{"id":"7a846e33-8093-41c6-b1d9-a7912c3d815c","arxiv_id":"2606.09153","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"In the proportional high-dimensional regime, sample canonical directions in finite-rank spiked Gaussian CCA retain a deterministic fraction of population directional information with explicit CLT fluctuations around that limit.","lead":"This paper derives deterministic limits and central limit theorems for how well sample canonical directions align with population directions in high-dimensional spiked CCA under Gaussian assumptions. A smart generalist might read it to gauge the reliability of directional recovery when analyzing relationships between two large variable sets.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the modeling hypotheses on which the deterministic limit and CLT rest. Because the abstract and claim description already condition the results on those hypotheses and employ standard RMT tools, the load-bearing conditions are transparent rather than concealed. The UNVERDICTED status is therefore unchanged; the absence of the full proof text prevents further technical verification but does not reveal an additional flaw in the stated argument.","tokens_in":1740,"tokens_out":283,"duration_ms":14302,"concrete_test":"Re-derive the limiting variance expression in §3 (or equivalent) from the joint resolvent of the two Gram matrices without invoking the simple-spike separation assumption; check whether the same closed-form variance is recovered when two population spikes coincide.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim establishes a deterministic limit and CLT for squared alignment of sample canonical directions under the stated Gaussian finite-rank spiked model in the proportional-growth regime, with variance via resolvent traces and consistent plug-in estimators obtained by inverting the outlier map. These steps are conditional on the explicitly declared assumptions (Gaussianity, simple spikes). The approach follows standard deterministic-equivalent techniques in high-dimensional RMT; no internal inconsistency, hidden circularity, or unstated regime violation is visible in the claim structure.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies the asymptotic behavior of sample canonical directions in a finite-rank spiked high-dimensional CCA model under Gaussian population assumptions and proportional growth of dimensions with sample size. It derives a deterministic first-order limit for the squared alignment between sample and population directions for each simple spike, proves a CLT for fluctuations around this limit with asymptotic variance expressed via resolvent trace functionals, and constructs consistent plug-in estimators for the limit and variance by inverting the deterministic outlier eigenvalue map.","tokens_in":1840,"tokens_out":377,"duration_ms":19756,"significance":"If the derivations hold, the work supplies explicit, computable measures of directional recovery quality in high-dimensional CCA, where sample directions are typically inconsistent. The deterministic-equivalent approach and resolvent-based variance, together with the consistency proof for the plug-in estimators, provide a practical tool for assessing retained population information; this extends standard RMT techniques to CCA and is supported by simulations and real-data examples.","major_comments":[],"minor_comments":[{"comment":"The introduction would benefit from an early, explicit statement of the main theorems (including the precise form of the deterministic limit and the CLT variance expression) to orient the reader before the technical sections.","section":null},{"comment":"Notation for the two data-block dimensions and the spike strengths should be introduced with a single consolidated table or display equation near the model definition to avoid repeated cross-references.","section":null},{"comment":"In the simulation section, the number of Monte Carlo replications and the precise parameter values used to generate the population covariance blocks should be stated explicitly so that the reported alignment histograms can be reproduced.","section":null},{"comment":"The real-data illustration would be strengthened by reporting the estimated spike strengths and the resulting plug-in estimates of alignment and variance alongside the raw canonical correlations.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the supportive review, accurate summary of our contributions, and recommendation for minor revision. The report correctly identifies the key results on the deterministic limit and CLT for squared alignments of sample canonical directions, as well as the plug-in estimators.","responses":[],"tokens_in":1210,"tokens_out":70,"duration_ms":8597,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key takeaway is that the paper works out deterministic first-order limits for how much squared alignment sample canonical directions retain with their population versions in a finite-rank spiked CCA setup, then adds a CLT for the fluctuations whose variance comes from resolvent trace functionals, plus consistent estimators obtained by inverting the outlier eigenvalue map.\n\nThis is a direct extension of existing spiked-model eigenvalue results to the directional part. The Gaussian population, proportional growth regime, and simple-spike assumptions are stated up front, and the approach follows the usual deterministic-equivalent machinery for these models. The plug-in estimators look like a standard, non-circular step that makes the limits usable on data.\n\nThe main limitation is the narrow regime: Gaussianity and simple spikes are required, so the results do not automatically cover non-Gaussian data or overlapping spikes. The abstract does not claim broader validity, and without the full proofs it is hard to judge how delicate the technical conditions turn out to be. No circularity or internal inconsistency shows up in the claim structure.\n\nThis is for people working on high-dimensional CCA or random-matrix applications in multivariate statistics who need quantitative statements about directional recovery rather than just correlation strength. It is incremental but cleanly executed on its own terms.\n\nI would send it to peer review. The explicit variance expressions and estimator consistency are concrete enough to merit referee attention even if revisions are needed on the proofs.","headline":"This paper derives explicit deterministic limits and CLTs for squared alignment of sample canonical directions in the Gaussian spiked CCA model, with consistent plug-in estimators.","tokens_in":2309,"tokens_out":356,"would_cite":false,"duration_ms":14596,"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":"In spiked high-dimensional Gaussian CCA, squared alignment between sample and population canonical directions converges to an explicit deterministic limit with fluctuations obeying a central limit theorem.","keywords":["high-dimensional CCA","spiked model","canonical directions","asymptotic distribution","central limit theorem","resolvent functionals","directional recovery","outlier eigenvalues"],"falsifier":"Empirical squared alignments computed on data generated from the model that deviate systematically from the predicted deterministic limit as dimensions and sample size increase.","tokens_in":2641,"feed_emoji":"📊","tokens_out":668,"duration_ms":13983,"temperature":0.7,"pith_summary":"This paper studies the directional recovery properties of sample canonical directions in a finite-rank spiked CCA model where the two data dimensions grow proportionally to sample size under Gaussian assumptions. Sample directions fail to be consistent for their population counterparts even when the associated canonical correlations separate from the bulk. The authors derive a deterministic limit for the squared alignment that quantifies retained population directional information and prove a central limit theorem for the fluctuations around this limit, with variance given by limits of resolvent trace functionals. They further construct consistent plug-in estimators for the limit and variance by inverting the deterministic outlier eigenvalue map.","feed_headline":"Sample CCA directions retain fixed alignment fraction in spiked model","feed_subtitle":"Squared alignment converges to deterministic limit with explicit CLT variance from resolvents; plug-in estimators recover both quantities fr","key_machinery":"Deterministic first-order limit of the squared alignment between sample and population canonical directions, together with the associated central limit theorem derived from resolvent trace functionals.","core_discovery":"For each simple population spike, the squared alignment between a sample canonical direction and its population counterpart admits a deterministic first-order limit that measures retained directional information at the population level. Fluctuations of this alignment around the limit obey a central limit theorem whose asymptotic variance is expressed through deterministic limits of resolvent trace functionals. Plug-in estimators for both the limiting mean and the asymptotic variance are obtained by inverting the deterministic outlier eigenvalue map and are shown to be consistent.","pith_inferences":["The explicit form of the alignment limit suggests a simple correction factor that could be applied to improve estimation of population directions from the sample ones.","Because the variance depends on resolvent traces, the same machinery may extend to other linear statistics of the sample canonical vectors.","The results indicate that directional recovery quality can be assessed and reported routinely in applied CCA analyses once the plug-in estimators are implemented."],"forward_implications":["The limiting alignment supplies an explicit quantitative measure of population directional information retained by each sample direction.","The central limit theorem supplies asymptotic normality that can be used for inference on directional recovery quality.","Consistent plug-in estimators allow computation of both the limit and its variance directly from observed data without knowledge of population parameters.","The same inversion technique that produces the estimators also yields computable expressions for the resolvent-based variance."],"fun_headline_variants":["Spiked CCA directions retain deterministic alignment limit","Alignment fluctuations in spiked CCA obey CLT from resolvents","Fixed alignment for sample CCA directions in high-dim spiked model","Resolvent trace functionals give CCA alignment variance limit","CCA sample directions approach alignment limit with explicit CLT"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The observations come from a Gaussian population with finite-rank spiked structure and the two block dimensions grow proportionally with sample size.","fun_headline_variants_meta":{"raw":{"variants":["Spiked CCA directions retain deterministic alignment limit","Alignment fluctuations in spiked CCA obey CLT from resolvents","Fixed alignment for sample CCA directions in high-dim spiked model","Resolvent trace functionals give CCA alignment variance limit","CCA sample directions approach alignment limit with explicit CLT"]},"model":"grok-4.3","cost_usd":0.005652,"raw_usage":{"total_tokens":2700,"prompt_tokens":663,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":56524500,"prompt_tokens_details":{"text_tokens":663,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1963,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":663,"tokens_out":74,"duration_ms":16677,"temperature":1.0,"reasoning_tokens":1963,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T14:54:36.659281+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Empirical squared alignments computed on data generated from the model that deviate systematically from the predicted deterministic limit as dimensions and sample size increase.","supporting_citations":[],"review_version":1}