{"id":"4a715eb7-1730-47d5-bb21-45243142a5b0","arxiv_id":"2501.15725","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For latent position random graphs with infinite-rank or indefinite link kernels, the paper proves refined eigenvector expansions, row-wise central limit theorems, and a rank-adaptive test for equality of latent positions.","lead":"Random graph models with infinitely many latent components are usually harder to analyze than low-rank ones. This paper proves sharp row-by-row error bounds, normal approximations, and a rank-adaptive hypothesis test for such models, extending spectral inference to general graphon-type networks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4's rank-adaptive N(0,2) claim rests on an unproved data-driven selector: condition (4.12) guarantees only gaps of order sqrt(r n rho_n) log^{3/4} n, below the sqrt(r n rho_n) log n required by Theorem 4.","rationale":"The reader's weakest-assumption focused on the indefinite-kernel setting and the lack of a high-probability spectral concentration bound. That is a real issue, but my stress-test found a more elementary and more broadly damaging gap: Corollary 4, which is the paper's advertised rank-adaptive testing result, is asserted without proof, and its data-driven selector (4.12) appears incompatible with the quantitative gap conditions used in Theorem 4 and Lemma 1. Even in the positive semidefinite infinite-rank case, the threshold in (4.12) is weaker by a logarithmic factor than the condition (4.11) needed to control residuals. Furthermore, the universal claim that b_r diverges for every infinite-rank kernel is false for spectra that decay faster than r^{-1/2}, because then no diverging r(n) can satisfy delta_r = omega(sqrt(r n rho_n) log n). This is not a matter of consensus; it is an internal incompatibility between the stated selector and the stated conditions. The main perturbation theorems 1-3 and Theorem 4 as conditional statements may still be correct, so I would not reject the paper outright, but the central inference claim cannot be accepted as stated. The concrete check above would settle whether the concern lands by exhibiting a natural kernel where the selector fails the theorem's hypotheses or by showing the claimed N(0,2) limit numerically.","tokens_in":66191,"tokens_out":10571,"duration_ms":106006,"concrete_test":"Choose a concrete infinite-rank positive semidefinite kernel with rapidly decaying spectrum, e.g. the Gaussian kernel kappa(x,y) = exp(-(x-y)^2/sigma^2) on a bounded interval, whose integral operator has eigenvalues mu_r = Theta(exp(-c r)) for some c > 0. For n in {10^3, 10^4, 10^5} with fixed rho_n (say rho_n = 0.4), compute b_r defined in (4.12) and the corresponding population gap delta_{b_r} = |lambda_{b_r}| - |lambda_{b_r+1}|. Check whether delta_{b_r} = omega(sqrt(b_r n rho_n) log n), as required by Lemma 1 and Theorem 4. If the ratio delta_{b_r} / (sqrt(b_r n rho_n) log n) tends to 0, the data-driven rank fails the theorem's hypotheses. As a complementary check, simulate T(hat X_i, hat X_j) under H0 for this kernel at increasing n and compare the empirical distribution to N(0,2); lack of convergence to normality would confirm the concern.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most load-bearing gap is Corollary 4 (Section 4.2). It is stated without proof: the text says \"Combining Theorem 4 and Lemma 1 yields...\" but gives no argument that the data-driven rank b_r in (4.12) satisfies the hypotheses of Theorem 4 or Lemma 1, nor that b_r -> infinity. Under H0, replacing r by b_r in the expansion (4.13) requires the perturbation residual and the estimation errors for tr M* and ||M*||_F to be negligible. Lemma 1's condition (4.11) requires, for growing r, essentially delta_r = omega(sqrt(r n rho_n) log n). The selector (4.12), however, uses thresholds of order log^{7/4} n, sqrt(j n rho_n) log^{3/4} n, and (n rho_n)^{3/4}; combined with Weyl's inequality it can guarantee at best delta_j >= sqrt(j n rho_n) log^{3/4} n, not log n. Thus b_r may overshoot the valid range and the residual epsilon_n in (4.13) is not controlled. Moreover, the claim that b_r -> infinity in probability for every infinite-rank kernel is not true without an eigenvalue decay condition: if the eigenvalues of K decay faster than r^{-1/2}, e.g. exponentially as for analytic kernels, then every diverging sequence r(n) fails delta_r = omega(sqrt(r n rho_n) log n) because delta_r = n rho_n (mu_r - mu_{r+1}) decays faster than sqrt(r) n rho_n; no valid sequence exists and Theorem 4 is vacuous. Remark 7 acknowledges that spectral gaps can be arbitrarily small but does not connect this to the rank-adaptive claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies spectral embeddings of latent position random graphs whose link function may have infinite rank and may be indefinite. Theorems 1–3 give two-to-infinity norm expansions of the form bU|bΛ|^{1/2}W(n) − U|Λ|^{1/2} = EU|Λ|^{-1/2} + Q, with explicit high-probability bounds on the main term and residual, for positive semidefinite and indefinite kernels; Theorem 5 is a deterministic perturbation counterpart. Corollary 2 provides row-wise normal approximations, Corollary 3 gives entrywise bounds for edge-probability estimation, and Theorem 4 with Lemma 1 and Corollary 4 propose a rank-adaptive test for equality of latent positions, claimed to converge to a weighted chi-square distribution, and to N(0,2) for infinite-rank kernels.","tokens_in":66555,"tokens_out":6214,"duration_ms":60405,"significance":"If the main theorems are correct, the paper is a substantial advance: it extends refined eigenvector fluctuation results from low-rank or positive semidefinite models to infinite-rank and indefinite kernels, supplies explicit constants in the main order term, and avoids population-level coherence assumptions. The proof machinery—leave-one-out analysis, matrix Bernstein bounds, and the deterministic perturbation framework of Theorem 5—is detailed and appears credible for Theorems 1–3, Corollaries 1–3, and the conditional statement in Theorem 4. No circularity or parameter fitting is present. However, the headline rank-adaptive inference claim in Corollary 4 is not proved, and as stated it is not compatible with the paper’s own assumptions for general infinite-rank indefinite kernels.","major_comments":[{"comment":"The assertion that the data-driven rank b_r defined in Eq. (4.12) satisfies the hypotheses of Theorem 4 and Lemma 1 is unproved, and the thresholds in (4.12) do not match condition (4.11). Lemma 1 requires δ_r = ω(max{log^{3/2} n, sqrt(r n ρ_n) log n, (n ρ_n)^{3/4}/(r^{1/4}+log^{1/4} n)}), while the selector b_r only enforces sample eigengaps at least max{log^{7/4} n, sqrt(j dave(A)) log^{3/4} n, (dave(A))^{3/4}}. By Weyl's inequality this yields at best a population gap of order sqrt(j n ρ_n) log^{3/4} n, lacking the log n factor required by (4.11). Consequently b_r can overshoot the range in which the residual ε_n in (4.13) is controlled, and no argument is given that ε_n → 0 for the random, data-dependent b_r.","section":"§4.2, Corollary 4"},{"comment":"The claim that for every infinite-rank kernel 'r(n) → ∞ in probability' and that (4.15) holds is not established and is false without an explicit lower bound on eigenvalue gaps. A diverging sequence r(n) must satisfy δ_r = ω(sqrt(r n ρ_n) log n), equivalently μ_r − μ_{r+1} = ω(sqrt(r) log n). For kernels with exponentially decaying eigenvalues, as considered in Remark 6, every diverging r(n) fails this condition because μ_r − μ_{r+1} decays exponentially while sqrt(r) log n grows polynomially. Remark 7 acknowledges that gaps can be arbitrarily small but does not connect that observation to the rank-adaptive N(0,2) claim.","section":"§4.2, Theorem 4 and Corollary 4"},{"comment":"For indefinite kernels the paper explicitly notes, after Eq. (3.21), that no high-probability spectral concentration inequality comparable to Eq. (2.5) is established; only O_p(n^{-1/2}) rates under conditions (3.22) or (3.23) are cited, and those conditions are acknowledged to be difficult to verify. Since Corollary 4 applies to general, possibly indefinite kernels and selects b_r from the eigenvalues of A, its proof would require a non-asymptotic lower bound on |λ_j| − |λ_{j+1}| in terms of the population spectrum. No such bound is provided, so the generality of the rank-adaptive test is not supported.","section":"§3.1 and §4.2"}],"minor_comments":[{"comment":"In item 2, 'probablity' should read 'probability'; in addition, since r(n) is a deterministic sequence in Theorem 4, the phrase 'r(n) → ∞ in probability' should be rephrased, for example as 'one may choose a diverging sequence r(n)'.","section":"§4.2, Corollary 4"},{"comment":"The displayed bound in Eq. (4.1) has an unmatched parenthesis, and the text 'we can reformulated these conditions' should read 'we can reformulate these conditions'.","section":"§4.1, Corollary 3"},{"comment":"The definition of b_r in Eq. (4.12) is an arg max over the set of j satisfying a threshold; if no such j exists the set is empty, and the convention for b_r should be stated explicitly.","section":"§4.2, Corollary 4"}],"recommendation":"major_revision","confidential_remarks":"The core perturbation results (Theorems 1–3, Corollaries 1–3, and Theorem 4 as a conditional statement) appear technically substantial and likely publishable. My concern is concentrated on Corollary 4, which is a highlighted application: the rank-adaptive selector is unproved and the claimed N(0,2) limit for all infinite-rank kernels is not compatible with the paper's own eigenvalue-gap discussion in Remarks 6–7 and Section 3.1. This is fixable by either proving the selector under explicit decay/gap conditions, restricting Corollary 4 to kernels satisfying such conditions, or downgrading the rank-adaptive claim to a conjecture. I would not reject on the basis of the perturbation theorems alone."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious paper. Theorems 1–3 with their main-term/residual decomposition for two-to-infinity norm bounds are a genuine advance over the low-rank, first-order bounds in the existing literature, and the extension to indefinite and infinite-rank kernels is the right level of generality. The proofs are detailed, with explicit constants and careful leave-one-out arguments; the comparison to [1, 39] in Section 3.3 is convincing. The entrywise graphon estimation bounds in Section 4.1 and the row-wise CLT in Corollary 2 are solid. My verdict is conditional, but the problem in Corollary 4 is more serious than a missing proof.\n\nCorollary 4 claims a data-driven rank selector b_r that is adaptive and yields asymptotic N(0,2) for every infinite-rank kernel. The step from Theorem 4 and Lemma 1 to Corollary 4 is asserted, not proved. The selector (4.12) uses thresholds of order log^{7/4} n, sqrt(j n rho_n) log^{3/4} n, and (n rho_n)^{3/4}. Lemma 1 requires, for growing r, essentially delta_r = omega(sqrt(r n rho_n) log n). By Weyl's inequality, the selected gaps can guarantee only delta_j of order sqrt(j n rho_n) log^{3/4} n at best, weaker than required by a log^{1/4} factor. Moreover, the claim that b_r -> infinity in probability for every infinite-rank kernel is not true as stated. If the kernel eigenvalues decay exponentially, e.g. mu_r ~ exp(-c r^beta), then delta_r = n rho_n (mu_r - mu_{r+1}) decays faster than sqrt(r) n rho_n, so no diverging r(n) satisfies delta_r = omega(sqrt(r n rho_n) log n); Theorem 4 is vacuous. Remark 7 notes that spectral gaps can be arbitrarily small but does not connect this to the rank-adaptive claim. This is a serious gap in the paper's advertised inference contribution.\n\nWhat holds up: the central eigenvector expansion theorems are proved in detail and the machinery appears sound. The numerics are plausible but no code is shipped. The paper fits no parameters, and the self-citations are to earlier technical tools, which is appropriate.\n\nWho this is for: researchers working on spectral inference for networks, especially graphon and infinite-rank latent position models. They will want Theorems 1–3 and Corollary 2. The rank-adaptive testing claim in Corollary 4 needs repair or a more careful statement with explicit eigenvalue decay assumptions. This paper deserves a serious referee; it should not be desk-rejected. I would recommend sending it to a statistical journal with a request to fix Corollary 4.","headline":"Strong two-to-infinity perturbation theory for infinite-rank graphons, but the rank-adaptive test's N(0,2) claim rests on an unproved selector and fails as stated for exponentially decaying kernels.","tokens_in":67099,"tokens_out":3180,"would_cite":true,"duration_ms":29882,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","60B20","62H12","62H15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For latent position graphs with infinite-rank, indefinite link functions, the leading sample eigenvectors are shown to decompose into a Gaussian noise term plus a controlled residual, yielding entrywise graphon bounds and a rank-adaptive…","keywords":["latent position graphs","infinite-rank kernels","indefinite kernels","two-to-infinity norm","spectral embedding","graphon estimation","hypothesis testing","random graphs"],"falsifier":"A decisive check is to simulate a latent position graph with an indefinite infinite-rank kernel such as $\\kappa(x,y)=\\cos(2\\pi(x-y))$ on $[0,1]^2$, set $X_n=X_1$, choose $r$ by the data-driven rule, and compare the empirical null distribution of $T(\\widehat X_1,\\widehat X_n)$ over many replicates with $N(0,2)$ and with the weighted chi-square approximation using plug-in weights; systematic departures beyond Monte Carlo error would falsify Corollary 4.","tokens_in":2087,"feed_emoji":"📊","tokens_out":2604,"duration_ms":95954,"temperature":0.7,"pith_summary":"This paper extends spectral perturbation theory for random graphs beyond low-rank, positive-semidefinite models to the general latent position model, where the link function may have infinitely many nonzero eigenvalues and may take both signs. The main theorem packages the alignment residual as $\\widehat U |\\widehat\\Lambda|^{1/2} W - U |\\Lambda|^{1/2} = E U |\\Lambda|^{-1/2} + Q$, with high-probability bounds in the maximum row norm, so the leading eigenvectors are not merely close to the population ones but have a Gaussian-dominated first-order behavior. On top of this expansion the paper derives entrywise estimation bounds for the edge probability matrix and a test for equality of latent positions whose null distribution is a weighted sum of independent chi-square variables; when the kernel has infinite rank the test statistic is asymptotically standard normal after centering. A sympathetic reader would say the upshot is that spectral embeddings remain statistically usable with growing dimension even when the kernel is indefinite and full rank.","feed_headline":"Rank-adaptive graph test turns chi-square for infinite-rank kernels","feed_subtitle":"New two-to-infinity eigenvector bounds make the test work for indefinite kernels and growing dimension.","key_machinery":"The central object is the expansion identity $\\widehat U |\\widehat\\Lambda|^{1/2} W^{(n)} - U |\\Lambda|^{1/2} = E U |\\Lambda|^{-1/2} + Q$ measured in the two-to-infinity norm $\\|M\\|_{2\\to\\infty} = \\max_i \\|M_{i\\cdot}\\|$, i.e., the maximum Euclidean row length. The identity separates a linear noise term, whose rows are sums of independent mean-zero Bernoulli deviations, from a residual $Q$ controlled by the eigenvalue gap $\\delta_r$ through sin-$\\theta$ subspace alignment and matrix concentration inequalities. A leave-one-out analysis on the adjacency matrix, where one row and column are replaced by the population values, is what controls the delicate term $E(I-UU^\\top)\\widehat U$. The same decomposition is repeated for unweighted and eigenvalue-weighted embeddings, and a deterministic perturbation theorem abstracts the expansion so that only quantities linear in the noise matrix $E$ need to be bounded for a given application.","core_discovery":"The central claim is that for independent-edge latent position graphs with kernels that may be indefinite and of infinite rank, the leading scaled sample eigenvectors can be aligned to the population eigenvectors by an orthogonal matrix $W$, after which the difference obeys $\\widehat U |\\widehat\\Lambda|^{1/2} W - U |\\Lambda|^{1/2} = E U |\\Lambda|^{-1/2} + Q$ with explicit high-probability bounds on both the main term and the residual. The same program is carried out for $\\widehat U W - U$ and $\\widehat U \\widehat\\Lambda W - U \\Lambda$. The results allow for repeated population eigenvalues, for the embedding dimension $r$ to grow with $n$, and for positive semidefinite as well as indefinite link functions; the indefinite case pays a heavier factor involving $|\\lambda_r|^{-1/2}$ because $|P|$ has no entrywise closed form. These fine-grained bounds are then used to obtain entrywise high-probability errors for estimating the edge probability matrix and a plug-in, rank-adaptive test statistic whose null limit is a weighted sum of independent chi-square variables, reducing to $N(0,2)$ when the kernel has infinite rank.","pith_inferences":["If the expansion is as sharp as claimed, the same main-term-plus-residual decomposition should carry over to general signal-plus-noise matrix models beyond Bernoulli graphs, because the deterministic perturbation result is linear in the noise matrix and does not use the graph structure itself.","The $N(0,2)$ limit for infinite-rank kernels is attractive but rests on spectral concentration that the paper does not fully establish for indefinite kernels; a safer practical route may be to use the weighted chi-square approximation with plug-in weights even when the data-driven rank does not grow.","Since repeated eigenvalues are explicitly allowed, the framework is plausibly applicable to kernels with symmetry, such as rotationally invariant kernels on spheres, where standard eigengap assumptions are known to fail; the paper's simulations with the Laplace kernel show near-zero gaps yet a small data-driven rank.","The numerical results suggest that, for smooth kernels, the effective dimension needed for hypothesis testing is far smaller than $n$, so the practical payoff of the infinite-rank theory may be to justify a low-dimensional approximation rather than to use a large number of spectral coordinates."],"forward_implications":["The expansion makes the leading embedding row-wise close to a linearized noise term, so entrywise statements about rows of spectral embeddings become feasible rather than only subspace-level or Frobenius-level statements.","For positive semidefinite kernels, the embedding dimension $r$ may grow with $n$ under explicit eigengap conditions, and entrywise estimation of the edge probability matrix achieves a rate involving $\\lambda_r^{-1/2}(r^{1/2}+\\log^{1/2} n)$ plus a bias term from truncation.","The test statistic $T(\\widehat X_i, \\widehat X_j)$ is computable from the adjacency matrix alone, adapts to the unknown kernel rank, and under the null hypothesis converges to a weighted sum of independent chi-square variables; for infinite-rank link functions it is asymptotically $N(0,2)$.","A row-wise central limit theorem for the scaled eigenvectors holds even for indefinite kernels, with covariance determined by the Bernoulli noise, as long as the eigenvalue gap satisfies the stated growth condition.","The data-driven rank selector based on the sample eigengap converges to the true finite rank when the kernel has finite rank and diverges when the kernel has infinite rank, removing the need for a user-specified embedding dimension in the testing problem."],"supporting_citations":[{"why":"It supplies the entrywise eigenvector perturbation framework whose residual is refined here into a main term plus a controlled residual.","marker":"[1]"},{"why":"It provides the two-to-infinity norm calculus and the Procrustes alignment lemmas used throughout the paper.","marker":"[15]"},{"why":"It supplies the sharp high-probability eigenvalue concentration for positive semidefinite kernels that allows the embedding dimension to grow with $n$.","marker":"[47]"},{"why":"It provides the preceding unified two-to-infinity eigenspace perturbation bound that this paper improves and extends.","marker":"[39]"},{"why":"It defines the infinite-rank indefinite kernel setting with a sup-norm assumption on the kernel eigenfunctions that this paper seeks to relax.","marker":"[38]"},{"why":"It supplies the low-rank test statistic for equality of latent positions whose distribution theory is extended to growing embedding dimension.","marker":"[25]"},{"why":"It gives another low-rank membership or profile test framework that the proposed test is compared against.","marker":"[27]"},{"why":"It provides the universal singular value thresholding Frobenius bounds for probability matrix estimation that Corollary 3 strengthens to entrywise bounds.","marker":"[61]"}],"fun_headline_variants":["Eigenvector bounds for infinite-rank indefinite kernels","Rank-adaptive test hits chi-square for growing dimension","Two-to-infinity norm bounds for graph eigenvectors","Infinite-rank kernels: new limits for graph eigenvectors","Graph eigenvector limits enable chi-square test"],"cache_read_input_tokens":69120,"weakest_assumption_plain":"The load-bearing premise is that, for indefinite infinite-rank kernels, high-probability eigenvalue concentration sharp enough to let the embedding dimension $r$ grow with $n$ actually holds; the paper notes in Section 3.1 that such a bound is unavailable and only $O_p(n^{-1/2})$ rates are known under conditions that are difficult to verify.","fun_headline_variants_meta":{"raw":{"variants":["Eigenvector bounds for infinite-rank indefinite kernels","Rank-adaptive test hits chi-square for growing dimension","Two-to-infinity norm bounds for graph eigenvectors","Infinite-rank kernels: new limits for graph eigenvectors","Graph eigenvector limits enable chi-square test"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000383,"raw_usage":{"total_tokens":2030,"prompt_tokens":950,"completion_tokens":1080,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":1008}},"tokens_in":566,"tokens_out":1080,"duration_ms":10363,"temperature":1.0,"reasoning_tokens":1008,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:00:55.083702+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check is to simulate a latent position graph with an indefinite infinite-rank kernel such as $\\kappa(x,y)=\\cos(2\\pi(x-y))$ on $[0,1]^2$, set $X_n=X_1$, choose $r$ by the data-driven rule, and compare the empirical null distribution of $T(\\widehat X_1,\\widehat X_n)$ over many replicates with $N(0,2)$ and with the weighted chi-square approximation using plug-in weights; systematic departures beyond Monte Carlo error would falsify Corollary 4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the entrywise eigenvector perturbation framework whose residual is refined here into a main term plus a controlled residual."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the two-to-infinity norm calculus and the Procrustes alignment lemmas used throughout the paper."},{"cited_title":"Rosasco, M","cited_arxiv_id":null,"evidence_quote":"It supplies the sharp high-probability eigenvalue concentration for positive semidefinite kernels that allows the embedding dimension to grow with $n$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It defines the infinite-rank indefinite kernel setting with a sup-norm assumption on the kernel eigenfunctions that this paper seeks to relax."},{"cited_title":"Du and M","cited_arxiv_id":null,"evidence_quote":"It supplies the low-rank test statistic for equality of latent positions whose distribution theory is extended to growing embedding dimension."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives another low-rank membership or profile test framework that the proposed test is compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the universal singular value thresholding Frobenius bounds for probability matrix estimation that Corollary 3 strengthens to entrywise bounds."}],"review_version":1}