{"id":"f9acc540-9b25-4c7f-a495-9022823707e1","arxiv_id":"2503.18831","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves a CLT for the empirical p-sliced Wasserstein distance centered at the population cost, providing the first valid inference framework for non-compact measures.","lead":"The paper establishes a central limit theorem for the empirical p-sliced Wasserstein distance (p>1) that allows centering at the population cost rather than the expected empirical cost. This enables asymptotically valid statistical inference for sliced Wasserstein distances even when the measures lack compact support.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Control of OT potentials across directions is the least secure step enabling population centering in the CLT","rationale":"The reader's weakest_assumption directly names the same technical step that must hold for the improved centering claim; because the original review had no access to the proof, the verdict remains UNVERDICTED pending verification of that control.","tokens_in":1671,"tokens_out":322,"duration_ms":75582,"concrete_test":"Extract the precise moment and tail conditions used to bound the OT potentials (likely in the section applying Efron-Stein); construct a pair of non-compact measures satisfying those conditions but with heavier tails in some directions, recompute the bias term numerically for increasing n, and check whether it remains o(n^{-1/2}).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline improvement is that the CLT for the empirical p-sliced Wasserstein (p>1) holds when centered at the population sliced distance rather than only at its expectation. This requires showing that E[empirical SW] - population SW = o_p(n^{-1/2}). The argument invokes Efron-Stein plus a non-trivial uniform control on the 1D OT potentials (or their derivatives) over the random directions; without compactness this control must rely on moment assumptions whose precise form and sufficiency are not visible from the abstract. If the potential bound fails to be uniform in the slicing measure or deteriorates with dimension or tail heaviness, the bias term may not be negligible and population centering is invalid.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript establishes a central limit theorem for the empirical p-sliced Wasserstein distance (p>1) centered at the population sliced Wasserstein distance rather than only at its expectation. The argument combines the Efron-Stein inequality with a uniform control on one-dimensional optimal transport potentials (or their derivatives) over random directions on the sphere. This enables asymptotically valid inference and generalizes existing one-dimensional results to possibly non-compact measures; the paper also treats Monte Carlo approximation of the slicing integral and consistent variance estimation.","tokens_in":1837,"tokens_out":523,"duration_ms":25521,"significance":"If the uniform control on transport potentials holds under the paper's moment assumptions, the result supplies the first asymptotically valid inference framework for sliced Wasserstein distances between non-compact measures. This is a meaningful advance for high-dimensional applications where full Wasserstein distances are intractable.","major_comments":[{"comment":"The non-trivial uniform control on the optimal transport potentials across directions (invoked to obtain E[SW_n] - SW = o_p(n^{-1/2})) is the load-bearing step that permits population centering. The manuscript should state the precise moment conditions on the measures that guarantee this control is uniform in the slicing measure and does not deteriorate with dimension or tail heaviness; without explicit bounds or a counter-example check, it is unclear whether the o_p(n^{-1/2}) claim holds for the full range of non-compact measures advertised.","section":"Proof of Theorem 2.1 / Section 3"},{"comment":"The Efron-Stein application yields a CLT centered at the expectation; the passage from expectation to population sliced distance therefore rests entirely on the potential-control argument. If that argument requires stronger integrability than the CLT itself, the claimed improvement over the general Wasserstein case is narrower than stated.","section":"Section 2.2 and Theorem 2.1"}],"minor_comments":[{"comment":"Notation for the slicing measure and the Monte Carlo approximation error should be introduced earlier and kept consistent between the theoretical statements and the numerical section.","section":"Section 4"},{"comment":"The variance estimator is asserted to be consistent, but the rate or the conditions under which the plug-in estimator converges are not displayed; a short remark or reference to an auxiliary lemma would clarify this.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We are grateful to the referee for their thorough review and constructive feedback on our work. We address each of the major comments in detail below, outlining how we plan to revise the manuscript accordingly.","responses":[{"response":"We thank the referee for this observation. The manuscript's moment assumptions (as stated prior to Theorem 2.1) are sufficient for the uniform control, but we acknowledge that the bound was not made fully explicit. In the revision, we will add a lemma deriving the uniform bound on the potentials, showing it holds uniformly over directions and does not deteriorate with dimension under these assumptions. We will also include a brief discussion confirming the o_p(n^{-1/2}) rate for the non-compact measures considered.","revision_made":"yes","referee_comment":"[Proof of Theorem 2.1 / Section 3] The non-trivial uniform control on the optimal transport potentials across directions (invoked to obtain E[SW_n] - SW = o_p(n^{-1/2})) is the load-bearing step that permits population centering. The manuscript should state the precise moment conditions on the measures that guarantee this control is uniform in the slicing measure and does not deteriorate with dimension or tail heaviness; without explicit bounds or a counter-example check, it is unclear whether the o_p(n^{-1/2}) claim holds for the full range of non-compact measures advertised."},{"response":"The potential-control argument uses precisely the same moment conditions as the Efron-Stein step and the CLT. We will add a remark in Section 2.2 clarifying that no additional integrability is needed, thereby maintaining the claimed scope of the improvement over the general Wasserstein setting.","revision_made":"yes","referee_comment":"[Section 2.2 and Theorem 2.1] The Efron-Stein application yields a CLT centered at the expectation; the passage from expectation to population sliced distance therefore rests entirely on the potential-control argument. If that argument requires stronger integrability than the CLT itself, the claimed improvement over the general Wasserstein case is narrower than stated."}],"tokens_in":1331,"tokens_out":461,"duration_ms":85732,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that the authors prove a central limit theorem for the empirical p-sliced Wasserstein distance (p>1) centered at the population sliced distance rather than only at its expectation. This removes a bias obstacle and opens the door to valid inference procedures when the measures lack compact support. They reach this by combining the Efron-Stein inequality with a uniform control on the one-dimensional optimal transport potentials across directions, then add Monte Carlo approximation of the slicing integral and consistent variance estimation as practical follow-ons. The work extends existing one-dimensional CLTs to the sliced setting in higher dimensions and states the result explicitly for non-compact measures. The motivation and the added implementation pieces are handled clearly. The soft spot sits in the uniform control on the potentials. Without compactness the argument rests on moment conditions, and it is not obvious from the abstract how strong those conditions must be or whether the bound stays uniform when dimension grows or tails get heavier. If that control slips, the o_p(n^{-1/2}) claim for the centering difference would not hold. The abstract treats the step as non-trivial but feasible, so the details decide how far the result reaches. This is written for statisticians who need asymptotic justification for sliced Wasserstein distances in applications with unbounded supports. A reader working on rigorous inference for optimal transport distances would find the technical development useful. It deserves a serious referee because the claimed advance targets a real gap that has limited practical use of these distances.","headline":"This paper gives a CLT for empirical p-sliced Wasserstein that supports centering at the population distance for non-compact measures, but the uniform potential control is the step that needs the closest check.","tokens_in":2303,"tokens_out":379,"would_cite":false,"duration_ms":52766,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"we establish a central limit theorem for the p-sliced Wasserstein distance, for p>1, centered at the expected empirical cost... using the Efron-Stein inequality and a non-trivial control of the optimal transport potentials across directions"},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/ArithmeticFromLogic.lean","rs_theorem":"embed_injective","paper_passage":"Theorem 4.1... SJα(P) < ∞ and moment assumptions... bias term vanishes"}],"headline":"CLT for sliced Wasserstein via Efron-Stein + OT-potential control; no RS cost or ratio structure","alignment":"orthogonal","rationale":"The paper derives a CLT for empirical p-sliced Wasserstein (p>1) centered at the population value, using Efron-Stein variance bounds plus uniform control of 1D OT potentials across directions (Propositions 2.1-2.2, Theorems 3.2-3.3, 4.1). This machinery is classical empirical-process / optimal-transport theory with no appearance of reciprocal cost J, golden-ratio fixed points, 8-tick periodicity, or any forcing from a single distinction. It therefore lies outside every RS module (AbsoluteFloorClosure, Cost/FunctionalEquation, AlexanderDuality, etc.).","tokens_in":67488,"confidence":"high","tokens_out":359,"duration_ms":11776,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The empirical p-sliced Wasserstein distance obeys a central limit theorem centered at the population distance for p greater than 1.","keywords":["sliced Wasserstein distance","central limit theorem","Efron-Stein inequality","optimal transport","statistical inference","empirical processes","non-compact measures"],"falsifier":"A sequence of non-compact measures for which the empirical p-sliced Wasserstein distance, properly normalized, fails to converge in distribution to a centered normal random variable.","tokens_in":2578,"feed_emoji":"","tokens_out":611,"duration_ms":69962,"temperature":0.7,"pith_summary":"The paper proves a central limit theorem for the empirical p-sliced Wasserstein distance when p exceeds 1. The limiting normal law can be centered at the true population value of the distance rather than at the expected value of the empirical estimator. This centering property, which fails for the ordinary Wasserstein distance, opens the door to asymptotically valid confidence intervals and tests. The argument combines the Efron-Stein inequality with a uniform bound on the optimal transport potentials taken over all projection directions. The result covers measures that need not have compact support and supplies consistent estimators for the asymptotic variance together with a Monte Carlo scheme for the slicing integral.","feed_headline":"Sliced Wasserstein distance obeys CLT centered at population value","feed_subtitle":"The limiting normal can use the true distance as center, enabling inference for measures without compact support.","key_machinery":"The Efron-Stein inequality applied after a uniform control on the optimal transport potentials across all slicing directions.","core_discovery":"We establish a central limit theorem for the p-sliced Wasserstein distance, for p>1, centered at the expected empirical cost. Unlike for the general Wasserstein distance, the centering can be replaced by the population cost, enabling valid statistical inference. This generalizes and refines existing one-dimensional results, providing the first asymptotically valid inference framework for the sliced Wasserstein distance between possibly non-compact measures.","pith_inferences":["Hypothesis tests that treat the sliced distance as a test statistic become feasible at the usual asymptotic level.","The same proof pattern may extend to other sliced integral probability metrics whose one-dimensional projections satisfy a one-dimensional CLT.","In applications the result reduces computational cost by replacing resampling methods with direct normal approximation."],"forward_implications":["Asymptotically valid confidence intervals for the sliced Wasserstein distance can be formed without bootstrap.","Consistent estimators of the limiting variance are available from the same samples.","Monte Carlo approximation of the integral over directions preserves the central limit theorem.","The framework applies directly to measures with unbounded support."],"fun_headline_variants":[],"cache_read_input_tokens":64,"weakest_assumption_plain":"The optimal transport potentials admit a uniform bound that does not deteriorate when the projection direction varies.","fun_headline_variants_meta":{"error":"xAI API error (429): The model is currently at capacity due to high demand. Please try again in a few minutes. For guaranteed processing and availability, please request Provisioned Throughput: https://docs.x.ai/developers/advanced-api-usage/provisioned-throughput"},"cache_creation_input_tokens":0},"created_at":"2026-05-22T22:49:24.583657+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A sequence of non-compact measures for which the empirical p-sliced Wasserstein distance, properly normalized, fails to converge in distribution to a centered normal random variable.","supporting_citations":[],"review_version":1}