{"id":"6a1c0b99-714b-4528-98a9-d14d54989d12","arxiv_id":"2506.05963","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Kernel tests xdHSIC, xLI, xSI detect high-order joint independence and factorisation without permutations, with a claimed standard normal null.","lead":"A new set of kernel-based statistical tests removes the need for slow permutation resampling when detecting interactions among three or more variables at once. The tests compute a single statistic with a claimed normal null distribution, potentially enabling 100x faster causal discovery and feature selection.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The N(0,1) null for d>2 is asserted, not proved: variance-estimator consistency is missing, and the cross-centring lemma does not justify Hadamard products in xLI/xSI.","rationale":"The reader's rejection is based on the absence of a derivation of the null distribution. My stress-test agrees with that assessment and identifies the same load-bearing condition: consistency of the variance estimators and the CLT for d>2. I add a specific technical reason why the cited d=2 result does not transfer: the cross-centring lemma operates on individual kernels, while the proposed statistics combine kernels with Hadamard products, and no proof shows that equivalence is preserved. The empirical null plots and power experiments are suggestive but cannot replace a theorem. The complexity statement in Proposition 3.8 (O(d n^2) for d subtests of cost O(d n^2)) is internally inconsistent and further weakens the manuscript, but it is not the primary issue. If the missing theorem is supplied and the corrupted formulas are corrected, a conditional acceptance would be plausible; as written, the central claim is unsupported, so I recommend REJECT.","tokens_in":18383,"tokens_out":14412,"duration_ms":149532,"concrete_test":"Analytical check: for d=3 under H0, write \\sqrt{n}\\tilde{xdHSIC} as \\sqrt{n}(\\bar{W}-\\bar{V})+o_p(1) via its Hajek projection, and compute both Var(\\bar{W}) and the probability limit of s_I^2. The claimed N(0,1) limit requires these to coincide (ratio 1) plus a Lyapunov CLT for \\bar{W}. Evaluate this with explicit Gaussian-kernel integrals or high-precision Monte Carlo (n=10^6 with 10^5 replications). If the ratio differs from 1, the central claim is refuted; if it equals 1, the missing theorem can be supplied.","verdict_should_be":"REJECT","load_bearing_attack":"The strongest claim is that xdHSIC, xLI, and xSI are asymptotically standard normal under their nulls after one data split. For d>2 this is asserted, not proved. Section 3.1 cites Shekhar et al. (2023) after Hypothesis 3.2, but that result is for the d=2 cross U-statistic; here the statistics are d-dimensional cross-block V-statistics with different variance estimators. Appendix A.1 only checks O(dn^2) complexity, and Appendix A.2 proves equivalence for a single cross-centred kernel. The Lancaster and Streitberg statistics use Hadamard products of individually cross-centred kernels, \\overline{K^m}\\odot\\bigodot_{i\\ne m}\\overline{K^i}; equivalence of individual factors does not imply equivalence of their Hadamard product, so the reduction of each subtest to a pairwise problem is not justified. No consistency proof for s_I^2, s_L^2, or s_S^2 appears, and the displayed definitions of s_L^2 and s_S^2 (Definitions 3.7 and 3.10) contain unreadable placeholders rather than a complete expression. Since the permutation-free property and the abstract headline claim are exactly this N(0,1) null, the missing derivation is load-bearing. This is a statement about absence of proof, not a claim that the statistics are wrong.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes three permutation-free kernel tests for high-order interactions: xdHSIC for joint independence, xLI for Lancaster factorisations, and xSI for Streitberg/complete factorisations. The statistics are formed by splitting the sample, cross-centring kernel matrices, and using V-statistic-type bilinear forms; the authors claim that each normalised statistic has a standard normal null distribution, so a single computation replaces p permutations. The paper reports O(dn^2) complexity, presents synthetic and real-data experiments (causal discovery, feature selection, stock returns), and provides code.","tokens_in":18687,"tokens_out":8809,"duration_ms":87793,"significance":"If the normality claim were established, the paper would make a useful contribution: it would extend permutation-free testing beyond pairwise independence to d>2, make high-order interaction testing practical in combinatorial searches, and the d=2 V-statistic reformulation xHSICV is a clean simplification. The empirical evaluation is broad, and public code is a strength. However, the central theoretical claim is not proved for d>2, and the variance estimators are incompletely specified; until that is fixed, the methodological contribution is not established.","major_comments":[{"comment":"The central claim that xdHSIC is asymptotically standard normal under H0 for d>2 is asserted, not proved. The citation to Shekhar et al. (2023) after Hypothesis 3.2 covers the pairwise cross U-statistic, whereas xdHSIC is a d-dimensional cross-block V-statistic with a different variance estimator s_I^2. No theorem states the limiting distribution for d>2, and Appendix A.1 only verifies the O(dn^2) complexity; it does not establish consistency of s_I^2 or asymptotic normality. Since the permutation-free property is exactly this null distribution, the missing derivation is load-bearing.","section":"Section 3.1, Definition 3.1 and Hypothesis 3.2"},{"comment":"The displayed definitions of s_L^2 and s_S^2 contain unreadable placeholder tokens rather than complete expressions, so the normalised statistics xLI and xSI cannot be checked from the manuscript. No consistency proof for either variance estimator is provided; the remarks in Sections 3.2.1 and 3.2.2 only assert the N(0,1) null. This must be repaired and accompanied by a proof.","section":"Definitions 3.7 and 3.10"},{"comment":"Lemma 3.5 establishes asymptotic equivalence of a single cross-centred kernel to the original kernel. The Lancaster and Streitberg statistics, however, are built from Hadamard products of separately cross-centred kernels, e.g. \\overline{K^m} \\odot \\bigodot_{i\\ne m}\\overline{K^i} in Definition 3.7. Equivalence of the individual factors does not imply equivalence of their Hadamard product, and no argument reduces the null distribution of such products to the pairwise xHSIC framework. Appendix A.3 and A.4 merely assert that the products can be viewed as single kernels; this addresses complexity, not distributional convergence.","section":"Lemma 3.5 and Section 3.2"},{"comment":"These statements conflict. Proposition 3.8 claims xLI for all H_pi_L is O(dn^2), while Table 1 reports O(d^2 n^2); Proposition 3.11 claims xSI is O(dn^2), while Table 1 reports O(2^d d n^2) for xSI+xLI. Since xLI runs d singleton subtests and xSI covers the non-singleton bipartitions, the all-subtests complexities in the table are the relevant ones, and the propositions should be corrected to state per-subtest complexity.","section":"Propositions 3.8, 3.11, and Table 1"}],"minor_comments":[{"comment":"The normalised statistics are written with a denominator s_I, s_L, or s_S even though the displayed objects are s_I^2, s_L^2, s_S^2; please clarify that the denominator is the square root of the displayed variance estimator.","section":"Definitions 3.1, 3.7, 3.10"},{"comment":"The displayed implications use corrupted arrow tokens (e.g., '/Leftr⫯g⊸tl⫯ne⇒') and need to be typeset correctly.","section":"Appendix B"},{"comment":"The sentence 'xSI+xLI can be computed in less that 1 second' contains a typo; it should read 'less than'.","section":"Section 5.1"},{"comment":"The claim that xHSICV has 'higher power' than xHSIC is based on a single simulation (Fig. 9); please qualify this as an empirical observation for the considered setting.","section":"Appendix C"}],"recommendation":"major_revision","confidential_remarks":"I am also concerned by the heavy reliance on the authors' own prior work (Liu et al. 2023b) for the vanishing properties of Lancaster and Streitberg interactions; this is not circular since those results are published, but the manuscript should state clearly which components are new relative to that work. The placeholder tokens in the main equations suggest the manuscript was compiled from an incomplete source; this should be checked before any further review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The new construction is real: extending the permutation-free trick to d>2 with V-statistics and cross-centring, and building xLI/xSI for Lancaster and Streitberg factorisations, is a genuine step beyond Shekhar et al. (2023), which only handles d=2. The empirical side is solid—null distributions look normal, type-I error is controlled, power tracks the permutation versions, and the speedups (100x) are demonstrated on synthetic and real data. Code is available. The paper also openly states its limitations (i.i.d. assumption, data-splitting loss at small n), which is honest.\n\nThe soft spot is exactly where the reader's report puts it: the central asymptotic normality claim is asserted, not proved. After Hypothesis 3.2, the text says the construction \"ensures\" N(0,1) under the null, citing Shekhar et al. (2023), but that result is for the d=2 U-statistic; here the statistics are d-dimensional cross-block V-statistics with different variance estimators. Appendix A.2 proves equivalence of a single cross-centred kernel to the original, but the xLI/xSI statistics are Hadamard products of such kernels, and that equivalence does not obviously survive the Hadamard product. No consistency proof for s_I^2, s_L^2, or s_S^2 appears anywhere. These are the load-bearing claims, and they're absent.\n\nThere are also presentation problems of the sort that should have been caught before submission. The displayed definitions of s_L^2 and s_S^2 in Definitions 3.7 and 3.10 contain unreadable placeholder glyphs, so as printed the formulas are meaningless. Proposition 3.8 claims O(dn^2) for xLI for all subhypotheses, but Table 1 lists xLI as O(d^2 n^2). Similarly Proposition 3.11 claims O(dn^2) for xSI, while Table 1 lists xSI+xLI as O(2^d dn^2). These may be reconcileable, but as written they contradict.\n\nThe reliance on Liu et al. (2023b) for the vanishing properties of Lancaster and Streitberg interactions is legitimate: it's published work, not circular.\n\nWho is this for? Someone working on scalable kernel independence tests or high-order interaction detection would want to know this approach exists. The paper deserves a serious referee, but the referee should demand a real proof of the distributional claim before acceptance. If the asymptotic normality holds under the stated conditions, it's a valuable tool; right now the evidence is empirical only. I'd send it to review, with the expectation of major revisions.","headline":"The permutation-free construction and the empirical work are genuinely useful, but the central N(0,1) claim for d>2 is asserted rather than proved, so the paper needs major theoretical revisions before acceptance.","tokens_in":19215,"tokens_out":3131,"would_cite":false,"duration_ms":28672,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G10","62H15","62G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that kernel high-order interaction tests can be made permutation-free through V-statistics and cross-centring, with null statistics that follow a standard normal distribution.","keywords":["permutation-free hypothesis tests","kernel mean embeddings","high-order interactions","joint independence","Lancaster interaction","Streitberg interaction","V-statistics","cross-centring"],"falsifier":"Generate i.i.d. data under a d=4 joint-independence null, compute xdHSIC over 10,000 replications at n=200 and n=500, and compare the empirical distribution with N(0,1) using a Q-Q plot and a Kolmogorov-Smirnov test; if the tails are heavier or the variance is not 1, the consistency claim for $s_I^{2}$ is falsified. The analogous check can be run for xLI and xSI under a null subhypothesis such as P1234 = P12 P34.","tokens_in":18201,"feed_emoji":"⚡","tokens_out":5301,"duration_ms":55647,"temperature":0.7,"pith_summary":"The paper introduces three permutation-free kernel tests—xdHSIC for joint independence, xLI for Lancaster (singleton) factorisations, and xSI for general two-block factorisations—that replace the usual 100–1000 permutation null with a single statistic claimed to converge to N(0,1) under the null. It argues that this removes the dominant computational bottleneck of kernel interaction tests while keeping the flexibility of reproducing kernel Hilbert space embeddings. The practical gain is that testing joint independence or complete factorisation of d variables scales as O(d $n^{2}$), with measured speedups of over 100-fold, making high-order interaction screening feasible where permutation schemes become infeasible. The central technical move is to split the sample in half and use only the off-diagonal cross-block kernel information, with a cross-centring that makes the statistic non-degenerate and self-normalizing.","feed_headline":"One computation replaces 100+ permutations in high-order tests","feed_subtitle":"Three kernel tests give standard normal p-values for d-way interactions, at up to 100x speedup.","key_machinery":"The load-bearing mechanism is cross-centring combined with V-statistics under data splitting. For 2n i.i.d. samples, the kernel matrix K is centred only across the two halves as K = C_u K C_l, where C_u and C_l subtract the mean of the upper and lower half respectively; this makes the cross-centred kernel asymptotically equivalent to the usual centred kernel while leaving the statistic with a non-degenerate limiting variance. The test statistics are quadratic forms 1^T (Hadamard product of cross-centred kernel blocks) 1 over the cross half-block, normalised by the empirical standard deviation of the per-row contributions. This self-normalised ratio is what converts the degenerate null of a V-statistic into a standard normal pivot, so a single computation replaces the permutation distribution.","core_discovery":"The central claim is that, for any d, joint independence and partial-factorisation hypotheses can be tested by V-statistics built from Hadamard products of cross-centred kernel matrices on two disjoint halves of the sample. With this sample split, each statistic takes the form $\\sqrt$(n) times an unnormalised cross-block sum divided by an empirical standard deviation, and the paper claims this ratio converges to N(0,1) under the corresponding null: joint independence for xdHSIC, each singleton-factorisation subhypothesis for xLI, and each two-block partition for xSI. xSI vanishes if and only if the joint distribution factorises into any two blocks, and combining xLI with xSI covers all 2^(d-1)-1 bipartitions needed for the complete factorisation test. The same construction specialised to d=2 yields a simpler pairwise independence statistic that the paper reports to have higher power than the existing permutation-free xHSIC.","pith_inferences":["An extension the paper leaves implicit is that the fixed N(0,1) null makes whole-family multiple-testing control over the exponential number of partitions straightforward: all subtests share one critical value, so composite rejection can be computed without generating a permutation null for each partition.","The data-splitting construction suggests a direct route to stationary time series and network data by replacing the two independent halves with two blocks separated by a gap; the paper notes permutation-based versions for such data exist but leaves the permutation-free version open.","Because the null pivot is a standard normal, formal power analysis could be derived analytically from the mean shift of the cross-block sum under contiguous alternatives; the paper does not provide such a power formula.","Combining cross-centring with low-rank kernel approximations such as random Fourier features or the Nystroem method should reduce the O(d n^2) cost to roughly O(d n m), which would make interaction screening feasible on very large feature sets."],"forward_implications":["xdHSIC, xLI, and xSI each reject their subhypotheses by comparing a single statistic with N(0,1), so p-values become deterministic and the computation is p-fold faster than using p permutations.","All three tests run in O(d n^2), and the xLI+xSI strategy keeps complexity quadratic in n regardless of d, whereas the permutation-based Streitberg test scales combinatorially with d.","For d=2, the V-statistic and cross-centring formulation gives a pairwise independence statistic that is simpler and reported to be more powerful than the earlier permutation-free xHSIC.","In score-based causal discovery, xdHSIC recovers the correct DAG with fewer samples than permutation-based dHSIC, and in the XOR-based feature selection example only xSI+xLI selects the ground-truth feature set.","On S&P 500 daily returns, the method screens 2- to 5-way interactions within and across sectors, finding systematically higher within-sector high-order interactions, especially for Utilities and Energy."],"supporting_citations":[{"why":"Supplies the permutation-free xHSIC template and the data-splitting plus self-normalisation route to a standard normal null that the paper extends to d>2.","marker":"Shekhar et al. (2023)"},{"why":"Supplies dHSIC, the V-statistic estimator, and the O(d n^2) computation argument reused in the new tests.","marker":"Pfister et al. (2018)"},{"why":"Supplies the three-variable Lancaster interaction kernel test and the V-structure datasets used for validation.","marker":"Sejdinovic et al. (2013a)"},{"why":"Supplies the d-order Lancaster and Streitberg interaction tests, the partition-lattice formulation, and the XOR simulation with ground-truth high-order interactions.","marker":"Liu et al. (2023b)"},{"why":"Provides the kernel mean embedding and Hilbert-Schmidt norm framework, including the centring used throughout the construction.","marker":"Gretton et al. (2007)"},{"why":"Defines the Streitberg interaction as the partition expansion whose vanishing conditions the xSI test targets.","marker":"Streitberg (1990)"},{"why":"Establishes the kernel equivalence argument used to justify the asymptotic validity of cross-centring.","marker":"Sejdinovic et al. (2013b)"},{"why":"Supports the claim that rejecting all subtests is less conservative than Bonferroni correction, which the composite factorisation tests rely on.","marker":"Rubenstein et al. (2016)"}],"fun_headline_variants":["No permutations needed for high-order interaction tests","Permutation-free kernel tests for d-way interactions","Standard normal p-values without permutations for high-order tests","One statistic replaces hundreds of permutations in interaction tests","V-statistics give permutation-free high-order independence tests"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The treatment assumes, without proof, that the empirical variance estimators $s_I^{2}$, $s_L^{2}$, and $s_S^{2}$ consistently estimate the asymptotic variance of the cross-block V-statistics for d>2, so that the normalised statistics actually converge to N(0,1).","fun_headline_variants_meta":{"raw":{"variants":["No permutations needed for high-order interaction tests","Permutation-free kernel tests for d-way interactions","Standard normal p-values without permutations for high-order tests","One statistic replaces hundreds of permutations in interaction tests","V-statistics give permutation-free high-order independence tests"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1529,"prompt_tokens":850,"completion_tokens":679,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":607}},"tokens_in":466,"tokens_out":679,"duration_ms":7275,"temperature":1.0,"reasoning_tokens":607,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:12:59.600928+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate i.i.d. data under a d=4 joint-independence null, compute xdHSIC over 10,000 replications at n=200 and n=500, and compare the empirical distribution with N(0,1) using a Q-Q plot and a Kolmogorov-Smirnov test; if the tails are heavier or the variance is not 1, the consistency claim for $s_I^{2}$ is falsified. The analogous check can be run for xLI and xSI under a null subhypothesis such as P1234 = P12 P34.","supporting_citations":[{"cited_title":"A permutation-free kernel independence test","cited_arxiv_id":null,"evidence_quote":"Supplies the permutation-free xHSIC template and the data-splitting plus self-normalisation route to a standard normal null that the paper extends to d>2."},{"cited_title":"u hlmann, P., Sch \\","cited_arxiv_id":null,"evidence_quote":"Supplies dHSIC, the V-statistic estimator, and the O(d n^2) computation argument reused in the new tests."},{"cited_title":"A kernel statistical test of independence","cited_arxiv_id":null,"evidence_quote":"Provides the kernel mean embedding and Hilbert-Schmidt norm framework, including the centring used throughout the construction."},{"cited_title":"Lancaster interactions revisited","cited_arxiv_id":null,"evidence_quote":"Defines the Streitberg interaction as the partition expansion whose vanishing conditions the xSI test targets."},{"cited_title":"A Kernel Test for Three-Variable Interactions with Random Processes","cited_arxiv_id":"1603.00929","evidence_quote":"Supports the claim that rejecting all subtests is less conservative than Bonferroni correction, which the composite factorisation tests rely on."}],"review_version":1}