{"id":"6011b4f9-aa3a-4c2c-b2ef-c0ff35becb13","arxiv_id":"2507.22170","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In the proportional high-dimensional limit, the paper derives exact squared-overlap formulas and phase transitions for Stack-SVD and SVD-Stack, and proves optimally weighted Stack-SVD always beats optimally weighted SVD-Stack.","lead":"The paper derives exact high-dimensional performance formulas for two common ways of combining noisy datasets: Stack-SVD, which concatenates data before a singular value decomposition, and SVD-Stack, which computes a consensus after per-dataset decompositions. A smart generalist should read it because it answers which integration order wins, provides optimal weighting schemes, and gives phase transition criteria for signal detection.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Lemma 1's conditioning argument needs only deterministic-w delocalization, not uniformity; Theorem 3's algebra checks out.","rationale":"The paper's main contribution is the asymptotic comparison of Stack-SVD and SVD-Stack. The reader accepted it with moderate confidence, flagging the use of [33] in Lemma 1 as the weakest assumption. In my view, that concern is not load-bearing. Lemma 1 requires only that, for each fixed realization of X2, the conditional probability that vhat1 has a large projection on the deterministic vector w2(ω) vanishes; Proposition 1 supplies exactly this statement. The dominated convergence theorem then transfers the pointwise conditional convergence to unconditional convergence. There is no step that needs uniform control over random orthogonal directions. I also verified the core dominance argument numerically and symbolically: the inequality (x+y)/(y+z)≥x/z for z≥x gives the termwise bound leading to f(S/(S+1))≥1, and since f is decreasing, the weighted Stack-SVD root is at least S/(S+1). The optimal weights in Theorem 1 can be recovered from the same KKT equations as the paper's reparameterization, which supports the mapping to [29]. The acknowledged edge cases and finite-M assumption are stated limitations rather than hidden flaws. The rank-r section is less fully detailed, especially the orthogonality of estimated vectors across components, but this is secondary to the rank-one Theorem 3 and is plausibly handled by the same delocalization machinery. Overall, I find no internal inconsistency or unsupported step that would change the reader's accept verdict.","tokens_in":35816,"tokens_out":36085,"duration_ms":405306,"concrete_test":"Simulate M=2 independent matrices with centered exponential (or Rademacher) noise at d=10^3 and d=10^4, for several θ,c across the phase transition, and compare the empirical |<vhat1,vhat2>|^2 to β1^2β2^2; this directly checks the one external delocalization input on which the SVD-Stack formulas rest.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central rank-one claims withstand scrutiny. The reader's weakest-assumption worry about Lemma 1 does not land: the proof conditions on X2, making w2(ω) a deterministic unit vector orthogonal to v for almost every ω; Proposition 1's second statement (from [33, Thm. 1(2)]) then gives pointwise conditional convergence, and the bounded indicator allows dominated convergence. No uniformity over random orthogonal directions is required. I also re-checked the dominant-weighting proof in Appendix C.2: Eq. (S40) with z=S≥(θ_i^4-c_i)/(θ_i^2+c_i) yields Eq. (S41), so f(S/(S+1))≥1 and monotonicity of f gives γ*≥S/(S+1). The optimal Stack-SVD weights in Theorem 1 are consistent with the KKT conditions of the reparameterized objective (Eqs. S31-S35), and the edge cases flagged in the paper (β1>0,β2=0; M finite) are acknowledged limitations rather than internal inconsistencies. The rank-r extension is less detailed, but it is not load-bearing for Theorem 3.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies two pipelines for estimating a shared rank-one (and, by extension, finite-rank) singular subspace across multiple noisy high-dimensional matrices: Stack-SVD (concatenate then SVD) and SVD-Stack (per-matrix SVD then aggregate). Under a proportional scaling regime with independent noise entries having zero mean, unit variance after scaling, and bounded fourth moment, the authors derive the limit in probability of the squared inner product between the estimated and true singular vector for both methods, including phase transitions and closed-form asymptotic performance expressions. They further derive optimal weighting schemes for both methods, show that optimally weighted Stack-SVD dominates optimally weighted SVD-Stack (and unweighted Stack-SVD), provide algorithms for estimating the signal strengths from data, extend to rank r under ordering assumptions, and support the theory with extensive simulations, including non-Gaussian noise, and a semi-synthetic single-cell RNA-seq experiment.","tokens_in":36010,"tokens_out":13020,"duration_ms":141177,"significance":"If correct, this is the first systematic random matrix treatment of SVD-Stack and a rigorous instance-dependent comparison of the two methods in the proportional regime. The explicit asymptotic formulas, weighting rules, and detectability thresholds are directly usable by practitioners, and the public code plus the non-Gaussian and semi-synthetic validations strengthen the paper's practical claims. The main theoretical result—dominance of optimally weighted Stack-SVD—is novel and goes beyond the minimax analyses available in the literature.","major_comments":[{"comment":"The claimed strict improvement over unweighted Stack-SVD when the θ_i are not all equal is false. Take M=2, θ=(1,2), c=(1,4). Then θ_i^2/c_i = 1 for both tables, so the optimal weights of Theorem 1, w_i^* ∝ θ_i/√(θ_i^2 + c_i), are equal, and both unweighted and optimally weighted Stack-SVD have asymptotic performance 2/3, although the θ_i are not all equal. The correct condition for strict improvement is that the θ_i^2/c_i are not all equal (equivalently, that the optimal weights are not all proportional to the unweighted choice). The proof in Appendix C.2 establishes only the inequality f(S/(S+1)) ≥ 1 and does not support the strictness over unweighted Stack-SVD; the statement of Theorem 3 and the sketch in Section 5 should be amended accordingly.","section":"Section 5, Theorem 3"}],"minor_comments":[{"comment":"There are spacing typos such as 'SVD-Stackfalls' in the discussion of Figure 4a and 'V ariance' in Section 7; these should be corrected.","section":"Throughout"},{"comment":"The variance-stabilizing transformation is written as X = 2√Y/d; since Y is a count matrix, this is dimensionally ambiguous. Please clarify whether the intended transformation is X = 2√(Y/d) or X = 2√Y/√d, and adjust the stated variance 1/d accordingly.","section":"Section 8"},{"comment":"The detectability condition for weighted SVD-Stack is written as β1>0 ⇐⇒ max_i θ_i^4/c_i ≥ 1; because β_i=0 at equality, the correct threshold is strict inequality, consistent with Proposition 1.","section":"Table 1"},{"comment":"Proposition 2 leaves the case β1>0, β2=0 to Appendix A.3; consider stating this edge case explicitly in the main text to avoid an apparent gap between the 'β2>0' and 'β1=0' clauses.","section":"Section 3.1, Proposition 2"}],"recommendation":"major_revision","confidential_remarks":"The false strictness claim in Theorem 3 is a fixable local error; the core dominance result, proofs, and experiments are convincing. The reader's report and stress-test note both conclude that the main technical concerns are resolved, and I concur. The theorem statement and the proof sketch need to be corrected to reflect the actual condition for strict improvement over unweighted Stack-SVD. I support publication after this correction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is the first proportional-regime analysis of SVD-Stack, and it settles a practical question that has been floating around unsupervised multi-table learning. The main results are the exact asymptotic performance and phase transition for SVD-Stack, the optimal weighting for both methods, and the dominance theorem showing optimally weighted Stack-SVD beats weighted SVD-Stack. Also nice: the paper shows neither unweighted method dominates, constructs instances where only weighted Stack-SVD works, and gives a consistent estimator for signal strengths that are individually below the detection threshold. The simulations are extensive, including non-Gaussian noise and a semi-synthetic scRNA-seq application, and the code is public.\n\nThe proofs borrow heavily from recent RMT results: [33] for spiked eigenvectors, [29] for optimally weighted PCA, [17] for heteroscedastic noise. That is not a flaw, but the reader who wants to verify the centrality of Theorem 3 will need to go into those papers. The algebra in the paper checks out; I rechecked the dominant-weighting proof and the KKT conditions, and the stress-test note is right that the Lemma 1 delocalization concern does not land. Conditioning on the second table makes the relevant direction deterministic, so pointwise conditional convergence plus dominated convergence is enough; no uniformity over random orthogonal directions is needed.\n\nThe soft spots are modest and mostly acknowledged. Unweighted SVD-Stack has an edge case when exactly one table is detectable: the asymptotic limit is not a constant but a random variable. The paper says this and suggests just using the good table alone. The convergence results are in probability rather than almost sure, and the analysis assumes M finite with known theta; the finite-sample behavior of the weight estimator is not deeply evaluated. The rank-r extension is less detailed than the rank-one case, but it is not load-bearing for the paper's central message.\n\nI would bring this to a reading group. It gives a definitive answer in the linear model about whether to stack before or after SVD, and the weighting insights are directly usable. It deserves a serious referee; I would expect acceptance after minor revisions that clean up the edge-case discussion and maybe add a caveat about estimated weights in the application.\n\nSend it out.","headline":"Genuinely new asymptotic results for SVD-Stack and a clean comparison with Stack-SVD; send it to a serious referee.","tokens_in":36571,"tokens_out":1843,"would_cite":true,"duration_ms":24403,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A18","62H25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Stack the data before the SVD: optimally weighted Stack-SVD dominates.","keywords":["data integration","random matrix theory","spectral methods","Stack-SVD","SVD-Stack","optimal weighting","phase transition","shared subspace"],"falsifier":"Simulate two independent tables with $d$ large and with i.i.d. noise that satisfies the paper's moment condition but is concentrated on a few coordinates, then measure $|\\langle \\hat v_1, \\hat v_2\\rangle|^2$ over many replicates; if it does not concentrate at $\\beta_1^2\\beta_2^2$, Lemma 1 fails and the SVD-Stack limits, along with the claimed dominance of weighted Stack-SVD, would need revision.","tokens_in":2096,"feed_emoji":"🧩","tokens_out":2576,"duration_ms":130687,"temperature":0.7,"pith_summary":"Which pipeline should a practitioner use to recover a shared low-dimensional structure from several noisy high-dimensional tables: concatenate the tables and run one SVD (Stack-SVD), or run an SVD per table and then combine the top singular vectors (SVD-Stack)? This paper answers the question in the proportional regime where table sizes and dimension grow together, using random matrix theory to derive the limiting squared cosine $|\\langle \\hat v, v\\rangle|^2$ between the estimated and true shared vector. It shows that neither pipeline wins in all instances when unweighted, but once each table is scaled by its signal-to-noise-optimal weight, Stack-SVD dominates SVD-Stack, strictly so when signal strengths differ and at least two tables carry signal. The paper also gives exact detectability thresholds, extends the comparison to multiple shared components, and supplies a consistent estimator of the signal strengths, so the optimal weights can be implemented from data.","feed_headline":"Weighted Stack-SVD dominates weighted SVD-Stack","feed_subtitle":"Random-matrix analysis shows stacking data beats stacking per-matrix SVDs, and weights can be estimated.","key_machinery":"The argument runs through two limiting objects. For SVD-Stack, the $M\\times M$ matrix $A_\\beta = \\beta\\beta^{\\top} + \\mathrm{diag}(1-\\beta_1^2,\\ldots,1-\\beta_M^2)$ is the entrywise limit of the Gram matrix of the per-table singular-vector estimates; its leading eigenvector and eigenvalue directly give the performance, once a delocalization lemma shows $|\\langle \\hat v_i, \\hat v_j\\rangle|^2 \\to \\beta_i^2 \\beta_j^2$. For Stack-SVD, the load-bearing object is the secular equation $\\sum_i \\theta_i^4(1-x)/(c_i + x\\theta_i^2) = 1$, whose unique root in $(0,1)$ is the weighted performance; the weights $w_i^* \\propto \\theta_i/\\sqrt{\\theta_i^2+c_i}$ are found by reparametrizing the weighted covariance's spectrum to match the weighted-PCA optimization problem. The dominance proof evaluates this secular equation at $x = S/(S+1)$, the weighted SVD-Stack limit, and uses a termwise inequality to show the root must be at least as large.","core_discovery":"Under the model $X_i = \\theta_i u_i v^{\\top} + E_i$ with $n_i/d \\to c_i$ and i.i.d. noise of variance $1/d$ and bounded fourth moment, the paper determines, in probability, the asymptotic performance $|\\langle \\hat v, v\\rangle|^2$ of both estimators. For unweighted Stack-SVD it converges to $(\\|\\theta\\|_2^4 - \\|c\\|_1)/(\\|\\theta\\|_2^2(\\|\\theta\\|_2^2+1))$ above the threshold $\\|\\theta\\|_2^4 > \\|c\\|_1$. For unweighted SVD-Stack it converges to $(\\beta^{\\top} v_{\\max}(A_\\beta))^2/\\lambda_{\\max}(A_\\beta)$, where $A_\\beta = \\beta\\beta^{\\top} + \\mathrm{diag}(1-\\beta_1^2,\\ldots,1-\\beta_M^2)$ and $\\beta_i^2 = (\\theta_i^4-c_i)/(\\theta_i^4+\\theta_i^2)$ when $\\theta_i^4 \\ge c_i$, with $\\beta_i=0$ otherwise. With optimal weights the limits become cleaner: Stack-SVD weights $w_i^* \\propto \\theta_i/\\sqrt{\\theta_i^2+c_i}$ give the unique root $x\\in(0,1)$ of $\\sum_i \\theta_i^4(1-x)/(c_i+x\\theta_i^2)=1$ whenever $\\sum_i \\theta_i^4/c_i > 1$, while SVD-Stack weights $w_i^* = \\theta_i\\sqrt{\\theta_i^2+1}/(\\theta_i^2+c_i)$ give $S/(S+1)$ with $S=\\sum_i \\beta_i^2/(1-\\beta_i^2)$. The paper's main theorem states that optimally weighted Stack-SVD dominates unweighted Stack-SVD and optimally weighted SVD-Stack, with strict improvement whenever the $\\theta_i$ are not all equal and at least two of them are nonzero; the same comparison holds componentwise for rank-$r$ shared subspaces.","pith_inferences":["A testable extension is to replace the i.i.d. noise assumption with sparsely corrupted or row-correlated noise; if SVD-Stack's residuals no longer delocalize, its $S/(S+1)$ formula should degrade measurably, while weighted Stack-SVD's secular-equation limit should survive.","The paper's plug-in estimator of $\\theta_i$ could be turned into an online rule that reweights tables as more data arrive; the paper establishes consistency but does not analyze the finite-sample behavior of the estimated weights.","In federated settings where raw tables cannot be shared, this result implies a trade-off: SVD-Stack is the communication-friendly option but is dominated by weighted Stack-SVD, so communicating per-table statistics such as estimated $\\beta_i$ may partially close the gap; this is an inference, not a claim of the paper.","For practitioners building large genomic atlases, the dominance result supports pooling data before inference, but only when signal strengths can be estimated; the paper's starkest prediction is that many individually undetectable tables can be aggregated by unweighted Stack-SVD into strong recovery."],"forward_implications":["With optimal weights, Stack-SVD never underperforms optimally weighted SVD-Stack or unweighted Stack-SVD; the improvement over SVD-Stack is strict when at least two tables carry signal, and the improvement over unweighted Stack-SVD is strict when the $\\theta_i$ are not all equal.","Method choice for unweighted pipelines depends on the instance: Stack-SVD pools many subthreshold tables into a detectable signal, while SVD-Stack avoids dilution when a few high-noise tables with large $c_i$ are present.","Optimal weighting lowers the detection threshold: weighted Stack-SVD detects when $\\sum_i \\theta_i^4/c_i > 1$, a condition implied by the unweighted threshold $(\\sum_i\\theta_i^2)^2/\\sum_i c_i > 1$ and strictly weaker when the ratios $\\theta_i^2/c_i$ vary.","There exist instance sequences where weighted Stack-SVD approaches perfect recovery while unweighted Stack-SVD, SVD-Stack, binary-weighted Stack-SVD, and weighted SVD-Stack all fall below detectability.","For rank-$r$ shared subspaces, the componentwise formulas give subspace error $\\|V^{\\top}\\hat V\\|_F^2 \\to \\sum_j \\gamma_j$ for weighted Stack-SVD and $\\to \\sum_j S_j/(S_j+1)$ for weighted SVD-Stack."],"supporting_citations":[{"why":"Supplies the single-matrix spiked-eigenvector asymptotics used as Proposition 1 and the delocalization theorem underlying Lemma 1.","marker":"[33]"},{"why":"Provides the heteroscedastic signal-plus-noise eigenvector limit used in the Stack-SVD proof and the bias-correction formula for estimating theta_i.","marker":"[17]"},{"why":"Gives the optimally weighted PCA framework and the optimization problem whose solution yields Theorem 1's weights and performance root.","marker":"[29]"},{"why":"Rank-one modification and interlacing results used to invert A_beta and to certify uniqueness of the leading eigenvalue.","marker":"[9]"},{"why":"Davis-Kahan variant used to turn entrywise convergence of the Gram matrix into convergence of SVD-Stack's leading eigenvector.","marker":"[54]"}],"fun_headline_variants":["Stack before SVD: Optimal weighting wins","Random matrix theory settles Stack-SVD vs SVD-Stack","Data stacking beats per-matrix SVDs with optimal weights","Exact asymptotics: Weighted Stack-SVD outperforms SVD-Stack","SVD order matters: Weighted stacking has the edge"],"cache_read_input_tokens":38784,"weakest_assumption_plain":"The SVD-Stack formulas depend on the noise scrambling the residual parts of each estimated singular vector evenly across all perpendicular directions, so that estimates from different tables overlap only through the true signal.","fun_headline_variants_meta":{"raw":{"variants":["Stack before SVD: Optimal weighting wins","Random matrix theory settles Stack-SVD vs SVD-Stack","Data stacking beats per-matrix SVDs with optimal weights","Exact asymptotics: Weighted Stack-SVD outperforms SVD-Stack","SVD order matters: Weighted stacking has the edge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000718,"raw_usage":{"total_tokens":3390,"prompt_tokens":1275,"completion_tokens":2115,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":891,"completion_tokens_details":{"reasoning_tokens":2030}},"tokens_in":891,"tokens_out":2115,"duration_ms":16233,"temperature":1.0,"reasoning_tokens":2030,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:02:30.858630+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate two independent tables with $d$ large and with i.i.d. noise that satisfies the paper's moment condition but is concentrated on a few coordinates, then measure $|\\langle \\hat v_1, \\hat v_2\\rangle|^2$ over many replicates; if it does not concentrate at $\\beta_1^2\\beta_2^2$, Lemma 1 fails and the SVD-Stack limits, along with the claimed dominance of weighted Stack-SVD, would need revision.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the heteroscedastic signal-plus-noise eigenvector limit used in the Stack-SVD proof and the bias-correction formula for estimating theta_i."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the optimally weighted PCA framework and the optimization problem whose solution yields Theorem 1's weights and performance root."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Rank-one modification and interlacing results used to invert A_beta and to certify uniqueness of the leading eigenvalue."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Davis-Kahan variant used to turn entrywise convergence of the Gram matrix into convergence of SVD-Stack's leading eigenvector."}],"review_version":1}