kernel Quantile Embeddings and Associated Probability Metrics

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that kernel quantile embeddings are injective for any continuous separating kernel on a sigma-compact Hausdorff space, making the associated kernel quantile discrepancies probability metrics under weaker conditions than…

desk verdict Genuinely new quantile-based distribution metrics with a sound metric theorem, but the implemented estimator is not the one the consistency proof covers. read the letter →

arxiv 2505.20433 v1 pith:MLUWJFUB submitted 2025-05-26 stat.ML cs.LGmath.STstat.TH

Masha Naslidnyk , Siu Lun Chau , François-Xavier Briol , Krikamol Muandet This is my paper · ORCID
classification stat.MLcs.LGmath.STstat.TH MSC 62G0562G1060B1046E22
keywords kernelquantileembeddingdiscrepancymaximummeanslicedWassersteindistancetwo-sampletestingquantile-characteristicGaussianmeasuresampling
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

The paper introduces kernel quantile embeddings (KQEs), which represent a probability distribution by the quantiles of its one-dimensional projections onto unit-norm directions in a reproducing kernel Hilbert space. Its central claim is that this representation is injective for any continuous separating kernel on a $\sigma$-compact Hausdorff space, so every such kernel is quantile-characteristic even when it is not mean-characteristic in the sense required by MMD. From these embeddings the paper builds kernel quantile discrepancies (KQDs), and proves they are probability metrics under those same mild assumptions. If correct, KQDs give practitioners a distance that separates every pair of distributions MMD separates, plus some pairs MMD cannot, with a near-linear-time estimator and an $O(n^{-1/2})$ sample rate.

What carries the argument

The load-bearing object is the kernel quantile embedding (KQE): for a unit-norm RKHS direction $u$, the embedding of $P$ is $\rho^{\alpha,u}_P(x) = \rho^{\alpha}_{u\#P} u(x)$, with $\rho^{\alpha}_{u\#P}$ the ordinary univariate $\alpha$-quantile of the scalar random variable $u(X)$ for $X \sim P$. Comparing these directional quantiles across all $\alpha$ and all $u$ yields the kernel quantile discrepancies, and the proof of injectivity rests on characteristic functionals $\varphi_P(f) = \int e^{i f(x)} P(dx)$, which determine a measure and let the authors extend the Cramér-Wold theorem from $\mathbb{R}^d$ to the RKHS under mild topological assumptions.

What would settle it

Exhibit two distinct Borel probability measures $P$ and $Q$ on a $\sigma$-compact Hausdorff space with a continuous separating kernel such that $\rho^{\alpha}_{u\#P} = \rho^{\alpha}_{u\#Q}$ for every $\alpha \in [0,1]$ and every unit-norm RKHS direction $u$; Theorem 1 asserts no such pair exists, so one such pair would settle the question directly.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the mean is not the only meaningful RKHS representation of a distribution: the full collection of directional quantile embeddings $\{\rho^{\alpha,u}_P : \alpha \in [0,1], u \in \mathbb{S}_{\mathcal{H}}\}$, where $\rho^{\alpha,u}_P = \rho^{\alpha}_{u\#P} u$, is a one-to-one encoding of probability measures under assumptions much weaker than those required for mean embeddings to be characteristic. Theorem 1 establishes this as an RKHS extension of the Cramér-Wold theorem; Theorem 2 shows every mean-characteristic kernel is quantile-characteristic but not conversely, so the polynomial-kernel example distinguishes distributions with matching low-order moments. Theorem 4 turns this into distances: the expected and supremum kernel quantile discrepancies are probability metrics when the averaging measure on quantile levels (and on the sphere, for the expected version) has full support. The paper also proves finite-sample consistency at rate $O(n^{-1/2})$ and shows the Gaussian expected version recovers kernelised sliced Wasserstein distances, linking the construction to optimal transport.

Load-bearing premise

The fast estimator samples directions from a finite-rank Gaussian $\gamma_m$ built from $m = \log n$ reference points, while the metric and consistency theorems assume a full-support Gaussian $\gamma$ on the infinite-dimensional unit sphere; no error bound controls the gap between $\gamma$ and $\gamma_m$.

Editorial extensions

If this is right

  • Any pair of distributions that MMD can separate with a kernel $k$ will also be separated by a KQD built from $k$, because every mean-characteristic kernel is quantile-characteristic.
  • Some pairs that MMD cannot separate, such as a Gaussian and a Laplace distribution with matching first two moments under a cubic polynomial kernel, are separated by KQDs.
  • KQDs are probability metrics under continuous separating kernels on sigma-compact Hausdorff spaces, without requiring the kernel to be characteristic in the mean-embedding sense.
  • With Lebesgue weighting over quantile levels, e-KQD and sup-KQD become kernelised expected and max sliced Wasserstein distances, reducing to classical sliced Wasserstein for a linear kernel with uniform directions.
  • The Gaussian e-KQD estimator runs in $O(n \log^2 n)$ with an $O(n^{-1/2} + l^{-1/2})$ sample-error rate, matching the rate of MMD estimators at lower cost.

Reading between the lines

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

  • Extension: if a bound on the gap between the full-support Gaussian measure $\gamma$ and the finite-rank approximation $\gamma_m$ used in Algorithm 1 can be established, the near-linear estimator would inherit the metric property and consistency guarantees, which currently hold only for the idealised full-support version.
  • Extension: because Theorem 1 needs only a continuous separating kernel, KQDs may provide probability metrics on discrete or structured spaces where characteristic kernels for MMD are known to fail, including the graph-kernel case the paper itself notes.
  • Extension: reweighting the quantile-level measure $\nu$ toward central quantiles gives a natural robustness dial, and the paper's experiments show the optimal weighting is problem-dependent, suggesting a tunable family rather than a single fixed distance.
  • Extension: the paper's future-work remark points toward conditional versions of KQEs, so a conditional quantile embedding for conditional distributions is a concrete and testable next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces kernel quantile embeddings (KQEs), defined as RKHS-valued directional quantiles of a distribution under all unit-norm RKHS directions, and uses them to construct two families of discrepancies, e-KQD and sup-KQD. The main theoretical contributions are Theorem 1 (quantile-characteristic kernels under mild topological assumptions), Theorem 2 (mean-characteristic implies quantile-characteristic, with a converse failure), Theorem 4 (the KQDs are probability metrics under full-support conditions on the quantile-level measure ν and, for e-KQD, the direction measure γ), and Theorem 5 (an O(n^{-1/2} + l^{-1/2}) consistency bound for the p=1 empirical e-KQD with a fixed direction measure γ and Monte Carlo directions γ_l). The paper also connects KQDs to sliced Wasserstein distances, proposes a near-linear-time estimator based on sampling from a Gaussian measure on the RKHS (Algorithm 1), and reports two-sample testing experiments with p=2. The central injectivity and metric results are clean and appear sound, but the practical estimator in Algorithm 1 uses a finite-rank Gaussian direction measure γ_m that is not covered by Theorems 4 and 5, and the proof of Theorem 5 contains a flawed inequality that requires an additional bounded-range assumption.

Significance. If fully established, this work would be a valuable contribution to kernel-based distribution testing: it shows that quantile-based embeddings separate distributions under conditions strictly weaker than those required for MMD, and it provides a conceptual bridge between kernel embeddings and sliced Wasserstein distances. The proofs of Theorems 1, 2, and 4 are elegant and reduce to standard external results (Vakhania's characteristic-functionals theorem, Cramér-Wold, Dvoretzky-Kiefer-Wolfowitz, Bobkov-Ledoux), with no fitted constants or circular reasoning. The paper includes a detailed appendix and a code repository, which are strengths. However, the advertised statistical and metric guarantees do not currently apply to the estimator actually implemented and benchmarked: the finite-rank γ_m used in Algorithm 1 is degenerate and data-dependent, and the p=2 rate used throughout the experiments is left as a conjecture. These gaps are substantial, but they are localized to the estimation theory and presentation, and are plausibly fixable within the manuscript's scope.

major comments (3)
  1. [Section 4, Algorithm 1, Proposition 1 and Theorem 4] The estimator implemented in Algorithm 1 replaces the full-support direction measure γ by γ_m, the projection onto S_H of the finite-rank Gaussian N(0,C_m) built from m=log n reference points z_{1:m}. Theorem 4 guarantees that e-KQD_p is a probability metric only when γ has full support on S_H, and Theorem 5 bounds the difference between the population quantity at a fixed γ and its Monte Carlo average over l directions sampled from that same γ. No result in the paper bounds the gap between e-KQD_p(·;γ_m) and e-KQD_p(·;γ), and γ_m is supported on an m-dimensional subspace of H, so it cannot satisfy the full-support condition of Theorem 4. Consequently, the implemented e-KQD is not proven to be a probability metric, and the consistency rate of Theorem 5 does not apply to it. The authors should either prove a bound on the finite-rank approximation error or explicitly reframe Algorithm 1 as estimating a different, finite-rank object and adjust the theoretical claims accordingly.
  2. [Appendix C.5, proof of Lemma 5] The proof of Lemma 5 contains the inequality ∫_{u(X)} |F_{u#P_n}(t) - F_{u#P}(t)| dt ≤ sup_{t∈u(X)} |F_{u#P_n}(t) - F_{u#P}(t)|, which is only valid if the Lebesgue measure of u(X) is at most one. In general u(X) ⊂ R is unbounded (for example, when k is an unbounded kernel on R^d), and the integral can be infinite even when the supremum is small. This invalidates the current proof of Lemma 5 and therefore the n^{-1/2} term in Theorem 5. The proof needs an additional assumption such as boundedness of u(X), or a truncation and dominated-convergence argument, before the stated consistency result can be accepted.
  3. [Section 5.1 and Conjecture 1] All main experiments in Section 5.1 use power p=2 for KQD-based discrepancies, yet Theorem 5 establishes the O(n^{-1/2}) rate only for p=1. The p>1 rate is explicitly left as Conjecture 1, with a proof sketch that relies on unverified control of the integral in Equation (19). Thus the statistical rate claimed in the abstract and introduction for the empirical estimators is not proven for the value of p used in the benchmark experiments. The authors should either prove Conjecture 1 under the stated regularity conditions, or restrict the rate claims to p=1 and present the p=1 experiments as the primary evidence for the statistical guarantees.
minor comments (5)
  1. [Section 4, Algorithm 1] The notation in Algorithm 1 is confusing: λ_{1:m} ∼ N(0, Id_m) is a vector, but the expressions λ_{1:m}^T k(z_{1:m}, x_{1:n})/√m and λ_{1:m}^T k(z_{1:m}, z_{1:m}) λ_{1:m}/m mix row/column conventions; clarifying whether k(z_{1:m}, x_{1:n}) is an m×n matrix and λ is an m×1 vector would improve reproducibility.
  2. [Theorem 5 and Section 4] Theorem 5 uses γ_l for the Monte Carlo approximation of a fixed γ, while Proposition 1 and Algorithm 1 introduce a different finite-rank measure γ_m based on m reference points. The two are conflated in the main text, which makes it easy for a reader to mistakenly believe that Theorem 5 covers Algorithm 1; the notation should be separated or unified with explicit caveats.
  3. [References] The reference list contains several duplicated entries with different years or suffixes (e.g., Chwialkowski et al. 2015a/b, Nienkötter and Jiang 2022/2023, Makigusa 2024a/b, Wang et al. 2024a/b), which should be consolidated.
  4. [Appendix D.2] The main text states that 'identical experiments for p=1 lead to the same conclusions', but Appendix D.2 reports that centered e-KQD1 performs noticeably worse than centered e-KQD2 in the Laplace-vs-Gaussian experiment; the wording should be softened to reflect this difference.
  5. [Section 5.1] Setting l = log n and m = log n leads to non-integer values for small n (e.g., n=32 gives log n ≈ 3.47); the text should specify that these are rounded to integers in the experiment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems reduce to external results (Vakhania, Gretton, Dvoretzky-Kiefer-Wolfowitz, Bobkov-Ledoux); the finite-rank gamma_m gap is a correctness gap, not a self-referential derivation.

full rationale

The paper's load-bearing mathematical claims do not reduce to their inputs by construction. Theorem 1 (quantile-characteristic kernels) is proved via characteristic functionals and Vakhania et al.'s theorem that characteristic functionals determine the measure, after showing the RKHS unit sphere separates points; this is an external, independent result, not a re-statement of the conclusion. Theorem 2 uses the MMD representer theorem from Gretton et al. (2012) to show a mean-characteristic kernel separates some projection's means, hence some projection's quantiles; again no fitted parameter or self-cited result is load-bearing. The metric property in Theorem 4 follows from Minkowski, full-support assumptions on nu and gamma, and the injectivity of Theorem 1; it is not assumed in advance. Theorem 5's n^{-1/2} rate is derived from the Dvoretzky-Kiefer-Wolfowitz inequality and McDiarmid's inequality, not from the quantity being estimated. The paper contains a genuine gap: Algorithm 1 draws directions from the finite-rank Gaussian gamma_m built from m = log n reference points z_{1:m} with xi = (P_n + Q_n)/2 in the experiments, while Theorem 4's metric statement and Theorem 5's consistency bound require gamma with full support on the RKHS unit sphere. This means the implemented estimator is not currently covered by the advertised guarantees, but this is a correctness/coverage gap, not circularity: gamma_m is not used to define the population e-KQD, no parameter is fitted to the predicted outcome, and the approximation error between gamma_m and gamma is unquantified rather than defined into the result. Self-citations (e.g., Bharti et al. 2023, Chau et al. 2025) appear only in contextual or benchmark-discussion roles and do not carry the argument. No uniqueness theorem from the authors' prior work is invoked to force a choice, and no known result is merely renamed as a prediction. The honest finding is therefore no significant circularity, score 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The theoretical results depend on standard topological and probabilistic theorems plus the explicitly stated A1, A2, and full-support assumptions on nu and gamma. The practical estimator introduces user-chosen parameters l, m, and sigma, but these do not enter the central metric theorems. No new physical or mathematical entities are postulated beyond the new distance definitions.

free parameters (3)
  • RBF bandwidth sigma = median pairwise distance
    Used in all main experiments; chosen by the median heuristic rather than optimized, but it is a data-dependent hyperparameter that affects test power.
  • Number of projections l = log n
    Set to log n in Algorithm 1 to achieve O(n log^2 n) complexity; user-chosen, not fitted to data.
  • Number of random features m = log n
    Set to log n in Algorithm 1; controls the rank of the covariance approximation and the complexity.
assumptions (6)
  • domain assumption A1: X is Hausdorff, separable, and sigma-compact
    Used in Theorems 1 and 4 to ensure the RKHS generates the Borel sigma-algebra via characteristic functionals.
  • domain assumption A2: the kernel k is continuous and separating on X
    Central to Lemma 1 and Theorems 1 and 4; separating kernels distinguish Dirac measures and generate the Borel sigma-algebra.
  • standard math Vakhania Theorem 2.2(a): characteristic functionals on a separating function space determine the Borel probability measure
    External theorem invoked in Appendix C.1 to prove Theorem 1.
  • standard math Dvoretzky-Kiefer-Wolfowitz inequality
    Used in Lemma 5 to obtain n^-1/2 convergence of empirical CDFs under no distributional assumptions.
  • standard math Bobkov-Ledoux Theorem 2.9: equality of integrated quantile differences and L1 CDF distances
    Used in Lemma 5 to relate quantile errors to CDF errors.
  • standard math Kukush Corollary 5.3: non-degenerate Gaussian measures on Hilbert space have full support
    Used to justify full support of gamma in Theorem 4.

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Cite this review

Pith. "Pith review of Kernel Quantile Embeddings and Associated Probability Metrics." pith.science (2026). https://pith.science/paper/MLUWJFUB

@misc{pith2026250520433,
  author       = {Pith},
  title        = {Pith review of: Kernel Quantile Embeddings and Associated Probability Metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLUWJFUB}},
  note         = {Machine review of arXiv:2505.20433}
}
read the original abstract

Embedding probability distributions into reproducing kernel Hilbert spaces (RKHS) has enabled powerful nonparametric methods such as the maximum mean discrepancy (MMD), a statistical distance with strong theoretical and computational properties. At its core, the MMD relies on kernel mean embeddings to represent distributions as mean functions in RKHS. However, it remains unclear if the mean function is the only meaningful RKHS representation. Inspired by generalised quantiles, we introduce the notion of kernel quantile embeddings (KQEs). We then use KQEs to construct a family of distances that: (i) are probability metrics under weaker kernel conditions than MMD; (ii) recover a kernelised form of the sliced Wasserstein distance; and (iii) can be efficiently estimated with near-linear cost. Through hypothesis testing, we show that these distances offer a competitive alternative to MMD and its fast approximations.

Figures

Figures reproduced from arXiv: 2505.20433 by the authors.

Figure 1
Figure 1. Illustration of bivariate quantiles. Left: Bivariate distribution P. Center: Density of the projection of P onto direction u on the unit circle, with ϕu(x) = ⟨u, x⟩. Right: different quantiles for all possible directions u. [y1 . . . yn] ⊤). The α-quantile of P, denoted ρ α P , can be estimated using [y1:n]⌈αn⌉ where ⌈·⌉ denotes the ceiling function. This estimator is known to converge at a rate of O(n −1/2 ) (Serfl… view at source ↗
Figure 2
Figure 2. Illustration of the impact of the slicing direction on KQEs. Suppose X ∼ P, the KQEs ρ α,u P (x) := ρ α u#P u(x) are obtained by considering the α th quantile of u(X). Clearly, these quantiles might vary significantly depend￾ing on the slicing direction used. for which k(x, ·) = k(y, ·) will not be able to distinguish between Dirac measures δx and δy. The proof of the follow￾ing result uses characteristic functional… view at source ↗
Figure 3
Figure 3. Experimental results comparing our proposed methods with baseline approaches. Methods represented by dotted [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Comparing the time (in seconds) required to com [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Type I control results for our experiment on CIFAR-10 v.s. CIFAR-10.1. We see all methods control their Type [PITH_FULL_IMAGE:figures/full_fig_p029_5.png]
Figure 6
Figure 6. Figure 6: The experiments in Figure [PITH_FULL_IMAGE:figures/full_fig_p029_6.png]
Figure 7
Figure 7. Figure 7: Gaussian KQD test power under different weighting measures. [PITH_FULL_IMAGE:figures/full_fig_p030_7.png]
Figure 8
Figure 8. Figure 8: All methods are cost O(n log2 n) unless specified otherwise. Left: Gaussian KQD compared with sliced Wasserstein with uniform or data-driven directions, on the power decay problem. Sliced Wasserstein fall well below KQD—consistent with their equivalence to KQD using a …

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