{"id":"05ac8be0-da18-4883-bd5d-06400579280d","arxiv_id":"2412.17792","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A multi-round distributed CCA algorithm is shown to match the pooled-data convergence rate with vector-only communication and a gap-free error bound that avoids explicit eigenvalue-gap assumptions.","lead":"The authors propose a communication-efficient distributed algorithm for canonical correlation analysis, where several machines jointly estimate correlation directions without pooling data. Its theoretical guarantee holds without assuming a gap between successive canonical correlations, a common but often unrealistic assumption.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma B.3's proof redefines κ as ||Ĥ1−Ĥ||, so Theorem 1's stated condition on ||T̂1−T̂|| does not by itself establish the inner-loop contraction that the linear-rate claim rests on.","rationale":"The reader's weakest assumption was the per-machine sample size requirement κ = op(γδρ̂1). I agree that this is a real limitation of the central claim, but the single most load-bearing technical hinge is more specific: Lemma B.3's proof conflates the spectral deviation of the whitened cross-covariance operator, κ = ||T̂1 − T̂||, with the Hessian discrepancy ||Ĥ1 − Ĥ||. The contraction factor in the inner loop (B.7) is derived from the Hessian discrepancy, yet the theorem's stated condition is only on the former. Since Theorem 1's proof applies Lemma B.3 to control ε_{T'} and then recurses, a failure here would invalidate the proof of linear convergence to the pooled estimator as written. The concern is not that the algorithm is wrong; it is that the argument as presented has a definitional gap. The repair would likely require an additional covariance concentration argument, which in turn makes the per-machine sample-size condition explicit and somewhat weakens the abstract's claim about the number of machines. This is consistent with the reader's conditional verdict: the main idea is plausible and likely fixable, but the proof currently does not rigorously support the central theorem without modification.","tokens_in":31729,"tokens_out":14264,"duration_ms":137700,"concrete_test":"Analytically check whether ||Ĥ1 − Ĥ|| ≤ C·||T̂1 − T̂|| holds under the paper's normalization max{σmax(Σx), σmax(Σy)} ≤ 1 and γ = min{σmin(Σx), σmin(Σy)} > 0. If yes, re-prove Lemma B.3 with the correct definitions and update the constants in Corollary 1 to absorb C. If no, exhibit a concrete data-generating process with small ||T̂1 − T̂|| but non-negligible ||Ĥ1 − Ĥ||, and verify numerically that the inner loop (Algorithm 1, step 9) stops contracting even when 2κ ≤ ω holds. This settles whether Theorem 1 needs a strengthened assumption on covariance concentration, not merely on the whitened cross-covariance operator.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1 depends on Lemma B.3, whose contraction factor is 8κ/(γω) in (B.7). The lemma statement explicitly recalls κ = ||T̂1 − T̂|| as in Theorem 1. However, the proof immediately writes '||Ĥ1 − Ĥ|| = κ' and bounds this Hessian discrepancy via covariance differences. These are different quantities: Ĥ is built from covariance blocks, while T̂ is the whitened cross-covariance. The bound ||I − M_{ρ1,1}M_ρ^{-1}|| ≤ ||Ĥ1^{-1}||·||Ĥ1 − Ĥ|| involves the whitened Hessians, and λ_min(Ĥ1) ≥ γω/4 does not convert a bound on ||T̂1 − T̂|| into a bound on ||Ĥ1 − Ĥ||. The map (Σx, Σy, Σxy) ↦ T = Σx^{-1/2}ΣxyΣy^{-1/2} can be well-conditioned even when the covariance blocks differ substantially in directions with little cross-covariance, so ||Ĥ1 − Ĥ|| is not controlled by κ. As written, Lemma B.3's conclusion (B.7) is therefore not proven under Theorem 1's assumptions. This is load-bearing because Theorem 1's proof invokes Lemma B.3 to set ε_{T'} and then recurses to obtain the final bound; without a valid inner-loop contraction in terms of the theorem's κ, Corollary 1's base 576κ/(γδρ̂1) is unsupported. The gap is likely repairable by adding a standard covariance concentration bound, but the repair would introduce an extra per-machine sample-size condition, reinforcing the reader's concern about the 'no stringent restrictions on number of machines' claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a multi-round distributed canonical correlation analysis (CCA) algorithm. The method constructs canonical directions through a convex, shift-and-invert preconditioned formulation and approximates the pooled Newton step using the first machine's Hessian together with averaged local gradients, requiring only O(dK) communication per inner iteration. The main theoretical contribution is a gap-free analysis: Theorem 1 and Corollary 1 claim that, under conditions involving the per-machine spectral deviation κ = ||T̂1 − T̂||, the distributed top canonical direction converges linearly to the pooled estimate with error Op((576κ/(γδρ̂1))^T), without requiring an explicit gap between canonical correlations. Corollaries 2–6 extend this to co-variability capture and top-L subspace guarantees. Simulations and three image benchmarks compare the method favorably with naive and whitened divide-and-conquer baselines.","tokens_in":32144,"tokens_out":13415,"duration_ms":123718,"significance":"If the proof gaps are repaired, the paper would make a useful contribution: a communication-efficient distributed CCA algorithm that transmits vectors rather than covariance matrices, combined with a gap-free analysis that avoids the commonly imposed eigengap assumption. The paper also provides an explicit concentration lemma (Lemma A.1) and is candid in the final remark that only the first machine's local sample size needs to be sufficiently large. The central claims are plausible and the numerical results support the algorithm's practical promise, but the proof of the main convergence theorem currently contains a load-bearing gap that must be fixed before the advertised rates are established.","major_comments":[{"comment":"Lemma B.3 is stated for κ = ||T̂1 − T̂||, exactly as in Theorem 1, but the proof contains the line '||Ĥ1 − Ĥ|| def = κ' and then bounds Ĥ1 − Ĥ by covariance deviations. No inequality is proved that controls ||Ĥ1 − Ĥ|| in terms of ||T̂1 − T̂||; the map from the covariance blocks to T = Σx^{-1/2}ΣxyΣy^{-1/2} does not have a dimension-free Lipschitz inverse in directions orthogonal to Σxy. Consequently the contraction factor 8κ/(γω) in (B.7) is not established under the theorem's assumptions. This is load-bearing because Theorem 1's proof uses ε_{T'} from Lemma B.3 in the recursion leading to (B.8), and Corollary 1's base 576κ/(γδρ̂1) depends on that factor. The gap appears repairable by adding a high-probability covariance-concentration bound, but the repair will introduce an explicit per-machine sample-size condition and should be reflected in the statements of Theorem 1 and Corollary 1.","section":"Appendix B, Lemma B.3 (Eq. B.7)"},{"comment":"The displayed identity '||I − M_{ρ1,1}M_ρ^{-1}|| = ||I − Ĥ1^{-1}Ĥ||' is not valid as written. From (B.1)–(B.2), M_{ρ1,1}M_ρ^{-1} = SĤ1^{-1}ĤS^{-1} with S = diag(Σ̂x^{1/2}, Σ̂y^{1/2}); S does not commute with Ĥ unless the cross-covariance is zero, so the spectral norm of I − SĤ1^{-1}ĤS^{-1} is not equal to that of I − Ĥ1^{-1}Ĥ. The proof needs an additional bound on ‖S(I − Ĥ1^{-1}Ĥ)S^{-1}‖, which introduces factors of the empirical covariance condition number and again affects the contraction factor in (B.7).","section":"Appendix B, Lemma B.3 proof"},{"comment":"The claim that the method 'does not impose stringent restrictions on the number of machines' needs qualification. Corollary 1's linear-rate condition is κ = op(γδρ̂1) with κ = Op((d log²d/n)^{1/2}); in the uniformly split setting n = N/K, this requires n ≫ d log²d/(γδρ̂1)² and hence K ≪ N(γδρ̂1)²/(d log²d). The final remark in Section 6 correctly notes that the first machine's sample size must be sufficiently large, but it does not state the fixed-total-sample-size consequence, and Table 1 shows the estimate deteriorating as K grows for fixed iteration counts. Please state this trade-off explicitly so that the advertised flexibility in K can be calibrated against the per-machine sample-size requirement.","section":"Section 3.1 (Corollary 1) and Section 6"}],"minor_comments":[{"comment":"The manuscript contains JMLR template artifacts, including 'Journal of Machine Learning Research 23 (2022) 1-35', 'Editor: My editor', and '©2022 Author One and Author Two'; these should be removed before any formal submission.","section":"Title page"},{"comment":"The inner-loop term in Theorem 1 has a constant 1, while the proof in (B.8) has a constant 512 in front of (64κ²/(γ²ω²))^{T'}; the constants should be reconciled.","section":"Theorem 1 and Eq. (B.8)"},{"comment":"The caption appears to be missing the panel label for MFEAT; it reads '(A): MMILL (B): MNIST MFEAT' rather than labeling all three panels.","section":"Figure 1 caption"},{"comment":"The matrix B = diag(Σ̂x, Σ̂y) is introduced in the proof of Lemma B.2 but not in the methodology section; defining it where the local gradients are first discussed would improve readability.","section":"Section 2.2 and Lemma B.2"},{"comment":"The abstract says the distributed estimate achieves 'the same convergence rate as if all observations were pooled together', but the theorems compare the distributed estimate with the pooled estimator, not with the population canonical directions; this should be clarified to avoid overstatement.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The main issue is a genuine but repairable proof gap in Lemma B.3: the contraction factor is not established in terms of the theorem's κ, and there is an additional unjustified similarity step involving covariance square roots. I recommend major revision rather than rejection because the approach is plausible and the missing arguments are likely standard concentration bounds. Please also ask the authors to explicitly reconcile the fixed-total-sample-size trade-off with the claim of no stringent restrictions on the number of machines."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a useful paper on distributed CCA, with a real algorithmic contribution and a proof gap in one lemma that is repairable but load-bearing as written. Don't desk-reject it; send it to a referee, but ask that referee to check Lemma B.3 carefully.\n\nWhat's new: a multi-round distributed algorithm for CCA that communicates only vectors, uses shift-and-invert with a local Hessian, and gives a gap-free convergence bound matching the pooled rate. The gap-free analysis is borrowed from Allen-Zhu and Li, but the distributed setting with local Hessian approximations is not in the prior literature. The bound in Corollary 1 is the kind of result people will want. The simulations and the three benchmark applications are honest and show the algorithm works.\n\nSoft spots, in order of seriousness.\n\n1. Lemma B.3. The theorem defines κ = ||T̂1 − T̂||. The lemma statement recalls that definition. The proof, however, writes ||Ĥ1 − Ĥ|| = κ and bounds the Hessian difference by covariance deviations. These are not the same quantity. The whitened cross-covariance T can be well-conditioned even when the covariance blocks differ in directions with little cross-covariance, so a bound on ||T̂1 − T̂|| does not control ||Ĥ1 − Ĥ||. As written, the inner-loop contraction (B.7) is not proven under the theorem's assumptions. That matters because Theorem 1 uses Lemma B.3 to set the inner-loop error and recurse. The fix looks straightforward: add a covariance concentration bound linking the two quantities, which will put an explicit per-machine sample-size condition into the theorem. But as written, the main bound doesn't follow.\n\n2. The abstract's claim of no stringent restrictions on the number of machines is overstated. The theory needs n large enough that κ = o(δ γ ρ1). For fixed N, that limits K. The paper's own remark at the end of Section 3 concedes the per-machine sample must be large; the abstract should say \"much weaker restrictions than one-shot methods\" rather than \"no stringent restrictions.\"\n\n3. Minor: no code or error bars in the experiments. The log-error tables are suggestive, not definitive.\n\nOverall: the core idea is sound, the writing is clear, and the citations are appropriate. The proof gap is the one thing that must be fixed before publication. I'd accept this for peer review and ask for a revision that repairs Lemma B.3 and recalculates the constants.","headline":"Worth a careful referee: a solid distributed CCA algorithm with a gap-free bound, but the main theorem's proof has a fixable yet load-bearing mismatch in Lemma B.3.","tokens_in":32669,"tokens_out":3322,"would_cite":true,"duration_ms":32026,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A distributed, communication-efficient algorithm estimates canonical correlations at pooled-data accuracy without assuming a gap between successive correlations.","keywords":["canonical correlation analysis","distributed estimation","gap-free bound","shift-and-invert preconditioning","communication-efficient algorithm","multi-round distributed optimization","top-L canonical directions"],"falsifier":"Simulate CCA data with a controlled spectral gap, split it across K machines, and on each run measure $\\kappa=\\|\\hat T_1-\\hat T\\|$ from the first machine's sample covariance versus the pooled one. Set the local sample size n so that $576\\kappa/(\\gamma\\delta\\hat\\rho_1)\\ge 1$; then Theorem 1's condition fails and the observed error curve should stop decreasing (or diverge) with outer iterations. A run with n just above the threshold should show the linear contraction predicted by Corollary 1, so the threshold itself is observable.","tokens_in":31514,"feed_emoji":"📊","tokens_out":11027,"duration_ms":97151,"temperature":0.7,"pith_summary":"Canonical correlation analysis finds the linear combinations of two variable sets that move together, and doing it on data spread over many machines normally forces a choice between communicating large covariance matrices or accepting biased one-shot estimates. This paper proposes a multi-round distributed algorithm whose rounds exchange only vectors, and proves that its output converges linearly to the estimate a single pooled machine would produce. The proof works without the usual requirement that the top canonical correlations be separated by a nonzero gap, an assumption the paper shows is violated in three common image datasets. A sympathetic reader can take the central claim as: distributed CCA can match pooled-data accuracy at moderate communication cost, and the guarantee degrades gracefully, through a tunable threshold $\\delta$, as the correlations cluster together.","feed_headline":"Distributed CCA matches pooled accuracy without eigen-gap assumption","feed_subtitle":"Vector-only communication matches pooled accuracy even with nearly tied canonical correlations.","key_machinery":"The machinery is shift-and-invert preconditioning for CCA, executed in a distributed star network. Shift-and-invert preconditioning is an iterative method that turns a top-eigenvector problem into repeated linear-system solves: the paper writes the CCA objective as a convex quadratic whose Hessian is $\\hat H=\\begin{pmatrix}\\bar\\rho_1\\hat\\Sigma_x & -\\hat\\Sigma_{x,y}\\\\ -\\hat\\Sigma_{x,y}^T & \\bar\\rho_1\\hat\\Sigma_y\\end{pmatrix}$, so one Newton step is applying $(\\bar\\rho_1 I-\\hat C)^{-1}$ to a vector rather than computing an SVD. In each outer round, the central machine collects the local gradient vectors $\\hat g_{k,j}$ (each of dimension $d=d_x+d_y$) and updates using the first machine's Hessian $\\hat H_1$ as a surrogate for the pooled Hessian; Lemma B.3 shows this inner loop contracts at rate $8\\kappa/(\\gamma\\omega)$. The gap-free component is the error metric (3.1): instead of requiring $\\hat\\rho_1-\\hat\\rho_2>0$, the paper measures error against all pooled directions with correlation at most $(1-\\delta)\\hat\\rho_1$, and Lemma B.2 shows each outer iteration shrinks the sum of those squared projections by a factor $128\\omega^2/(\\delta^2\\hat\\rho_1^2)$. That ratio, together with the surrogate-Hessian approximation, is what produces the contraction base $576\\kappa/(\\gamma\\delta\\hat\\rho_1)$.","core_discovery":"The paper establishes that top canonical directions can be computed in a distributed star network so that the estimator matches the pooled estimator up to the same statistical rate, with no explicit gap between successive canonical correlations. The main result (Theorem 1, simplified in Corollary 1) bounds the distance between the distributed and pooled estimates by a gap-free error: for any $\\delta\\in(0,1)$, the squared projections of the distributed direction onto pooled directions whose correlations are at most $(1-\\delta)\\hat\\rho_1$ are $O_p\\{(128\\omega^2/(\\delta^2\\hat\\rho_1^2))^T + (1/(1-128\\omega^2/(\\delta\\hat\\rho_1)^2))(64\\kappa^2/(\\gamma^2\\omega^2))^{T'}\\}$. Choosing $\\omega=(\\kappa\\delta\\hat\\rho_1/\\gamma)^{1/2}/3$ and $T'=T$ reduces the bound to $O_p\\{(576\\kappa/(\\gamma\\delta\\hat\\rho_1))^T\\}$, a linear contraction whenever the first machine's spectral deviation $\\kappa=\\|\\hat T_1-\\hat T\\|$ is small. Corollary 2 converts this into a co-variability guarantee: $(\\hat u^\\text{dist})^T\\hat\\Sigma_{x,y}(\\hat v^\\text{dist})\\ge (1-\\delta)(1-\\varepsilon)\\hat\\rho_1$, so the distributed pair captures almost all of the pooled top-pair co-variability even when consecutive correlations coincide. Theorem 2 and its corollaries extend the same linear rate to the top-$L$ subspaces, replacing the classical $\\sin\\Theta$ gap condition with an enlarged-subspace statement.","pith_inferences":["The paper's 'no stringent restriction on K' is conditional: for fixed total sample size N, increasing K eventually makes n too small to keep $576\\kappa/(\\gamma\\delta\\hat\\rho_1)<1$. An implicit consequence is that the method has a maximum number of usable machines at a given N, and the paper does not spell out its closed form.","The same enlargement-of-target trick, measuring error against all pooled directions within a $\\delta$-neighborhood of the top correlation, could be transplanted to distributed PCA and generalized eigendecomposition, where explicit eigen-gaps are also commonly assumed; that transfer is not explored in the paper.","A practitioner-facing diagnostic follows directly: estimate $\\hat\\kappa=\\|\\hat T_1-\\hat T\\|$ from local and pooled covariances; if $576\\hat\\kappa/(\\hat\\gamma\\delta\\hat\\rho_1)$ is near or above 1, the central machine should pull more data before trusting the output, or $\\delta$ should be increased."],"forward_implications":["After $O(\\log(1/\\varepsilon))$ outer rounds, the distributed estimate reaches a pooled-level error; each round transmits only $d$-dimensional vectors, so the communication cost is $O(TT'dK)$ and the algorithm qualifies as communication-efficient under the vector-transmission standard.","The gap-free bound remains meaningful when the top canonical correlations are tied: the error is controlled by the tunable threshold $\\delta$, and Corollary 2 guarantees the distributed pair captures at least $(1-\\delta)(1-\\varepsilon)$ of the pooled top-pair co-variability.","For the top-$L$ setting, Theorem 2 extends the same linear rate to subspaces without requiring $\\hat\\rho_L-\\hat\\rho_{L+1}>0$; Corollaries 4-6 provide a gap-free analogue of the classical $\\sin\\Theta$ theorem and show each estimated direction's captured co-variability is within a $(1-\\delta)$ factor of the pooled one.","The proof conditions hold with overwhelming probability under sub-Gaussian data, and the same algorithm admits a fully stochastic initialization (no local SVD), so the convergence guarantee is not tied to an expensive initial eigen-decomposition."],"supporting_citations":[{"why":"Supplies the doubly accelerated framework for CCA and generalized eigendecomposition, the gap-free analysis style, and the lemma (Lemma C.4 / Theorem F.1) that Theorem 2's subspace proof adapts.","marker":"Allen-Zhu and Li (2017)"},{"why":"Origin of shift-and-invert preconditioning: turns the top-eigenvector CCA problem into repeated linear-system solves.","marker":"Golub and Van Loan (1983)"},{"why":"Developed the globally convergent shift-and-invert stochastic CCA method and the initial-estimate/upper-bound construction the distributed algorithm inherits.","marker":"Wang et al. (2016)"},{"why":"Provides the stochastic CCA framework and matrix-concentration bounds (Corollary 7) used to show $\\kappa=O_p\\{(d\\log^2 d/n)^{1/2}\\}$.","marker":"Gao et al. (2019)"},{"why":"Variant Davis-Kahan theorem used to establish the initial condition and the sin-Θ-type distance comparison for top-L subspaces.","marker":"Yu et al. (2015)"},{"why":"The distributed PCA baseline whose bias problem motivates multi-round schemes; the paper contrasts its convergence with the present algorithm.","marker":"Fan et al. (2019)"},{"why":"One-shot divide-and-conquer CCA, the main communication-bias competitor the paper improves upon.","marker":"Lv et al. (2020)"},{"why":"Produces the theoretical eigen-gap lower bounds that the real benchmark image data visibly violate, motivating the gap-free analysis.","marker":"Cheng et al. (2021)"},{"why":"Establishes the communication-efficiency standard (O(d) vector transmissions per round) used to argue the distributed algorithm is communication-efficient.","marker":"Jordan et al. (2019)"}],"fun_headline_variants":["Distributed CCA hits pooled accuracy with no eigen-gap","No eigen-gap? Distributed CCA still matches pooled rates","Gap-free distributed CCA reaches pooled statistical rates","Vector-only CCA achieves pooled accuracy, gaps or not","Distributed CCA: pooled-level accuracy without gap conditions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the first machine's local sample is large enough that the difference between its covariance structure and the pooled one (the paper's $\\kappa$) stays small relative to the target correlation strength $\\gamma\\delta\\hat\\rho_1$, since otherwise the contraction ratio $576\\kappa/(\\gamma\\delta\\hat\\rho_1)$ exceeds one and the linear convergence to pooled accuracy is not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Distributed CCA hits pooled accuracy with no eigen-gap","No eigen-gap? Distributed CCA still matches pooled rates","Gap-free distributed CCA reaches pooled statistical rates","Vector-only CCA achieves pooled accuracy, gaps or not","Distributed CCA: pooled-level accuracy without gap conditions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000696,"raw_usage":{"total_tokens":3175,"prompt_tokens":1001,"completion_tokens":2174,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":2092}},"tokens_in":617,"tokens_out":2174,"duration_ms":13419,"temperature":1.0,"reasoning_tokens":2092,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:09:43.117170+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate CCA data with a controlled spectral gap, split it across K machines, and on each run measure $\\kappa=\\|\\hat T_1-\\hat T\\|$ from the first machine's sample covariance versus the pooled one. Set the local sample size n so that $576\\kappa/(\\gamma\\delta\\hat\\rho_1)\\ge 1$; then Theorem 1's condition fails and the observed error curve should stop decreasing (or diverge) with outer iterations. A run with n just above the threshold should show the linear contraction predicted by Corollary 1, so the threshold itself is observable.","supporting_citations":[{"cited_title":"Doubly accelerated methods for faster CCA and generalized eigendecomposition","cited_arxiv_id":null,"evidence_quote":"Supplies the doubly accelerated framework for CCA and generalized eigendecomposition, the gap-free analysis style, and the lemma (Lemma C.4 / Theorem F.1) that Theorem 2's subspace proof adapts."},{"cited_title":"Efficient globally convergent stochastic optimization for canonical correlation analysis","cited_arxiv_id":null,"evidence_quote":"Developed the globally convergent shift-and-invert stochastic CCA method and the initial-estimate/upper-bound construction the distributed algorithm inherits."},{"cited_title":"Stochastic Canonical Correlation Analysis","cited_arxiv_id":null,"evidence_quote":"Provides the stochastic CCA framework and matrix-concentration bounds (Corollary 7) used to show $\\kappa=O_p\\{(d\\log^2 d/n)^{1/2}\\}$."},{"cited_title":"Distributed estimation of principal eigenspaces","cited_arxiv_id":null,"evidence_quote":"The distributed PCA baseline whose bias problem motivates multi-round schemes; the paper contrasts its convergence with the present algorithm."},{"cited_title":"Tackling Small Eigen-Gaps : Fine-Grained Eigenvector Estimation and Inference Under Heteroscedastic Noise","cited_arxiv_id":null,"evidence_quote":"Produces the theoretical eigen-gap lower bounds that the real benchmark image data visibly violate, motivating the gap-free analysis."}],"review_version":1}