{"id":"d84d9319-e945-4ca4-8310-d972be3d7a90","arxiv_id":"2605.25173","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Nyström acceleration of KSD GoF tests preserves asymptotic level and local consistency under mild conditions and matches quadratic-time performance at lower runtime on spherical and functional data.","lead":"The paper proves that Nyström approximation preserves the asymptotic level and local consistency of bootstrapped Kernel Stein Discrepancy goodness-of-fit tests while cutting runtime from quadratic to linear in sample size. A smart generalist might read it to understand how to run reliable statistical tests on large modern datasets without paying the usual computational cost.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly flags the reliance on the known Nyström fact as the potential pivot point, but the paper claims to have closed the bootstrap gap with a proof. Absent any visible flaw in the argument structure or mismatch between claim and stated contribution, the UNVERDICTED verdict requires no adjustment.","tokens_in":1713,"tokens_out":249,"duration_ms":26172,"concrete_test":"Locate the main theorem stating preservation of asymptotic level (typically in the results section) and confirm that its hypotheses are exactly the mild conditions from the cited Nyström-KSD consistency result plus any additional bootstrap-specific requirements; if the hypotheses match without extra restrictions, the claim holds as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a proof that asymptotic level and local consistency of the bootstrapped KSD GoF test carry over to its Nyström version. The abstract explicitly positions this as an extension of a known result on estimator accuracy under mild conditions, with the bootstrap analysis as the novel part. No internal inconsistency, unstated assumption, or gap in the high-level logic is detectable from the given material.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript develops a Nyström-accelerated variant of the bootstrapped quadratic-time Kernel Stein Discrepancy (KSD) goodness-of-fit test. It proves that the asymptotic level and local consistency of the original test are preserved under the Nyström approximation (under mild conditions already known to control estimator accuracy), and reports numerical experiments on spherical and functional data showing statistical parity with substantially reduced runtime.","tokens_in":1783,"tokens_out":369,"duration_ms":17646,"significance":"The preservation proof for bootstrap-based KSD testing under Nyström acceleration directly addresses the quadratic scaling barrier while retaining the test's theoretical guarantees. This is a useful extension for practitioners working with large samples on general domains; the explicit statement that the bootstrap analysis is the novel component, together with the numerical parity demonstration, strengthens the contribution.","major_comments":[],"minor_comments":[{"comment":"§3 (or the theorem statement on preservation): the precise statement of the 'mild conditions' under which the Nyström estimator accuracy carries over should be recalled or referenced explicitly so that the bootstrap extension is self-contained.","section":"Theorem on asymptotic level"},{"comment":"Numerical section: the choice of Nyström rank (or number of landmarks) is described only qualitatively; adding a short sensitivity table or explicit rule used in the experiments would improve reproducibility.","section":"Experiments"},{"comment":"Notation: the distinction between the full KSD estimator and its Nyström version is clear in the text but the bootstrap resampling step could use a single consistent symbol (e.g., B_n vs. B_n^N) throughout to avoid any momentary ambiguity.","section":"Bootstrap procedure"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of the manuscript, recognition of its contribution in preserving bootstrap-based KSD test guarantees under Nyström acceleration, and recommendation for minor revision. No major comments were raised in the report.","responses":[],"tokens_in":1208,"tokens_out":63,"duration_ms":11607,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core result is that the Nyström version keeps the bootstrap-based KSD test valid at the same asymptotic level and local consistency as the full quadratic-time version. This closes the specific open question on the bootstrap step that prior Nyström work on KSD estimation had left unaddressed.\n\nThe paper does the preservation proof cleanly and states the conditions explicitly. The numerical experiments on spherical and functional data show the accelerated estimator and bootstrap match the original in power while cutting runtime, which is the practical payoff. That combination of a targeted proof and matching numerics is the main value.\n\nThe soft spots are limited. Everything hinges on the mild conditions for the Nyström approximation already established in earlier work; if those do not hold for a particular kernel or data regime, the guarantees do not apply, though the paper does not overclaim. The experiments are narrow in domain, so the practical robustness across more settings is not yet shown.\n\nThis is for people already running KSD tests who need to handle larger samples. It is a precise, incremental extension rather than a new framework. The presence of the preservation proof plus reproducible numerics makes the central claim checkable, so it deserves referee time.","headline":"The paper proves that Nyström approximation preserves asymptotic level and local consistency for bootstrapped KSD GoF tests under the same mild conditions already known for the estimator.","tokens_in":2259,"tokens_out":319,"would_cite":false,"duration_ms":22066,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Nyström acceleration preserves asymptotic level and local consistency of bootstrapped KSD goodness-of-fit tests.","keywords":["Nyström approximation","kernel Stein discrepancy","goodness-of-fit test","bootstrap","asymptotic level","local consistency","computational acceleration"],"falsifier":"Generate samples from the null distribution, run the Nyström-accelerated bootstrap KSD test at fixed significance level, and check whether the empirical rejection rate converges to that level as sample size grows.","tokens_in":2624,"feed_emoji":"📊","tokens_out":606,"duration_ms":29435,"temperature":0.7,"pith_summary":"The paper proves that accelerating kernel Stein discrepancy estimation via the Nyström method does not compromise the asymptotic level or local consistency of the associated bootstrap goodness-of-fit test. Classical KSD tests suffer from quadratic runtime and rely on bootstrapping whose distribution is hard to compute directly. Showing that the fast version inherits the same guarantees means practitioners can test model fit on larger datasets with the same reliability. Experiments illustrate that the accelerated tests match the original in power while running much faster on spherical and functional data examples.","feed_headline":"Nyström keeps KSD test validity while slashing runtime","feed_subtitle":"Bootstrapped goodness-of-fit tests retain asymptotic level and local consistency after acceleration, enabling use on bigger samples.","key_machinery":"Nyström approximation of the kernel Stein discrepancy estimator, which reduces quadratic kernel evaluations to a lower-rank form while keeping the bootstrap null distribution and power properties unchanged.","core_discovery":"We prove that the key properties of the quadratic-time bootstrapped KSD-based GoF test (asymptotic level and local consistency) are preserved by its Nyström acceleration. This holds because the Nyström method permits accelerating KSD estimation with no loss of statistical accuracy under mild conditions.","pith_inferences":["The same Nyström argument may apply to other kernel discrepancies that currently rely on quadratic-time bootstrap tests.","The accelerated procedure could support sequential or streaming goodness-of-fit monitoring where fresh data arrives continuously.","Combining Nyström with other low-rank approximations such as random features might yield further speed-ups whose bootstrap properties remain provable."],"forward_implications":["Bootstrapped KSD tests become feasible for sample sizes where full quadratic computation exceeds available resources.","The same asymptotic validity proof covers both the original and accelerated estimators, so no separate bootstrap analysis is required.","Local consistency carries over, so the accelerated test detects local alternatives at the same rate as the full method.","Numerical results on spherical and functional data confirm that statistical performance stays on par while runtime drops substantially."],"fun_headline_variants":["Nyström maintains KSD test asymptotics","Nyström speeds KSD tests without accuracy loss","Bootstrapped KSD validity holds after Nyström","Nyström acceleration preserves GoF test consistency"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The mild conditions that let the Nyström method accelerate KSD estimation without loss of statistical accuracy continue to hold when the estimator enters the bootstrap test.","fun_headline_variants_meta":{"raw":{"variants":["Nyström maintains KSD test asymptotics","Nyström speeds KSD tests without accuracy loss","Bootstrapped KSD validity holds after Nyström","Nyström acceleration preserves GoF test consistency"]},"model":"grok-4.3","cost_usd":0.005439,"raw_usage":{"total_tokens":2530,"prompt_tokens":655,"num_sources_used":0,"completion_tokens":56,"cost_in_usd_ticks":54390500,"prompt_tokens_details":{"text_tokens":655,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1819,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":655,"tokens_out":56,"duration_ms":7078,"temperature":1.0,"reasoning_tokens":1819,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T23:38:00.481929+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Generate samples from the null distribution, run the Nyström-accelerated bootstrap KSD test at fixed significance level, and check whether the empirical rejection rate converges to that level as sample size grows.","supporting_citations":[],"review_version":1}