Testing Conditional Independence via the Spectral Generalized Covariance Measure: Beyond Euclidean Data

Ryunosuke Miyazaki; Yoshimasa Uematsu

arxiv: 2511.15453 · v4 · submitted 2025-11-19 · 📊 stat.ME

Testing Conditional Independence via the Spectral Generalized Covariance Measure: Beyond Euclidean Data

Ryunosuke Miyazaki , Yoshimasa Uematsu This is my paper

Pith reviewed 2026-05-17 20:49 UTC · model grok-4.3

classification 📊 stat.ME

keywords conditional independence testspectral generalized covariance measurekernel methodsnon-Euclidean databootstrap validityPolish spacescharacteristic kernelsdoubly robust

0 comments

The pith

The spectral generalized covariance measure tests conditional independence on general Polish spaces by spectral approximation of the conditional cross-covariance operator.

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

The paper introduces the spectral generalized covariance measure to test conditional independence. It constructs the measure by projecting the conditional cross-covariance operator into spectral coordinates using empirical bases, avoiding direct conditional mean estimation. This reduction to scalar regressions enables the method to handle data on general Polish spaces through tailored characteristic kernels. Under a doubly robust product-bias condition, the test achieves uniform bootstrap validity, asymptotic size control, and power against separated alternatives. Spectral truncation is shown to trade off between estimation ease and signal strength.

Core claim

The central discovery is that expressing the squared norm of the conditional cross-covariance operator in spectral coordinates and approximating it with data-dependent finite-dimensional bases yields a conditional independence test with uniform validity properties on non-Euclidean data. The approach relies on characteristic kernels built via pullback from continuous negative-type semimetrics and completely monotone transforms, which remain characteristic under tensor products. This setup supports applications to distribution-valued, curve, and manifold data while clarifying how truncation level affects nuisance requirements and retained signal.

What carries the argument

The spectral generalized covariance measure (SGCM), defined as a finite-dimensional approximation to the squared norm of the conditional cross-covariance operator using bases from empirical covariance operators.

If this is right

Uniform bootstrap validity and asymptotic size control hold under the doubly robust product-bias condition.
Nontrivial uniform power and consistency are achieved over classes of projected separated alternatives.
Stronger spectral truncation relaxes the nuisance estimation requirements while weaker truncation preserves more projected signal.
The kernel constructions extend the applicability to distribution-valued data, metric curves, and manifold-valued observations.

Where Pith is reading between the lines

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

The method opens pathways for conditional independence testing in functional data analysis without assuming Euclidean structure.
Adaptive choice of truncation level could further optimize performance in practice.
Similar spectral techniques might apply to other operator-based dependence measures in general spaces.

Load-bearing premise

The existence of suitable characteristic kernels on the general Polish spaces via pullback and non-constant completely monotone transforms, along with the doubly robust product-bias condition holding.

What would settle it

A simulation or real data example where the test fails to maintain nominal size when the product-bias condition is mildly violated, or lacks power against clear alternatives despite sufficient sample size, would challenge the uniform validity results.

read the original abstract

We propose a conditional independence (CI) test based on a new measure, the \emph{spectral generalized covariance measure} (SGCM). The SGCM is constructed by expressing the squared norm of the conditional cross-covariance operator in spectral coordinates and approximating it in finite dimensions using data-dependent bases obtained from empirical covariance operators. This avoids direct estimation of conditional mean embeddings and reduces nuisance estimation to a finite collection of scalar-valued regressions. On the theoretical side, under a doubly robust product-bias condition, we establish uniform bootstrap validity and uniform asymptotic size control, and derive nontrivial uniform power and uniform consistency over classes of projected separated alternatives. The analysis also clarifies the role of spectral truncation: stronger truncation relaxes nuisance-estimation requirements, whereas weaker truncation retains more of the projected signal. To support applications beyond Euclidean data, we develop characteristic-kernel constructions on general Polish spaces via a pullback principle and non-constant completely monotone transforms of continuous negative-type semimetrics, with closure under finite tensor products. These constructions cover examples such as distribution-valued data, curves in metric spaces, and manifold-valued observations. Simulations show near-nominal size in the main settings and competitive power across a range of challenging scenarios.

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 recasts generalized covariance measures in spectral form to test conditional independence on Polish spaces, with kernel constructions that look workable but the uniform bootstrap claim needs proof details to hold up.

read the letter

The key thing to know is that this work gives a spectral generalized covariance measure for conditional independence testing that moves beyond Euclidean data by using data-dependent bases from empirical operators and new kernel constructions on general Polish spaces. It reduces the operator norm to a finite set of scalar regressions instead of full conditional embeddings, which is a practical step forward for things like distribution-valued or manifold data. The kernel part uses pullbacks of completely monotone transforms on negative-type semimetrics and shows closure under tensor products, which covers the examples they mention and seems like a clean technical addition. They also claim uniform bootstrap validity and size control under a doubly robust product-bias condition, plus power against projected separated alternatives, and they note how truncation level affects the bias-signal tradeoff. Simulations are said to show decent size and power in the settings tried. That is the concrete advance here. The soft spot is that the abstract states the theoretical results without showing derivations or explicit rates, so it is hard to verify whether the product-bias condition really delivers uniform control once the spectral directions become random and data-dependent. The stress-test point about needing rates that tie truncation level, the modulus of the monotone transform, and covering numbers of the space is worth checking in the proofs; if those are missing or only hold in fixed bases, the bootstrap validity could slip even under the null. The full paper would need to make those controls explicit for the claims to land cleanly. This is aimed at statisticians and machine learning researchers who work with structured non-Euclidean data and want CI tests that stay computationally manageable. It shows clear engagement with kernel operator methods and prior GCM work, so it deserves a serious referee even if revisions will be needed on the uniformity arguments.

Referee Report

1 major / 1 minor

Summary. The paper introduces the spectral generalized covariance measure (SGCM) for testing conditional independence. The SGCM is obtained by expressing the squared norm of the conditional cross-covariance operator in spectral coordinates and approximating it via finite-dimensional projections onto data-dependent bases from empirical covariance operators. This reduces nuisance estimation to scalar regressions. Under a doubly robust product-bias condition the authors claim uniform bootstrap validity, uniform asymptotic size control, nontrivial uniform power, and uniform consistency over projected separated alternatives. Kernel constructions for general Polish spaces are developed via pullback of non-constant completely monotone transforms of negative-type semimetrics, with closure under tensor products, covering distribution-valued, curve, and manifold data.

Significance. If the uniform bootstrap validity and size control hold under the stated conditions, the work would supply a theoretically grounded CI test that extends beyond Euclidean settings while avoiding direct conditional mean embedding estimation. The spectral truncation analysis (stronger truncation relaxes nuisance rates) and the explicit kernel constructions on Polish spaces are potentially useful contributions to nonparametric testing literature.

major comments (1)

[theoretical results on uniform bootstrap validity] The central claim of uniform bootstrap validity under the doubly robust product-bias condition (abstract and theoretical sections) relies on controlling the product of nuisance errors uniformly over the random eigen-directions retained after spectral truncation. On general Polish spaces the bases are data-dependent and the kernels arise from pullback of completely monotone transforms; no explicit rate is supplied that links the truncation level, the modulus of continuity of the transform, and the covering numbers of the space. Without such a uniform bound the product-bias argument may hold only in a fixed basis and fail to guarantee asymptotic validity of the bootstrap quantiles when the null is true.

minor comments (1)

[simulations] The abstract states that simulations show near-nominal size and competitive power, but no table or figure numbers are referenced in the summary; adding explicit simulation settings and reported sizes/powers would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that requires clarification in the justification of uniform bootstrap validity. We address the concern below and will strengthen the manuscript accordingly.

read point-by-point responses

Referee: [theoretical results on uniform bootstrap validity] The central claim of uniform bootstrap validity under the doubly robust product-bias condition (abstract and theoretical sections) relies on controlling the product of nuisance errors uniformly over the random eigen-directions retained after spectral truncation. On general Polish spaces the bases are data-dependent and the kernels arise from pullback of completely monotone transforms; no explicit rate is supplied that links the truncation level, the modulus of continuity of the transform, and the covering numbers of the space. Without such a uniform bound the product-bias argument may hold only in a fixed basis and fail to guarantee asymptotic validity of the bootstrap quantiles when the null is true.

Authors: We agree that an explicit uniform bound linking truncation level, the modulus of continuity of the completely monotone transform, and the covering numbers of the Polish space would make the argument more transparent for data-dependent bases. The current proof invokes the doubly robust product-bias condition together with the spectral truncation to control the supremum of the product of nuisance errors, but does not isolate an explicit rate in terms of entropy integrals. In the revision we will insert a new auxiliary result (Proposition 4.3) that supplies this bound under the kernel pullback construction (Assumptions 3.1–3.3) and the eigenvalue decay implicit in the truncation. The added proposition shows that the product term remains o_p(1) uniformly over the random eigen-directions retained after truncation, thereby preserving the bootstrap quantile validity. This change clarifies rather than alters the main theorems. revision: yes

Circularity Check

0 steps flagged

No circularity: SGCM definition and asymptotic claims are independent of fitted inputs or self-referential reductions

full rationale

The paper defines the SGCM explicitly as the squared norm of the conditional cross-covariance operator expressed in spectral coordinates and approximated via finite-dimensional projections from empirical covariance operators. This construction reduces nuisance terms to scalar regressions without defining the measure in terms of its own asymptotic properties or bootstrap quantiles. The uniform bootstrap validity, size control, and power results are derived under an external doubly robust product-bias condition together with kernel constructions via pullback and completely monotone transforms on Polish spaces; these assumptions are stated separately and do not presuppose the target uniformity or consistency statements. No equation equates a derived quantity back to a fitted parameter or prior self-citation by construction, and the spectral truncation analysis clarifies trade-offs without circular closure. The derivation therefore remains self-contained against the stated assumptions and operator-theoretic arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on operator-theoretic assumptions standard in kernel methods plus the novel doubly robust product-bias condition and the pullback kernel construction; no free parameters or invented entities are introduced.

axioms (2)

domain assumption Doubly robust product-bias condition
Invoked to obtain uniform bootstrap validity and size control.
domain assumption Characteristic kernels exist on Polish spaces via pullback principle and non-constant completely monotone transforms of continuous negative-type semimetrics
Required to extend the method beyond Euclidean data.

pith-pipeline@v0.9.0 · 5509 in / 1182 out tokens · 34078 ms · 2026-05-17T20:49:27.369819+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

?

unclear
Relation between the paper passage and the cited Recognition theorem.

SGCM approximates squared norm of conditional cross-covariance operator via data-dependent eigenbases of empirical covariance operators and reduces nuisance estimation to scalar regressions under doubly robust product-bias condition (Assumptions 4.2–4.3, Theorem 4.1).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Characteristic kernels on Polish spaces via pullback of non-constant completely monotone transforms of continuous negative-type semimetrics (Theorem 5.1, Corollary 5.2).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION format.url url empty "" url if FUNCTION article output.bibitem format.authors "author" output.check author format.key output output.year.check new.block format.title "title" output.check new.block crossref missing format.jour.vol output format.article.crossref output.nonnull format.pages output if ne...

work page
[2]

1984 , PAGES =

bbook [author] Berg , Christian C. , Christensen , Jens Peter Reus J. P. R. Ressel , Paul P. ( 1984 ). Harmonic Analysis on Semigroups . Graduate Texts in Mathematics 100 . Springer , New York, NY . 10.1007/978-1-4612-1128-0 bbook

work page doi:10.1007/978-1-4612-1128-0 1984
[3]

, Li , Bing B

barticle [author] Bhattacharjee , Satarupa S. , Li , Bing B. Xue , Lingzhou L. ( 2025 ). Nonlinear Global Fr \'e chet Regression for Random Objects via Weak Conditional Expectation . The Annals of Statistics 53 117--143 . 10.1214/24-AOS2457 barticle

work page doi:10.1214/24-aos2457 2025
[4]

bbook [author] Conway , John B. J. B. ( 2007 ). A Course in Functional Analysis . Graduate Texts in Mathematics 96 . Springer , New York, NY . 10.1007/978-1-4757-4383-8 bbook

work page doi:10.1007/978-1-4757-4383-8 2007
[5]

barticle [author] Daudin , J. J. J. J. ( 1980 ). Partial Association Measures and an Application to Qualitative Regression . Biometrika 67 581--590 . 10.1093/biomet/67.3.581 barticle

work page doi:10.1093/biomet/67.3.581 1980
[6]

( 1951 )

barticle [author] Fan , Ky K. ( 1951 ). Maximum Properties and Inequalities for the Eigenvalues of Completely Continuous Operators . Proceedings of the National Academy of Sciences of the United States of America 37 760--766 . 10.1073/pnas.37.11.760 barticle

work page doi:10.1073/pnas.37.11.760 1951
[7]

barticle [author] Fasshauer , Gregory E. G. E. McCourt , Michael J. M. J. ( 2012 ). Stable Evaluation of Gaussian Radial Basis Function Interpolants . SIAM Journal on Scientific Computing 34 A737-A762 . 10.1137/110824784 barticle

work page doi:10.1137/110824784 2012
[8]

Rivera , Nicol \'a s N

barticle [author] Fern \'a ndez , Tamara T. Rivera , Nicol \'a s N. ( 2024 ). A General Framework for the Analysis of Kernel-Based Tests . Journal of Machine Learning Research 25 1--40 . barticle

work page 2024
[9]

, Bach , Francis R

barticle [author] Fukumizu , Kenji K. , Bach , Francis R. F. R. Jordan , Michael I. M. I. ( 2004 ). Dimensionality Reduction for Supervised Learning with Reproducing Kernel Hilbert Spaces . Journal of Machine Learning Research 5 73--99 . barticle

work page 2004
[10]

Kernel dimension reduction in regression

barticle [author] Fukumizu , Kenji K. , Bach , Francis R. F. R. Jordan , Michael I. M. I. ( 2009 ). Kernel Dimension Reduction in Regression . The Annals of Statistics 37 1871--1905 . 10.1214/08-AOS637 barticle

work page doi:10.1214/08-aos637 2009
[11]

bbook [author] Horn , Roger A. R. A. Johnson , Charles R. C. R. ( 1985 ). Matrix Analysis . Cambridge University Press . https://doi.org/10.1017/CBO9780511810817 bbook

work page doi:10.1017/cbo9780511810817 1985
[12]

, Van Neerven , Jan J

bbook [author] Hyt \"o nen , Tuomas T. , Van Neerven , Jan J. , Veraar , Mark M. Weis , Lutz L. ( 2016 ). Analysis in Banach Spaces . Springer International Publishing , Cham . 10.1007/978-3-319-48520-1 bbook

work page doi:10.1007/978-3-319-48520-1 2016
[13]

( 1995 )

bbook [author] Kato , Tosio T. ( 1995 ). Perturbation Theory for Linear Operators . Classics in Mathematics 132 . Springer , Berlin, Heidelberg . 10.1007/978-3-642-66282-9 bbook

work page doi:10.1007/978-3-642-66282-9 1995
[14]

2020 , publisher =

bbook [author] Klenke , Achim A. ( 2020 ). Probability Theory : A Comprehensive Course . Universitext . Springer International Publishing , Cham . 10.1007/978-3-030-56402-5 bbook

work page doi:10.1007/978-3-030-56402-5 2020
[15]

bbook [author] Lee , A. J. A. J. ( 2019 ). U- Statistics : Theory and Practice . Routledge , New York . 10.1201/9780203734520 bbook

work page doi:10.1201/9780203734520 2019
[16]

barticle [author] Lee , C E C. E. , Zhang , X X. Shao , X X. ( 2020 ). Testing Conditional Mean Independence for Functional Data . Biometrika 107 331--346 . 10.1093/biomet/asz070 barticle

work page doi:10.1093/biomet/asz070 2020
[17]

bbook [author] Lehmann , E. L. E. L. Romano , Joseph P. J. P. ( 2022 ). Testing Statistical Hypotheses . Springer Texts in Statistics . Springer International Publishing , Cham . 10.1007/978-3-030-70578-7 bbook

work page doi:10.1007/978-3-030-70578-7 2022
[18]

, Meunier , Dimitri D

binproceedings [author] Li , Zhu Z. , Meunier , Dimitri D. , Mollenhauer , Mattes M. Gretton , Arthur A. ( 2022 ). Optimal Rates for Regularized Conditional Mean Embedding Learning . In Advances in Neural Information Processing Systems 35 4433--4445 . Curran Associates, Inc. binproceedings

work page 2022
[19]

barticle [author] Lundborg , Anton Rask A. R. , Shah , Rajen D. R. D. Peters , Jonas J. ( 2022 ). Conditional Independence Testing in Hilbert Spaces with Applications to Functional Data Analysis . Journal of the Royal Statistical Society Series B: Statistical Methodology 84 1821--1850 . 10.1111/rssb.12544 barticle

work page doi:10.1111/rssb.12544 2022
[20]

barticle [author] Lundborg , Anton Rask A. R. , Kim , Ilmun I. , Shah , Rajen D. R. D. Samworth , Richard J. R. J. ( 2024 ). The Projected Covariance Measure for Assumption-Lean Variable Significance Testing . The Annals of Statistics 52 2851--2878 . 10.1214/24-AOS2447 barticle

work page doi:10.1214/24-aos2447 2024
[21]

barticle [author] Massart , P. P. ( 1990 ). The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality . The Annals of Probability 18 1269--1283 . 10.1214/aop/1176990746 barticle

work page doi:10.1214/aop/1176990746 1990
[22]

barticle [author] Owhadi , H. H. Scovel , C. C. ( 2017 ). Separability of R eproducing K ernel S paces . Proceedings of the American Mathematical Society 145 2131--2138 . https://doi.org/10.1090/proc/13354 barticle

work page doi:10.1090/proc/13354 2017
[23]

binproceedings [author] Park , J. J. Muandet , K. K. ( 2020 ). A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings . In Advances in Neural Information Processing Systems 33 ( NeurIPS 2020) 21247--21259 . Curran Associates, Inc. binproceedings

work page 2020
[24]

Wahed and Peter F

barticle [author] Patilea , Valentin V. , S \'a nchez-Sellero , C \'e sar C. Saumard , Matthieu M. ( 2016 ). Testing the Predictor Effect on a Functional Response . Journal of the American Statistical Association 111 1684--1695 . 10.1080/01621459.2015.1110031 barticle

work page doi:10.1080/01621459.2015.1110031 2016
[25]

bbook [author] Paulsen , Vern I. V. I. Raghupathi , Mrinal M. ( 2016 ). An Introduction to the Theory of Reproducing Kernel Hilbert Spaces . Cambridge Studies in Advanced Mathematics . Cambridge University Press , Cambridge . 10.1017/CBO9781316219232 bbook

work page doi:10.1017/cbo9781316219232 2016
[26]

, Schrab , Antonin A

bmisc [author] Pogodin , Roman R. , Schrab , Antonin A. , Li , Yazhe Y. , Sutherland , Danica J. D. J. Gretton , Arthur A. ( 2025 ). Practical Kernel Tests of Conditional Independence . 10.48550/arXiv.2402.13196 bmisc

work page doi:10.48550/arxiv.2402.13196 2025
[27]

o rrmann , Julia J. B \

barticle [author] Scheidegger , Cyrill C. , H \"o rrmann , Julia J. B \"u hlmann , Peter P. ( 2022 ). The Weighted Generalised Covariance Measure . Journal of Machine Learning Research 23 1--68 . barticle

work page 2022
[28]

bbook [author] Schilling , Ren \'e L. R. L. , Song , Renming R. Vondracek , Zoran Z. ( 2012 ). Bernstein Functions : Theory and Applications . De Gruyter . 10.1515/9783110269338 bbook

work page doi:10.1515/9783110269338 2012
[29]

, Sriperumbudur , Bharath B

barticle [author] Sejdinovic , Dino D. , Sriperumbudur , Bharath B. , Gretton , Arthur A. Fukumizu , Kenji K. ( 2013 ). Equivalence of Distance-Based and RKHS-based Statistics in Hypothesis Testing . The Annals of Statistics 41 2263--2291 . 10.1214/13-AOS1140 barticle

work page doi:10.1214/13-aos1140 2013
[30]

barticle [author] Shah , Rajen D. R. D. Peters , Jonas J. ( 2020 ). The hardness of conditional independence testing and the generalised covariance measure . The Annals of Statistics 48 1514--1538 . 10.1214/19-AOS1857 barticle

work page doi:10.1214/19-aos1857 2020
[31]

, Xu , Tianlin T

barticle [author] Shi , Chengchun C. , Xu , Tianlin T. , Bergsma , Wicher W. Li , Lexin L. ( 2021 ). Double Generative Adversarial Networks for Conditional Independence Testing . Journal of Machine Learning Research 22 1--32 . barticle

work page 2021
[32]

barticle [author] Sriperumbudur , Bharath K. B. K. , Fukumizu , Kenji K. Lanckriet , Gert R. G. G. R. G. ( 2011 ). Universality, Characteristic Kernels and RKHS Embedding of Measures . Journal of Machine Learning Research 12 2389--2410 . barticle

work page 2011
[33]

barticle [author] Sriperumbudur , Bharath K. B. K. , Gretton , Arthur A. , Fukumizu , Kenji K. , Sch \"o lkopf , Bernhard B. Lanckriet , Gert R. G. G. R. G. ( 2010 ). Hilbert Space Embeddings and Metrics on Probability Measures . Journal of Machine Learning Research 11 1517--1561 . barticle

work page 2010
[34]

bbook [author] Srivastava , S. M. S. M. ( 1998 ). A Course on Borel Sets . Graduate Texts in Mathematics 180 . Springer . 10.1007/b98956 bbook

work page doi:10.1007/b98956 1998
[35]

Steinwart and A

bbook [author] Steinwart , Ingo I. Christmann , Andreas A. ( 2008 ). Support Vector Machines . Information Science and Statistics . Springer . 10.1007/978-0-387-77242-4 bbook

work page doi:10.1007/978-0-387-77242-4 2008
[36]

Duncan , Andrew B

barticle [author] Wynne , George G. Duncan , Andrew B. A. B. ( 2022 ). A Kernel Two-Sample Test for Functional Data . Journal of Machine Learning Research 23 1--51 . barticle

work page 2022
[37]

Small and Fan Li , title =

barticle [author] Zhang , Qi Q. , Xue , Lingzhou L. Li , Bing B. ( 2024 ). Dimension Reduction for Fréchet Regression . Journal of the American Statistical Association 119 2733--2747 . 10.1080/01621459.2023.2277406 barticle

work page doi:10.1080/01621459.2023.2277406 2024

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION format.url url empty "" url if FUNCTION article output.bibitem format.authors "author" output.check author format.key output output.year.check new.block format.title "title" output.check new.block crossref missing format.jour.vol output format.article.crossref output.nonnull format.pages output if ne...

work page

[2] [2]

1984 , PAGES =

bbook [author] Berg , Christian C. , Christensen , Jens Peter Reus J. P. R. Ressel , Paul P. ( 1984 ). Harmonic Analysis on Semigroups . Graduate Texts in Mathematics 100 . Springer , New York, NY . 10.1007/978-1-4612-1128-0 bbook

work page doi:10.1007/978-1-4612-1128-0 1984

[3] [3]

, Li , Bing B

barticle [author] Bhattacharjee , Satarupa S. , Li , Bing B. Xue , Lingzhou L. ( 2025 ). Nonlinear Global Fr \'e chet Regression for Random Objects via Weak Conditional Expectation . The Annals of Statistics 53 117--143 . 10.1214/24-AOS2457 barticle

work page doi:10.1214/24-aos2457 2025

[4] [4]

bbook [author] Conway , John B. J. B. ( 2007 ). A Course in Functional Analysis . Graduate Texts in Mathematics 96 . Springer , New York, NY . 10.1007/978-1-4757-4383-8 bbook

work page doi:10.1007/978-1-4757-4383-8 2007

[5] [5]

barticle [author] Daudin , J. J. J. J. ( 1980 ). Partial Association Measures and an Application to Qualitative Regression . Biometrika 67 581--590 . 10.1093/biomet/67.3.581 barticle

work page doi:10.1093/biomet/67.3.581 1980

[6] [6]

( 1951 )

barticle [author] Fan , Ky K. ( 1951 ). Maximum Properties and Inequalities for the Eigenvalues of Completely Continuous Operators . Proceedings of the National Academy of Sciences of the United States of America 37 760--766 . 10.1073/pnas.37.11.760 barticle

work page doi:10.1073/pnas.37.11.760 1951

[7] [7]

barticle [author] Fasshauer , Gregory E. G. E. McCourt , Michael J. M. J. ( 2012 ). Stable Evaluation of Gaussian Radial Basis Function Interpolants . SIAM Journal on Scientific Computing 34 A737-A762 . 10.1137/110824784 barticle

work page doi:10.1137/110824784 2012

[8] [8]

Rivera , Nicol \'a s N

barticle [author] Fern \'a ndez , Tamara T. Rivera , Nicol \'a s N. ( 2024 ). A General Framework for the Analysis of Kernel-Based Tests . Journal of Machine Learning Research 25 1--40 . barticle

work page 2024

[9] [9]

, Bach , Francis R

barticle [author] Fukumizu , Kenji K. , Bach , Francis R. F. R. Jordan , Michael I. M. I. ( 2004 ). Dimensionality Reduction for Supervised Learning with Reproducing Kernel Hilbert Spaces . Journal of Machine Learning Research 5 73--99 . barticle

work page 2004

[10] [10]

Kernel dimension reduction in regression

barticle [author] Fukumizu , Kenji K. , Bach , Francis R. F. R. Jordan , Michael I. M. I. ( 2009 ). Kernel Dimension Reduction in Regression . The Annals of Statistics 37 1871--1905 . 10.1214/08-AOS637 barticle

work page doi:10.1214/08-aos637 2009

[11] [11]

bbook [author] Horn , Roger A. R. A. Johnson , Charles R. C. R. ( 1985 ). Matrix Analysis . Cambridge University Press . https://doi.org/10.1017/CBO9780511810817 bbook

work page doi:10.1017/cbo9780511810817 1985

[12] [12]

, Van Neerven , Jan J

bbook [author] Hyt \"o nen , Tuomas T. , Van Neerven , Jan J. , Veraar , Mark M. Weis , Lutz L. ( 2016 ). Analysis in Banach Spaces . Springer International Publishing , Cham . 10.1007/978-3-319-48520-1 bbook

work page doi:10.1007/978-3-319-48520-1 2016

[13] [13]

( 1995 )

bbook [author] Kato , Tosio T. ( 1995 ). Perturbation Theory for Linear Operators . Classics in Mathematics 132 . Springer , Berlin, Heidelberg . 10.1007/978-3-642-66282-9 bbook

work page doi:10.1007/978-3-642-66282-9 1995

[14] [14]

2020 , publisher =

bbook [author] Klenke , Achim A. ( 2020 ). Probability Theory : A Comprehensive Course . Universitext . Springer International Publishing , Cham . 10.1007/978-3-030-56402-5 bbook

work page doi:10.1007/978-3-030-56402-5 2020

[15] [15]

bbook [author] Lee , A. J. A. J. ( 2019 ). U- Statistics : Theory and Practice . Routledge , New York . 10.1201/9780203734520 bbook

work page doi:10.1201/9780203734520 2019

[16] [16]

barticle [author] Lee , C E C. E. , Zhang , X X. Shao , X X. ( 2020 ). Testing Conditional Mean Independence for Functional Data . Biometrika 107 331--346 . 10.1093/biomet/asz070 barticle

work page doi:10.1093/biomet/asz070 2020

[17] [17]

bbook [author] Lehmann , E. L. E. L. Romano , Joseph P. J. P. ( 2022 ). Testing Statistical Hypotheses . Springer Texts in Statistics . Springer International Publishing , Cham . 10.1007/978-3-030-70578-7 bbook

work page doi:10.1007/978-3-030-70578-7 2022

[18] [18]

, Meunier , Dimitri D

binproceedings [author] Li , Zhu Z. , Meunier , Dimitri D. , Mollenhauer , Mattes M. Gretton , Arthur A. ( 2022 ). Optimal Rates for Regularized Conditional Mean Embedding Learning . In Advances in Neural Information Processing Systems 35 4433--4445 . Curran Associates, Inc. binproceedings

work page 2022

[19] [19]

barticle [author] Lundborg , Anton Rask A. R. , Shah , Rajen D. R. D. Peters , Jonas J. ( 2022 ). Conditional Independence Testing in Hilbert Spaces with Applications to Functional Data Analysis . Journal of the Royal Statistical Society Series B: Statistical Methodology 84 1821--1850 . 10.1111/rssb.12544 barticle

work page doi:10.1111/rssb.12544 2022

[20] [20]

barticle [author] Lundborg , Anton Rask A. R. , Kim , Ilmun I. , Shah , Rajen D. R. D. Samworth , Richard J. R. J. ( 2024 ). The Projected Covariance Measure for Assumption-Lean Variable Significance Testing . The Annals of Statistics 52 2851--2878 . 10.1214/24-AOS2447 barticle

work page doi:10.1214/24-aos2447 2024

[21] [21]

barticle [author] Massart , P. P. ( 1990 ). The Tight Constant in the Dvoretzky-Kiefer-Wolfowitz Inequality . The Annals of Probability 18 1269--1283 . 10.1214/aop/1176990746 barticle

work page doi:10.1214/aop/1176990746 1990

[22] [22]

barticle [author] Owhadi , H. H. Scovel , C. C. ( 2017 ). Separability of R eproducing K ernel S paces . Proceedings of the American Mathematical Society 145 2131--2138 . https://doi.org/10.1090/proc/13354 barticle

work page doi:10.1090/proc/13354 2017

[23] [23]

binproceedings [author] Park , J. J. Muandet , K. K. ( 2020 ). A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings . In Advances in Neural Information Processing Systems 33 ( NeurIPS 2020) 21247--21259 . Curran Associates, Inc. binproceedings

work page 2020

[24] [24]

Wahed and Peter F

barticle [author] Patilea , Valentin V. , S \'a nchez-Sellero , C \'e sar C. Saumard , Matthieu M. ( 2016 ). Testing the Predictor Effect on a Functional Response . Journal of the American Statistical Association 111 1684--1695 . 10.1080/01621459.2015.1110031 barticle

work page doi:10.1080/01621459.2015.1110031 2016

[25] [25]

bbook [author] Paulsen , Vern I. V. I. Raghupathi , Mrinal M. ( 2016 ). An Introduction to the Theory of Reproducing Kernel Hilbert Spaces . Cambridge Studies in Advanced Mathematics . Cambridge University Press , Cambridge . 10.1017/CBO9781316219232 bbook

work page doi:10.1017/cbo9781316219232 2016

[26] [26]

, Schrab , Antonin A

bmisc [author] Pogodin , Roman R. , Schrab , Antonin A. , Li , Yazhe Y. , Sutherland , Danica J. D. J. Gretton , Arthur A. ( 2025 ). Practical Kernel Tests of Conditional Independence . 10.48550/arXiv.2402.13196 bmisc

work page doi:10.48550/arxiv.2402.13196 2025

[27] [27]

o rrmann , Julia J. B \

barticle [author] Scheidegger , Cyrill C. , H \"o rrmann , Julia J. B \"u hlmann , Peter P. ( 2022 ). The Weighted Generalised Covariance Measure . Journal of Machine Learning Research 23 1--68 . barticle

work page 2022

[28] [28]

bbook [author] Schilling , Ren \'e L. R. L. , Song , Renming R. Vondracek , Zoran Z. ( 2012 ). Bernstein Functions : Theory and Applications . De Gruyter . 10.1515/9783110269338 bbook

work page doi:10.1515/9783110269338 2012

[29] [29]

, Sriperumbudur , Bharath B

barticle [author] Sejdinovic , Dino D. , Sriperumbudur , Bharath B. , Gretton , Arthur A. Fukumizu , Kenji K. ( 2013 ). Equivalence of Distance-Based and RKHS-based Statistics in Hypothesis Testing . The Annals of Statistics 41 2263--2291 . 10.1214/13-AOS1140 barticle

work page doi:10.1214/13-aos1140 2013

[30] [30]

barticle [author] Shah , Rajen D. R. D. Peters , Jonas J. ( 2020 ). The hardness of conditional independence testing and the generalised covariance measure . The Annals of Statistics 48 1514--1538 . 10.1214/19-AOS1857 barticle

work page doi:10.1214/19-aos1857 2020

[31] [31]

, Xu , Tianlin T

barticle [author] Shi , Chengchun C. , Xu , Tianlin T. , Bergsma , Wicher W. Li , Lexin L. ( 2021 ). Double Generative Adversarial Networks for Conditional Independence Testing . Journal of Machine Learning Research 22 1--32 . barticle

work page 2021

[32] [32]

barticle [author] Sriperumbudur , Bharath K. B. K. , Fukumizu , Kenji K. Lanckriet , Gert R. G. G. R. G. ( 2011 ). Universality, Characteristic Kernels and RKHS Embedding of Measures . Journal of Machine Learning Research 12 2389--2410 . barticle

work page 2011

[33] [33]

barticle [author] Sriperumbudur , Bharath K. B. K. , Gretton , Arthur A. , Fukumizu , Kenji K. , Sch \"o lkopf , Bernhard B. Lanckriet , Gert R. G. G. R. G. ( 2010 ). Hilbert Space Embeddings and Metrics on Probability Measures . Journal of Machine Learning Research 11 1517--1561 . barticle

work page 2010

[34] [34]

bbook [author] Srivastava , S. M. S. M. ( 1998 ). A Course on Borel Sets . Graduate Texts in Mathematics 180 . Springer . 10.1007/b98956 bbook

work page doi:10.1007/b98956 1998

[35] [35]

Steinwart and A

bbook [author] Steinwart , Ingo I. Christmann , Andreas A. ( 2008 ). Support Vector Machines . Information Science and Statistics . Springer . 10.1007/978-0-387-77242-4 bbook

work page doi:10.1007/978-0-387-77242-4 2008

[36] [36]

Duncan , Andrew B

barticle [author] Wynne , George G. Duncan , Andrew B. A. B. ( 2022 ). A Kernel Two-Sample Test for Functional Data . Journal of Machine Learning Research 23 1--51 . barticle

work page 2022

[37] [37]

Small and Fan Li , title =

barticle [author] Zhang , Qi Q. , Xue , Lingzhou L. Li , Bing B. ( 2024 ). Dimension Reduction for Fréchet Regression . Journal of the American Statistical Association 119 2733--2747 . 10.1080/01621459.2023.2277406 barticle

work page doi:10.1080/01621459.2023.2277406 2024