{"id":"cba952fb-49d2-48f1-8cf8-92b41bb93574","arxiv_id":"2509.00562","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"SANVI uses an extended surrogate likelihood and Gaussian variational inference to estimate latent positions in GRDPGs asymptotically efficiently and much faster than MCMC.","lead":"This paper introduces SANVI, a fast variational inference method for estimating hidden node positions in generalized random dot product graphs, a broad network model family. It replaces slow Monte Carlo sampling with stochastic gradient optimization that keeps comparable accuracy while cutting computation time, with proofs of asymptotic efficiency.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.5's efficiency claim rests on Theorem A.7 (one-step estimator), which is imported from Xie (2024) without proof for the sparse GRDPG/indefinite-signature setting; if that remainder bound fails, the asymptotic normality of MESLE and the VI posterior mean collapse.","rationale":"The reader's verdict is CONDITIONAL, with the main weaknesses being the unverified imported one-step estimator result, the possible Assumption 1(e)/simulation mismatch, and the absence of code. My stress-test pass confirms that the most load-bearing point is the imported Theorem A.7: it is explicitly unproved ('We omit the proof and use the theorem directly'), it is a generalization to exactly the regime the paper needs (sparse GRDPG with indefinite signature), and Theorem 3.2's asymptotic normality proof uses it as a black box. The reader's weakest_assumption field focused on Theorem A.1, but Theorem A.7 is more directly in the chain from spectral embedding to MESLE efficiency; Theorem A.1 is actually proved in Appendix A, whereas Theorem A.7 is not. This does not by itself prove the paper wrong: the theorem may well be a correct generalization of existing work, in which case the gap is presentational/verificational. But the central claim cannot be certified from the manuscript alone. The numerical experiments are extensive and show SANVI matching MCMC and beating ASE, which supports the finite-sample utility but cannot settle the asymptotic normality claim. Therefore I do not recommend changing the verdict; it should remain conditional on verification of Theorem A.7. My agreement with the reader is 'partial' because the reader named Theorem A.1 in the weakest_assumption field but also flagged the one-step estimator result in the rationale, so we identify the same family of concerns but weight the one-step estimator as the more load-bearing element.","tokens_in":50104,"tokens_out":15230,"duration_ms":180204,"concrete_test":"Independently derive Theorem A.7 for the GRDPG/indefinite-signature case from Theorem A.1 and the Xie (2024) proof, tracking the signature matrix I_{p,q} and the block-sign alignment W in every step. Specifically, verify that the first-order expansion term has coefficient G^{-1/2}_{0in} \\rho_n^{1/2} I_{p,q} x_{0j} / [p_{0ij}(1-p_{0ij})] and that the remainder satisfies \\|r_{in}^{(OS)}\\| \\le C t^2/(n\\rho_n^{1/2}) under only Assumption 1, especially (log n)/(n\\rho_n) \\to 0. If the derivation requires an extra factor \\rho_n^{-1/2} in the remainder, or an additional condition such as (log n)^4/(n\\rho_n) \\to 0 for the single-node CLT, then Theorems 3.2 and 3.5 must be restated with that condition. A secondary computational check: simulate a sparse GRDPG with q>0 and n\\rho_n = (log n)(\\log\\log n)^2, and compare the empirical distribution of \\sqrt{n} G^{1/2}_{0in}(W^T \\hat x_i - \\rho_n^{1/2} x_","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central efficiency claim (Theorem 3.5) is a chain: Theorem 3.2 gives asymptotic normality of the MESLE; Theorem 3.3 gives BvM for the ESL posterior; Theorem 3.4 transfers it to the VI posterior; Theorem 3.5 transfers it to the VI mean. The link that carries the asymptotic-normality burden is Theorem 3.2, whose proof (Appendix B.1) invokes Theorem A.7, the one-step estimator expansion, as the source of the first-order representation of \\hat x_i. Theorem A.7 is stated for the sparse GRDPG with indefinite signature I_{p,q}, but Remark 3 says it is 'a generalization ... with slight modifications' of Theorem 4.7 in Xie (2024) and 'We omit the proof and use the theorem directly.' This omitted proof is load-bearing: the expansion\n\nG^{1/2}_{0in}(W^T \\hat x_i^{(OS)} - \\rho_n^{1/2} x_{0i}) = (1/n) \\sum_j (A_{ij}-p_{0ij}) G^{-1/2}_{0in} \\rho_n^{1/2} I_{p,q} x_{0j} / [p_{0ij}(1-p_{0ij})] + r_{in}\n\nand the remainder bound \\|r_{in}\\| \\le C t^2/(n\\rho_n^{1/2}) are exactly what make the MESLE asymptotically equivalent to the oracle MLE. If the indefinite-signature/sparse generalization introduces extra factors in the alignment W or requires a stronger sparsity condition than Assumption 1(d), then the CLT in Theorem 3.2 does not follow, and the BvM/efficiency results in Theorems 3.3-3.5 collapse. The paper's own proof of Theorem A.1 is a partial substitute, but Theorem A.7 is not derived from it here.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes SANVI, a Gaussian variational inference method for generalized random dot product graphs (GRDPGs). The method replaces unknown latent positions and the signature by the signature-adjusted adjacency spectral embedding and uses an extended surrogate likelihood (ESL) that is separable in vertices and defined on the whole Euclidean space. The main theoretical results are: strong convexity of the variational objective (Theorem 3.1), consistency, existence/uniqueness, asymptotic normality and efficiency of the maximum ESL estimator (Theorem 3.2), a Bernstein-von Mises result for the ESL posterior (Theorem 3.3), an analogous BvM result for the Gaussian variational posterior (Theorem 3.4), and asymptotic efficiency of the variational posterior mean (Theorem 3.5). Numerical experiments compare SANVI with ASE, OSE, and MCMC-based Bayes estimation on simulated GRDPGs and a political-blog network, reporting that SANVI is competitive in accuracy and substantially faster than MCMC.","tokens_in":50635,"tokens_out":17448,"duration_ms":196282,"significance":"If the main theorems hold, the paper provides a useful and nontrivial contribution: a computationally scalable likelihood-informed Bayesian procedure for GRDPG latent positions with oracle-type asymptotic efficiency and a BvM guarantee. The ESL construction is clean, the algorithm is parallelizable, and the empirical comparisons support the computational advantage over MCMC. The paper also contains detailed proofs of several new auxiliary lemmas (e.g., Lemmas A.2--A.6, A.9) and an explicit proof of the spectral-embedding approximation Theorem A.1. However, the central efficiency and BvM claims depend heavily on the one-step estimator expansion in Theorem A.7, which is imported from prior work without proof for the sparse indefinite-signature setting. This makes the asymptotic guarantees conditional on an unverified generalization, and the BvM results are proved only for the dense case ρ_n = 1. These issues are load-bearing for the advertised claims.","major_comments":[{"comment":"Theorem A.7 is the key first-order expansion used in the proof of Theorem 3.2 (Appendix B.1) to obtain the CLT for the MESLE, and through Theorems 3.3--3.5 it underpins the BvM and efficiency results. Remark 3 states that the theorem is a generalization of Theorem 4.7 of Xie (2024) to sparse GRDPGs with indefinite signature, but 'we omit the proof and use the theorem directly.' The paper does not verify the nontrivial aspects of this generalization: the indefinite signature I_{p,q}, the sparse scaling ρ_n, and the corresponding alignment matrix W. Since the remainder bound in Theorem A.7 is exactly what makes the MESLE asymptotically equivalent to the oracle MLE, this is a load-bearing gap. The manuscript should either provide a full proof of Theorem A.7 in the stated generality, or give a precise verification that the hypotheses of the theorem in Xie (2024) are satisfied in this setting","section":"Appendix A, Theorem A.7 and Remark 3"},{"comment":"The BvM theorem for the ESL posterior and all VI posterior results are stated only under ρ_n = 1, although the paper motivates the ESL construction with sparse GRDPGs and Theorem 3.2 allows ρ_n → 0 under Assumption 1(d). The abstract and conclusion claim asymptotic efficiency and BvM for the variational posterior without this qualification. As written, the sparse regime has no BvM or VI-efficiency guarantee. The authors should either extend the proofs to ρ_n → 0 or clearly and prominently restrict the efficiency claims to the dense case. This is not merely a presentation issue, because the finite-sample advantages of SANVI are most relevant in sparse networks.","section":"Theorems 3.3--3.5 and Section 5"},{"comment":"The proof concludes that √n(x*_i − bxi) = o_P(1) from the bound |exp(iu^T√n x*_i) − exp(iu^T√n bxi)| ≲ (log n)^{−1/2} for all ||u||_2 = 1. This does not follow as stated. In dimension d = 1 the sup over unit vectors is only over u = ±1, and a vector of the form 2πm is not detected; in higher dimensions the argument also needs a uniform bound over a continuum of directions with controlled norms, not just the unit sphere, because the Gaussian factor exp(−||u||^2/2 λ_min) degenerates for large u. The step can likely be repaired by using the supremum over all t in a ball or over u with ||u|| ≤ c, together with the growth of the Gaussian characteristic functions, but the proof as written is incomplete.","section":"Appendix B.4, proof of Theorem 3.5"}],"minor_comments":[{"comment":"The displayed full log-likelihood ℓ_A(X) has log(x_i^T I_{p,q} x_j) in both Bernoulli terms; the second term should be log(1 − x_i^T I_{p,q} x_j).","section":"Equation (2.1)"},{"comment":"The sampled z variables are indexed z_1,...,z_s, but the sums use z_s^{(t−1)} in each term; they should use z_k^{(t−1)}.","section":"Algorithm 1, line 13"},{"comment":"The text refers to 'Table 4.3' when it appears to mean Table 5; please correct the cross-reference.","section":"Section 4.3"},{"comment":"The proof of Theorem 3.3 bounds the L1 difference of unnormalized densities, but the passage to normalized densities requires an explicit statement that the normalizing constants are comparable via Z_h ≍ n^{-d/2}. Lemma B.1 says 'Theorem 3.3 implies' the ratio of normalizing constants is close; that implication actually uses the stronger unnormalized bound from the proof of Theorem 3.3. Please make this explicit.","section":"Appendix B.2--B.3"},{"comment":"The simulations set τ_n = min(0.001, e^{1.5}/n), but the paper does not discuss whether this choice satisfies Assumption 1(e) for the implied values of ρ_n and δ in each example. A brief verification or comment would be helpful.","section":"Section 4 and Assumption 1(e)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The main gap is the unproved Theorem A.7, which is imported from the authors' own prior work. This is not necessarily disqualifying if the cited theorem indeed covers the sparse indefinite-signature setting, but the manuscript should state and verify the required hypotheses in detail. The dense-only restriction of Theorems 3.3--3.5 should also be made explicit in the abstract and conclusions. The paper is within the scope of the journal and the algorithmic contribution is potentially valuable, but the asymptotic claims need to be placed on a verified footing before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a useful extension of the Wu–Xie surrogate-likelihood idea from RDPG to GRDPG with an indefinite signature, and it replaces MCMC with Gaussian VI. The paper does real work: it defines an extended surrogate likelihood on the whole Euclidean space, proves existence/uniqueness and asymptotic normality of the MESLE, and establishes Bernstein–von Mises results that transfer to the variational posterior and its mean. The numerics are honest: SANVI matches the MCMC-based Bayes estimate in error and runs much faster, while OSE is unstable in several of the settings tried. That matches the claims.\n\nThe main soft spot, and I agree with the stress-test note here, is Theorem A.7. That one-step estimator expansion is imported from Xie (2024), and the paper explicitly says it is used directly without proof (Remark 3). It is the load-bearing link in Theorem 3.2's proof of asymptotic normality, and everything downstream—Theorems 3.3 through 3.5—inherits it. The generalization to sparse GRDPG with indefinite signature may well be routine, but the remainder bound and the alignment matrix need checking. This should be fixed either by proving the theorem or by stating it as an assumption. As it stands, the chain of proofs has one unverified link right at the critical point.\n\nA second, more minor issue: the simulation protocol uses tau_n = min(0.001, e^{1.5}/n), which does not satisfy Assumption 1(e) when rho_n = 1 and n grows. That assumption requires tau_n bounded between delta^2 and delta/2, whereas the simulation sends tau_n to zero. This mismatch is not fatal, but it deserves an acknowledgment.\n\nMinor: no code is shipped. The algorithm is described in enough detail to reproduce, but code would help.\n\nOverall, the paper is in good shape. The contribution is new relative to the MCMC-based Bayes method, the theory is detailed where it is actually proved, and the numerical evidence supports the main claims. I'd send this to a serious referee. I'd cite it if I worked on network variational inference.","headline":"Solid extension of surrogate-likelihood VI to GRDPG; main caveat is an unproved imported one-step theorem that carries the efficiency claims.","tokens_in":51067,"tokens_out":2546,"would_cite":true,"duration_ms":33644,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F12","62F15","05C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"SANVI's variational posterior mean is asymptotically normal and efficient for generalized random dot product graphs, matching the oracle maximum likelihood estimator.","keywords":["generalized random dot product graphs","extended surrogate likelihood","variational Bayes","stochastic gradient descent","network variational inference","Bernstein-von Mises","adjacency spectral embedding","asymptotic efficiency"],"falsifier":"Simulate a GRDPG with a deliberately small spectral gap in the Gram matrix (two nearly identical latent positions) and run SANVI against the oracle MLE that uses the true other-vertex positions; Theorem 3.5 predicts the aligned errors differ by o_P(n^{−1/2}). If the entrywise embedding convergence used in Theorem A.1 degrades, the difference will grow with the inverse gap, and nominal 95% credible intervals from the variational covariance will undercover. A second check: set ρ_n = C log n/n with C large enough that (log n)/(nρ_n)→0 but (log n)^4/(nρ_n)→∞, and test whether the global error cons","tokens_in":50039,"feed_emoji":"🕸️","tokens_out":9403,"duration_ms":102139,"temperature":0.7,"pith_summary":"Bayesian inference for generalized random dot product graphs—network models where each vertex carries a low-dimensional latent position and edge probabilities are generalized inner products—is usually done with MCMC and is slow. This paper proposes SANVI, which replaces the exact likelihood with an extended surrogate likelihood: unknown latent positions are swapped for signature-adjusted adjacency spectral embeddings, and the logarithms are smoothed and extended from the constrained interval to all of R^d. The result is a separable, log-concave objective, so each vertex's posterior can be approximated by a Gaussian via stochastic gradient descent. The paper proves the variational posterior mean is asymptotically as accurate as the oracle maximum likelihood estimator—sqrt(n)-consistent, normal, and efficient—and that the variational posterior approaches the oracle-centered normal posterior in total variation. If correct, this gives network analysts a fast, parallelizable, optimization-based route to near-MCMC-quality posterior estimates.","feed_headline":"Variational inference on graphs matches oracle MLE accuracy","feed_subtitle":"A spectral surrogate plus Gaussian VI gives MCMC-quality estimates in about a fifth of the time.","key_machinery":"The extended surrogate likelihood (ESL): for each vertex i, b_l_in(x_i) = Σ_j [A_ij ψ_n(x_i^T x_tilde_j) + (1−A_ij)ψ_n(1−x_i^T x_tilde_j)], where x_tilde_j is the signature-adjusted adjacency spectral embedding of vertex j and ψ_n(t) equals log(t) on [τ_n, 1] and continues as a bounded quadratic outside. It decouples the generalized inner product between latent positions, relaxes the domain from the constraint set to all of R^d, and keeps the objective separable, smooth, and strongly concave. The Fisher information G_{0in} of the oracle likelihood appears as the asymptotic covariance; the signature-adjusted spectral embedding supplies the plug-in that converts the oracle likelihood into a co","core_discovery":"The paper's central claim is that the extended surrogate likelihood makes Gaussian variational inference for generalized random dot product graphs simultaneously fast and asymptotically efficient. The maximum extended surrogate likelihood estimator is consistent at rate sqrt(log n/n), uniquely defined with high probability, and satisfies sqrt(n)G_{0in}^{1/2}(W^T x_hat_i − rho_n^{1/2} x_{0i}) → N(0_d, I_d). The variational posterior mean inherits that limit, so the VI estimator matches the oracle likelihood's precision. In total variation, both the full posterior and the Gaussian variational posterior converge to the same normal distribution centered at the estimator, the variational one at r","pith_inferences":["Beyond the paper: the same ESL-plus-Gaussian-VI template could be used for edge prediction or vertex classification, where the variational covariance would yield predictive intervals; a natural test is whether such intervals achieve nominal coverage at the rates predicted by Theorems 3.4–3.5.","Beyond the paper: because the BvM rate for the variational posterior is O(1/sqrt(log n)) while the MCMC posterior is O(1/log n), the variational posterior's covariance—not its mean—is likely the quantity that degrades first at moderate n; experiments reporting coverage rather than point error would reveal this.","Beyond the paper: Theorem 3.3 is proven for ρ_n=1, and the paper states that relaxing to ρ_n→0 is possible but does not carry it out; a concrete extension is to verify the variational BvM bound under the same sparsity condition used in Theorem 3.2.","Beyond the paper: the truncation threshold τ_n must sit between δ^2ρ_n and (δ/2)ρ_n, so finite-sample performance may be sensitive to τ_n; sweeping τ_n in the simulation setups would give practitioners a practical calibration rule."],"forward_implications":["The variational posterior mean from SANVI is asymptotically efficient: after aligning the orthogonal ambiguity, sqrt(n)G_{0in}^{1/2}(W^T x^*_i − rho_n^{1/2} x_{0i}) converges to a standard normal, so the VI estimator loses no accuracy relative to the oracle MLE.","The variational posterior itself has a Bernstein-von Mises limit: it converges in total variation to the same normal posterior the oracle likelihood would produce, at rate O(1/sqrt(log n)), so the variational covariance can be used for approximate uncertainty quantification.","Because the ESL is separable across vertices, SANVI's computation decomposes into n O(n) subproblems, giving O(n^2) total time; the numerical comparison places it at roughly 20% of MCMC runtime while matching its squared error.","Under the stronger sparsity condition (log n)^4/(nρ_n)→0, the global squared error of the aligned ESL estimator is consistent for (1/n)Σ_i tr(G_{0in}^{-1}), the oracle Cramér-Rao-type lower bound.","The method applies to generalized random dot product graphs, including indefinite-signature cases such as rank-three GRDPG, and to sparse graphs with ρ_n→0 as long as nρ_n/log n→∞."],"supporting_citations":[{"why":"Establishes the asymptotic normality of the oracle log-likelihood estimator and the one-step estimator, defining the efficiency baseline and the OSE competitor.","marker":"Xie and Xu (2023)"},{"why":"Introduces the surrogate likelihood for random dot product graphs whose extension this paper builds, supplying the Bayesian method and framework being generalized.","marker":"Wu and Xie (2025)"},{"why":"Supplies Theorem A.1, the entrywise two-to-infinity convergence of the signature-adjusted adjacency spectral embedding that underpins the surrogate approximation.","marker":"Xie (2024)"},{"why":"Gives the generalized random dot product graph spectral embedding interpretation and signature handling used to define the embedding.","marker":"Rubin-Delanchy et al. (2022)"},{"why":"Provides the spectral-norm concentration of A−P used in Theorem A.1 and other high-probability bounds.","marker":"Lei and Rinaldo (2015)"},{"why":"Motivates replacing an intractable likelihood by a statistical criterion function in the Bayes formula.","marker":"Chernozhukov and Hong (2003)"},{"why":"Supplies the reparameterized Gaussian variational inference objective and automatic differentiation framework adapted here.","marker":"Kucukelbir et al. (2017)"},{"why":"Provides the Cholesky reparameterization and the modified gradient trick used in the SANVI stochastic gradient descent algorithm.","marker":"Xu and Campbell (2023)"}],"fun_headline_variants":["Spectral surrogate accelerates Bayesian graph inference","Gaussian VI for graphs hits asymptotic efficiency","SANVI: fast Bayesian inference for random dot product graphs","MLE-accuracy graph inference at fraction of MCMC cost","Bayesian graph inference without MCMC: SANVI method"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The argument stands or falls on the signature-adjusted adjacency spectral embedding converging to the true latent positions uniformly across vertices (in two-to-infinity norm) at rate about sqrt(log n/n); this entrywise convergence is imported from earlier eigenvector limit theory rather than re-proved in full generality here, and without it the surrogate likelihood does not track the oracle likelihood.","fun_headline_variants_meta":{"raw":{"variants":["Spectral surrogate accelerates Bayesian graph inference","Gaussian VI for graphs hits asymptotic efficiency","SANVI: fast Bayesian inference for random dot product graphs","MLE-accuracy graph inference at fraction of MCMC cost","Bayesian graph inference without MCMC: SANVI method"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000305,"raw_usage":{"total_tokens":1569,"prompt_tokens":706,"completion_tokens":863,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":788}},"tokens_in":450,"tokens_out":863,"duration_ms":9930,"temperature":1.0,"reasoning_tokens":788,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T13:28:16.470919+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a GRDPG with a deliberately small spectral gap in the Gram matrix (two nearly identical latent positions) and run SANVI against the oracle MLE that uses the true other-vertex positions; Theorem 3.5 predicts the aligned errors differ by o_P(n^{−1/2}). If the entrywise embedding convergence used in Theorem A.1 degrades, the difference will grow with the inverse gap, and nominal 95% credible intervals from the variational covariance will undercover. A second check: set ρ_n = C log n/n with C large enough that (log n)/(nρ_n)→0 but (log n)^4/(nρ_n)→∞, and test whether the global error cons","supporting_citations":[{"cited_title":"and Wu, D","cited_arxiv_id":null,"evidence_quote":"Establishes the asymptotic normality of the oracle log-likelihood estimator and the one-step estimator, defining the efficiency baseline and the OSE competitor."},{"cited_title":"and Rinaldo, A","cited_arxiv_id":null,"evidence_quote":"Provides the spectral-norm concentration of A−P used in Theorem A.1 and other high-probability bounds."}],"review_version":1}