{"id":"a102d323-abb4-4ddb-af23-ec583e7e4bea","arxiv_id":"2512.13997","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"With unequal sample sizes, MMD estimators converge under min(nX,nY) scaling, so tests can use all available data instead of discarding surplus samples.","lead":"This paper derives how the maximum mean discrepancy (MMD) two-sample test behaves when the two groups have different numbers of samples, and gives a rule for choosing the test kernel without throwing data away. A generalist might read it because unequal group sizes are common in practice, and most existing MMD theory silently assumes equal sizes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Power criterion assumes non-degenerate alternative; unproven for continuous non-analytic kernels with disjoint supports, a case the paper explicitly leaves open.","rationale":"The paper's main theorems (3.7, 3.9) appear mathematically sound under their stated conditions: the generalized U-statistic framework, the null chi-square-type limit, and the non-degenerate normal limit are derived consistently, and spot-checks of the variance formula match exact calculations. The proof of Theorem 3.4 contains an inconsistent kernel definition (Appendix D writes h as an inner product that differs from the kernel in (6)), but the subsequent conditional-expectation computations use the correct kernel and the final variance formula seems correct; this is a typo to fix, not a fatal flaw. The false 'characteristic' claim in Theorem 2.7(ii) is not needed for the example's validity. The absence of an equal-subsampling baseline in experiments weakens the empirical support but not the theory. The most substantive concern is the scope of the power criterion: it is conditional on non-degeneracy, and the paper explicitly leaves open the case of continuous non-analytic kernels with disjoint supports. Because this limitation is acknowledged and the central claims are otherwise supported, the reader's CONDITIONAL verdict is appropriate; no verdict change is needed.","tokens_in":27553,"tokens_out":31233,"duration_ms":222515,"concrete_test":"Construct two distributions on R with disjoint supports (e.g., P uniform on {1,2}, Q uniform on {3,4}) and a continuous non-analytic kernel with compact support, such as the triangular kernel k(x,y)=max(1−|x−y|,0). Compute ζX = Var_X(μ_P(X)−μ_Q(X)) and ζY = Var_Y(...), then σ^2 = 4ρXζX + 4ρYζY for sample-size ratios ρX,ρY. If σ^2=0 while μ_P≠μ_Q, the estimator is degenerate and the power criterion MMD^2/σ is undefined; this would confirm the concern. Additionally, simulate √min(n)(MMD^2−MMD^2) for this setting and compare its empirical distribution to the normal predicted by Corollary 3.9; a non-normal or zero-variance limit confirms the failure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim includes a new power criterion MMD^2/σ for kernel selection with unequal sample sizes. This criterion rests on Corollary 3.9, which gives √min(n)(MMD^2−MMD^2) → N(0, 4ρXζX + 4ρYζY) only when the estimator is non-degenerate and σ^2 > 0. The paper proves (Theorem 2.7(v)) that in Setting B — real-analytic kernels and positive-measure supports — ζX and ζY have the same sign, so non-degeneracy implies σ^2 > 0. But for continuous non-analytic kernels (e.g., typical neural network kernels) with disjoint supports, the paper states in Section 5 that it has not determined whether ζX and ζY are positive or match signs. In such cases ζX or ζY may be zero (as in the triangular-kernel example of Theorem 2.7(ii), where both are zero), so σ^2 can vanish and the sqrt(n) normal approximation fails; the estimator can be first-order degenerate even under the alternative. Thus the power-optimization criterion is not established in this regime. This is a genuine limitation, but it is explicitly acknowledged, so it does not invalidate the paper's claims within the stated assumptions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the asymptotic theory of the unbiased Maximum Mean Discrepancy (MMD) estimator to settings with unequal sample sizes by representing it as a generalized U-statistic. The main theoretical results are: (i) a null limiting distribution for min(nX,nY)·MMD² when MMD²=0, valid also in non-proportional regimes such as nY = nX²; (ii) asymptotic normality of √min(nX,nY)(MMD²−MMD²) under a non-degenerate alternative, with variance 4ρXζX + 4ρYζY; (iii) a characterization of degeneracy showing that a nonzero MMD can still produce a degenerate estimator, though not in typical analytic-kernel settings; and (iv) a proposed power-optimization criterion based on the ratio MMD²/σ. The paper also gives an exact finite-sample variance formula (Theorem 3.4) and reports experiments on Gaussian data and CIFAR-10 vs. CIFAR-10.1.","tokens_in":27845,"tokens_out":32938,"duration_ms":241167,"significance":"If the main results are made fully correct, the paper fills a real gap: MMD-based kernel choice has mostly assumed equal sample sizes, wasting data when sizes differ, and previous proportional-regime asymptotics do not cover cases like nY = nX². The paper is also valuable for its transparent treatment of degeneracy, including an explicit example of first-order degeneracy under a nonzero MMD, and for clearly stating the unresolved continuous non-analytic disjoint-support case. Detailed proofs are provided in appendices. However, the exact variance claim in Theorem 3.4 is false as stated, and the derivation of the power approximation contains a recurring algebraic error. These issues require a substantial revision before the paper can be accepted.","major_comments":[{"comment":"The proof uses h(x,x';y,y') = ⟨k(x,·)−k(y,·), k(x',·)−k(y',·)⟩, which is not the block-symmetric kernel (6) and is not symmetric under per-block permutations. Sen's variance formula (7) does not apply to this h. The resulting variance formula is incorrect. Counterexample: let k(x,y)=xy, P=N(1,1), Q=N(0,1), nX=nY=3. Direct expansion of (1) gives Var = 4/3δ²(σP²+σQ²)+1/3(σP⁴+σQ⁴)+4/9σP²σQ² = 34/9 for δ=1, σP=σQ=1. Theorem 3.4 evaluates to 8/9+8/9+1/3+1/3+4/9+4/9 = 30/9. The leading O(1/n) terms are correct, but the exact finite-sample formula is false. This invalidates Theorem 3.4 and any finite-sample variance corrections built on it.","section":"Section D / Theorem 3.4"},{"comment":"The power approximation contains a repeated algebraic error. From the normal limit, n·MMD² > cα is equivalent to T_n > (cα − n·MMD²)/(σ√n) = cα/(σ√n) − √n·MMD²/σ. The manuscript instead writes (cα√n − √n·MMD²)/σ and consequently derives the CDF as Φ(√n·MMD²/σ − cα√n/σ). The correct expression is Φ(√n·MMD²/σ − cα/(σ√n)). The asymptotic dominance of MMD²/σ is unaffected, but the displayed formula is not correct as written.","section":"Sections 2.4 and 3.3"},{"comment":"The power criterion MMD²/σ is presented as a general method, but it is only established when the estimator is non-degenerate with positive limiting variance. The manuscript explicitly leaves open the case of continuous non-analytic kernels with disjoint supports, relevant to neural-network kernels. In that regime ζX or ζY can vanish (e.g., the triangular-kernel example in Theorem 2.7(ii)), so the sqrt(n) normal approximation and the power criterion are not guaranteed. This is acknowledged in the conclusion, but the abstract and introduction state the power-optimization result more broadly than the assumptions currently justify.","section":"Section 5 / Corollary 3.9"}],"minor_comments":[{"comment":"The second variance term is typographically wrong: ζY should be ⟨µP−µQ, C_Q(µP−µQ)⟩, not ⟨µP−µQ, C_P(µP−µQ)⟩. The plug-in formulas in Section 3.3 use the correct definition.","section":"Corollary 3.9"},{"comment":"The proof says 'so we seek to use Theorem 3.7' but the intended reference is Theorem E.5. Also, the displayed definition of ˜hYY contains a typo: it should be ⟨φ(Y)−µP, φ(Y')−µP⟩, not an expression involving X and Y.","section":"Proof of Theorem 3.7"},{"comment":"The statement of the lemma uses \\widehat{U}_{n,r+1}, but the proof concludes a bound involving \\widehat{U}_{n,r}. The notation should be aligned to avoid confusion about which projection is being controlled.","section":"Appendix E, Lemma E.3"},{"comment":"The kernel h is written with i and j reused for both X and Y blocks, which is easy to misread. Using four distinct argument names (e.g., x1,x2;y1,y2) would make the block-wise symmetry and the cross-term structure clearer.","section":"Equation (6)"}],"recommendation":"major_revision","confidential_remarks":"The false variance theorem is a serious issue, but it appears fixable by recomputing with the correct block-symmetric kernel (6). The main asymptotic distribution results (Theorems 3.7, 3.8, and the leading term of Corollary 3.9) do not rely on the exact finite-sample variance formula, so the central testing methodology may survive after correction. I would ask the authors to correct or remove Theorem 3.4, re-examine all statements that depend on it, and fix the power-approximation algebra before the paper is reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: This is a solid, useful paper that extends MMD asymptotics to unequal sample sizes, and the central theorems look right. But the printed proof of the variance theorem has a kernel mismatch that needs a rewrite, and the power criterion is only proven under non-degeneracy; the paper explicitly leaves open the continuous non-analytic disjoint-support case, where its own triangular-kernel example shows the variance can vanish.\n\nThe genuinely new pieces are Theorem 3.7 (null distribution with min(nX,nY) scaling, including non-proportional regimes), Corollary 3.9 (non-degenerate normality), and the exact finite-sample variance in Theorem 3.4. The generalized U-statistic results (Theorem 3.8, E.5) appear novel, and the paper deserves credit for working out the c>2 and asymmetric-kernel details. The degeneracy result — zero MMD implies degenerate, but nonzero MMD can also be degenerate — is a real correction to informal claims in the literature. The paper ships detailed appendices, and the algebra spot-checks in limits and in the equal-size reduction.\n\nThe soft spots are real but reparable. Appendix D defines h as the paired estimator kernel (2) but then computes conditional expectations as if it were the generalized kernel (6). The two agree after conditioning on X-only or Y-only variables, so many zeta terms are right, but zeta_XY and the higher-order terms differ; the proof as printed does not establish Theorem 3.4. The final formula may still be correct, but the section needs to be redone with the correct kernel. Corollary 3.9 has a typo (zeta_Y defined with C_P instead of C_Q). The triangular kernel example in Theorem 2.7 claims that kernel is characteristic, which is false (its Fourier transform has zeros); the conclusion mu_P != mu_Q is still true and can be shown by direct evaluation. The experiments never compare against the equal-subsampling baseline they claim to improve, so the practical advantage is not demonstrated. And the power criterion MMD^2/sigma rests on sigma>0; for continuous non-analytic kernels with disjoint supports the paper leaves open whether zeta_X and zeta_Y are positive or even match signs, and the triangular example shows sigma can be zero. That caveat is stated in the conclusion, so it is a limitation, not an oversight.\n\nI read the central argument as basically sound. The variance proof needs fixing, the examples need small corrections, and the empirical section needs a fair baseline. This is refereeable and would be a useful addition after revision. I'd send it to peer review.","headline":"Solid extension of MMD asymptotics to unequal sample sizes, but the variance proof has a kernel mismatch and the power criterion has an acknowledged gap for non-analytic kernels with disjoint supports.","tokens_in":28302,"tokens_out":10720,"would_cite":true,"duration_ms":83351,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G10","62G20","62E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes the asymptotic distributions of the MMD estimator for two-sample tests with unequal sample sizes, scaling with the smaller sample, and thereby enables kernel selection that uses all available data.","keywords":["Maximum Mean Discrepancy","generalized U-statistics","two-sample testing","unequal sample sizes","asymptotic distribution","degenerate U-statistics","kernel selection","test power"],"falsifier":"Run a simulation with P = N(0,1), Q = N(0,1.2), a Gaussian kernel, and nX = 100, nY = 10,000; if the empirical distribution of sqrt(min(nX,nY)) * (MMD^2 - MMD^2) does not match a normal distribution with variance 4 zetaX + 0.04 zetaY (zetaX, zetaY estimated from the samples), the normality claim fails. Alternatively, with P = Q, the quantiles of min(nX,nY) * MMD^2 must match the weighted shifted chi-square of Theorem 3.7.","tokens_in":27453,"feed_emoji":"📊","tokens_out":9104,"duration_ms":65470,"temperature":0.7,"pith_summary":"This paper establishes the asymptotic theory needed to run Maximum Mean Discrepancy (MMD) two-sample tests when the two sample sizes are unequal, removing the usual requirement of equal subsamples or proportional sample sizes. The key finding is that the standard unbiased MMD estimator, scaled by the smaller sample size n = min(nX, nY), has a well-defined null limit—a weighted shifted chi-square distribution—and a normal limit under the alternative, even when one sample is asymptotically dominant. This yields a practical power criterion, MMD^2 divided by the asymptotic standard deviation, which can be maximized to choose kernels without discarding any data. The paper also clarifies a subtle degeneracy property: zero MMD forces the estimator to be degenerate, but a degenerate estimator can sometimes occur with nonzero MMD, though not under common analytic or overlapping-support conditions.","feed_headline":"Drop the subsampling: MMD tests handle unequal sample sizes","feed_subtitle":"New limit laws scale with the smaller sample, so MMD tests gain power without discarding data.","key_machinery":"The central object is the MMD estimator written as a two-sample generalized U-statistic with kernel h(x, x'; y, y') = k(x,x') + k(y,y') - (1/2)[k(x,y') + k(x',y) + k(x,y) + k(x',y')]. This lets the authors apply a new asymptotic theory for generalized U-statistics that allows sample sizes to grow at different rates, with n = min(nX, nY) as the natural scaling. The variance decomposition in Theorem 3.4, expressed through the covariance operators and mean embeddings of the two distributions, identifies the leading terms and yields the degeneracy conditions; the same machinery yields the first-order degenerate limit via an SVD of the conditional kernels.","core_discovery":"On the authors' own terms, the central discovery is that the ordinary unbiased MMD estimator, viewed as a two-sample generalized U-statistic, remains asymptotically tractable when nX does not equal nY, provided the scaling is min(nX, nY). Under the null MMD = 0, min(nX, nY) times the estimator converges to (rhoX + rhoY) times a weighted shifted chi-square sum, where rhoX and rhoY are the limiting ratios of the smaller sample size to each sample size. Under a non-degenerate alternative, sqrt(min(nX, nY)) times the centered estimator converges to a normal distribution with variance 4 rhoX zetaX + 4 rhoY zetaY. These limit laws justify choosing a kernel by maximizing the signal-to-noise ratio M","pith_inferences":["Inference: the asymptotic variance being governed by the smaller sample implies that collecting additional data from the larger group has diminishing returns once one sample dominates; the paper does not draw this practical conclusion explicitly.","Inference: the variance decomposition in Theorem 3.4 could be used to construct unbiased variance estimators for unequal sample sizes, paralleling existing unbiased estimators for equal sizes; this would likely improve finite-sample kernel selection.","Inference: the degeneracy counterexample (disjoint supports with a compact kernel) suggests that practitioners using kernels whose support is disjoint from the data (for instance in anomaly detection) may need to verify that the normal approximation holds, because the leading variance terms can behave unexpectedly.","Inference: the generalized U-statistic theory for first-order degeneracy may transfer to other two-sample statistics beyond MMD, such as energy distance, enabling unequal-sample-size analysis for those methods."],"forward_implications":["Two-sample MMD tests remain valid with unequal sample sizes: no need to subsample or discard data, and power increases as more data is used.","Kernel choice for MMD tests can be optimized under unequal sample sizes by maximizing the estimated signal-to-noise ratio MMD^2/(sigma_hat + lambda), with sigma_hat computed via simple plug-in estimators from the kernel matrices.","The null limiting distribution (a weighted shifted chi-square) supports the common practice of permutation thresholds, while the alternative normal limit gives a consistent test as long as MMD is positive and the non-degeneracy condition holds.","When sample sizes are equal, the new formulas reduce to the previously known U-statistic results, unifying equal and unequal cases.","The degeneracy results correct the assumption that nonzero MMD always yields a non-degenerate estimator; a degenerate estimator with nonzero MMD can occur, but not under real-analytic kernels with positive-mass supports or continuous kernels with overlapping supports."],"fun_headline_variants":["MMD tests now work with unequal sample sizes","No more discarding data: MMD handles uneven groups","Generalized U-statistics unlock MMD for uneven sample sizes","Unequal samples? MMD has new limit laws","MMD gains power with unequal samples via U-statistics"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The power-criterion and normality results hinge on the estimator being non-degenerate under the alternative, a condition the paper proves in common settings but leaves open for continuous non-analytic kernels with disjoint supports.","fun_headline_variants_meta":{"raw":{"variants":["MMD tests now work with unequal sample sizes","No more discarding data: MMD handles uneven groups","Generalized U-statistics unlock MMD for uneven sample sizes","Unequal samples? MMD has new limit laws","MMD gains power with unequal samples via U-statistics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000842,"raw_usage":{"total_tokens":3504,"prompt_tokens":744,"completion_tokens":2760,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":2691}},"tokens_in":488,"tokens_out":2760,"duration_ms":17443,"temperature":1.0,"reasoning_tokens":2691,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T16:19:02.747007+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a simulation with P = N(0,1), Q = N(0,1.2), a Gaussian kernel, and nX = 100, nY = 10,000; if the empirical distribution of sqrt(min(nX,nY)) * (MMD^2 - MMD^2) does not match a normal distribution with variance 4 zetaX + 0.04 zetaY (zetaX, zetaY estimated from the samples), the normality claim fails. Alternatively, with P = Q, the quantiles of min(nX,nY) * MMD^2 must match the weighted shifted chi-square of Theorem 3.7.","supporting_citations":[],"review_version":1}