{"id":"b6c97b48-e186-4363-975a-97539092ab96","arxiv_id":"2508.10782","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Gaussian matrices, generalized first-order methods and AMP are close to a conditionally Gaussian comparison process at dimension-free error, under Lipschitz and moment-matching conditions.","lead":"Galen Reeves proves that a wide class of iterative algorithms for Gaussian data, including gradient methods and approximate message passing, can be coupled to a simpler Gaussian process with an error that stays bounded as the dimension grows. This yields finite-sample, non-asymptotic guarantees without separability or asymptotic assumptions, and a matching lower bound shows the guarantee cannot be improved in general.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition-1 matching does not imply ψ1=0: a T=2 linear example gives ψ1≈√n, so Theorem 1 is not a corollary of Theorem 5.","rationale":"The paper's central contribution is Theorem 1, which is derived as an immediate corollary of Theorem 5 under the assertion that Definition-1 matching gives ψ1=ψ2=0. The reader flagged this as unproven; the stress-test shows it is not merely unproven but false in a simple linear, Lipschitz example with T=2. The calculation uses only the paper's definitions: f1 constant, f2=c+αy1, g=0, and the resulting Σ and b_{1,2}. The residual M-G-FB has a nonzero column proportional to c/α + w1, whose expected norm grows like √n, so ψ1 is not zero and is not dimension-independent. Consequently Theorem 5 cannot yield the dimension-free bound claimed in Theorem 1 through the stated proof. The framework may be repairable by changing the matching construction or the definition of B in Theorem 5, but as written the central claim lacks support. I retain the reader's CONDITIONAL verdict because the overall coupling idea is plausible and the flaw is localized and testable, but the revision needed is substantive: either prove a different relation that makes ψ1=0 under a corrected matching rule, or restate Theorem 1 without relying on ψ1=0 and incorporate the mismatch term into the constant. The concrete T=2 check above will determine which repair is needed.","tokens_in":24019,"tokens_out":31274,"duration_ms":323703,"concrete_test":"Implement the T=2 example exactly: choose n, c with ||c||^2=n, α≠0, f1=c, f2(y1)=c+αy1, g=0. Formally evaluate Definition 1 to obtain Σ=[[1,1],[1,1+α^2]], b_{1,2}=(1+α^2)/α, m2=b_{1,2}c. Then compute B=Σ^{-1}E[(Y-M)^T F] and ψ1=E||M-G-FB||_{2,1}||Ω||^{-1}. If the paper's claim is correct this is zero, but the calculation gives a nonzero value at least √n/(|α|(1+√(1+α^2))). This single check settles whether Definition 1 actually nulls ψ1.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 3.2 asserts that if (m,Σ) are matched per Definition 1 then Theorem 5 holds with ψ1=ψ2=0, and that Theorem 1 follows. This is the load-bearing step. It is false. Take T=2, fix c∈R^n with ||c||^2=n, set f1=c, f2(y1)=c+α y1, g1=g2=0. Definition 1 gives m1=0, Σ=[[1,1],[1,1+α^2]], and b_{1,2}=(Σ^{-1})_{1,1}C_{1,2}=(1+α^2)/α, so m2=(1+α^2)c/α. The matrix B used in Theorem 5 is B=Σ^{-1}E[(Y-M)^T F]=[[0,(1+α^2)/α],[0,-1/α]]. The second column of M-G-FB equals m2 - c B_{1,2} - (c+α w1)B_{2,2} = c/α + w1. Hence ψ1 ≥ ||c/α||/||Ω||_{2,1} = √n/(|α|(1+√(1+α^2))) > 0, and this grows with n. The mismatch is structural: Definition 1 computes b_st with Σ_{≤t}^{-1}, while B in Theorem 5 is built from the full T×T inverse Σ^{-1}; these agree only in special cases. Since the dimension-free bound of Theorem 1 depends on ψ1=ψ2=0, the claimed corollary is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a coupling-based framework for non-asymptotic analysis of generalized first-order methods (1) with Gaussian data matrices. The main object is a conditionally Gaussian comparison process (2) whose mean and covariance are prescribed by a state-evolution recursion (Definition 1). For any Lipschitz f_t,g_t and positive-definite Σ, the authors construct an explicit coupling (Definition 2) between the true iterates x_t and the Gaussian process y_t, and prove (Theorems 4 and 5) high-probability bounds on the coupling error in terms of deterministic mismatch parameters ψ1,ψ2. They then claim that under the state-evolution matching of Definition 1 these mismatches vanish, so that a dimension-free bound follows (Theorem 1). A separate lower bound (Theorem 6) computes the exact quadratic Wasserstein distance in a linear setting and is used to argue tightness of the upper bounds. The paper also claims improved scaling T=o(log n) for the number of iterations and an extension to pseudo-Lipschitz nonlinearities.","tokens_in":24320,"tokens_out":20419,"duration_ms":201704,"significance":"If the main theorem were established, the paper would be a significant advance: it would give the first dimension-free, non-asymptotic coupling bounds for full-memory, non-separable nonlinearities in AMP-type algorithms, going beyond the row-separable and asymptotic results in the literature. The Gaussian-conditioning construction and the exact Wasserstein lower bound in Theorem 6 are valuable and self-contained contributions in their own right. However, the central upper-bound claim is not established in the current version. The proof of Theorem 5 has a false step in the claim that Definition 1 matching implies ψ1=ψ2=0, and Theorem 3's proof has an unaddressed completeness gap in the Gram-Schmidt construction. These issues undermine the paper's main advertised result, although the general sample-mismatch bound in Theorem 4 may remain of independent interest.","major_comments":[{"comment":"The assertion that Definition 1 matching gives ψ1=ψ2=0 is false. Consider T=2, α>0, f1=c with ||c||^2=n, f2(y1)=c+α y1, and g1=g2=0. Definition 1 gives m1=0, Σ=[[1,1],[1,1+α^2]], b_{1,2}=(1+α^2)/α, and m2=(1+α^2)c/α. However, the matrix B defined just before Theorem 5 equals Σ^{-1}E[(Y-M(Y))^T F(Y)]/n = [[0,α],[0,0]], because f2(y1)=c+α y1 and Cov(w1,w2)=I. Hence the second column of M(Y)-G(Y)-F(Y)B is m2-αc = c/α, so ψ1 = (√n/α)/(1+√(1+α^2)), which grows as √n. The root cause is that Definition 1 (Eq. 6b) uses the leading principal inverse Σ_{≤t}^{-1}, whereas B in Theorem 5 uses the full T×T inverse Σ^{-1}; these agree only in special cases. Consequently Theorem 5 does not imply Theorem 1 via ψ1=ψ2=0, and the central dimension-free upper bound is unsupported as written.","section":"§4.2, proof of Theorem 3"},{"comment":"The proof of Theorem 3 relies on the identity Σ_{t=1}^T u_t u_t^T = I. This identity holds only if the vectors q1,...,qT from Definition 2 are extended to an orthonormal basis of R^n; for T<n, Definition 2 as written defines only T vectors. The fallback rule in Definition 2 references 'the span of {q1,...,qn}' before q_n has been defined, so the construction is not well-defined. The authors should specify a completion of the Gram-Schmidt process to n vectors (for example, using the fixed basis v1,...,vn) and verify that the independence of z1,...,zT and identity (10) are independent of the chosen completion. As it stands, Theorem 3 is not fully proved.","section":"§4.2, proof of Theorem 3"},{"comment":"Theorem 1 assumes only that f_t and g_t are L-Lipschitz, but the proof route through Theorem 5 requires each m_t to be L_m-Lipschitz, and the final bound depends on L_m through L = log2(2T) + √T(Lf+Lg+Lm)(1+Lm)^{T-1}κ(Σ)^{1/2}. No argument is given that the matched m_t defined by (6a)-(6b) are Lipschitz with constants bounded independently of n; the coefficients b_st depend on the conditioning of Σ_{≤t} and may be large. Theorem 1 should either state the Lipschitz condition on m_t explicitly or prove that it follows from the matching construction with a dimension-free constant; otherwise the theorem is not a consequence of Theorem 5.","section":"§1, Theorem 1; §3.2, Theorem 5"}],"minor_comments":[{"comment":"In the display after solving for X and Y, the second equality reads Y = ZΩ(I−Λ)^{-1}, but it should be Y = ZΩ(I−Γ)^{-1} since Y = YΓ + ZΩ.","section":"§4.5"},{"comment":"Theorem 5 defines L := log2(2T) + ... while the proof in Section 4.4 defines L = log2(T) + ...; the two definitions should be made consistent.","section":"§3.2, Theorem 5; §4.4"},{"comment":"The inequality Lip(g_t) ≤ (Σ_{s<t}|b_st|^2)^{1/2} L_f ≤ √T L_f^2 appears to assume |b_st| ≤ L_f, which is not established in the text; please justify or remove this step.","section":"§3.2.1, Eq. (13)"},{"comment":"The pseudo-Lipschitz extension argument applies Theorem 5 to Lipschitz extensions and then uses the resulting bound on ∥X−Y∥ to infer that both X and Y lie in the ball B_n(ρ); this is circular unless ρ is chosen first and the Lipschitz-extension constants are shown to be acceptable for the resulting probability bounds. Please spell out the order of quantifiers.","section":"§3.2.2"}],"recommendation":"major_revision","confidential_remarks":"The counterexample in major comment 1 is simple and decisive: under the definitions as written, Definition 1 does not make the residual M-G-FB vanish, and ψ1 is of order √n. The authors need to either repair the matching condition so that the mismatch term is genuinely zero (for example, by using block-causal coefficients in the definition of B in Theorem 5) or revise the main theorem to include the resulting mismatch term. The current version does not establish the advertised dimension-free upper bound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline claim is not proven. The paper's main theorem (Theorem 1) is a dimension-free coupling bound, but the step that gets you there—Section 3.2's assertion that Definition 1 matching sets ψ1=ψ2=0—is false. A T=2 linear example with f1=c, f2(y1)=c+α y1, g1=g2=0 gives a matched (m,Σ) yet ψ1 ~ √n. The reason is structural: Definition 1 builds the coefficients b_st from inverses of leading principal submatrices Σ_{≤t}, while Theorem 5's B uses the full inverse Σ^{-1}. They agree only in special cases, and the example shows a mismatch of order √n. So the dimension-free conclusion does not follow from Theorem 5.\n\nWhat is genuinely new is the coupling construction itself: an independent copy of A used as a stochastic correction, with a Gram-Schmidt style basis and a direct comparison to a conditionally Gaussian process. That is a fresh approach for non-separable, full-memory nonlinearities, and the paper explains clearly how it unifies AMP and gradient-based schemes. The Wasserstein lower bound in Theorem 6 is also a useful complement.\n\nThe other soft spots are real but less fatal. The proof of Theorem 3 uses the identity \\sum_{t=1}^T u_t u_t^T = I_n, which requires extending the q's to a full orthonormal basis; the construction as written does not do that. Also, the paper assumes a matched fixed point (m,Σ) exists without proof; under the stated Lipschitz assumptions it is not obvious that Definition 1 yields square-integrable y's. The abstract's promised improvement in T-scaling, from o(log n/log log n) to o(log n), rests on the unproven ψ1=ψ2=0 and is therefore overstated as written.\n\nThe stress-test note you showed me is on target, though its B matrix is not quite what the paper's definition computes. The conclusion still holds: even with the correct B, the example gives ψ1 = Θ(√n). So Theorem 1 is not a corollary of Theorem 5.\n\nAll of this is fixable. The coupling idea is worth keeping; the proof needs a sharper condition that actually zeros out the mismatch, perhaps by defining B recursively rather than via a full inverse, or by adding a correction term to m. I would send this to a serious referee; the right referee can tell whether the framework can be repaired. I would not cite the main theorem in its current form, but I would follow the paper's development. For readers in AMP and high-dimensional inference, the construction is worth study even while the theorems are under repair.","headline":"A genuinely new coupling idea, but the main theorem is unproven because Definition-1 matching does not make the mismatch term ψ1 vanish.","tokens_in":24833,"tokens_out":11254,"would_cite":false,"duration_ms":105046,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that iterates of generalized first-order methods on Gaussian matrices can be coupled to a conditionally Gaussian proxy, with maximum deviation over $T$ steps bounded by a constant independent of the ambient dimension $n$.","keywords":["generalized first-order methods","approximate message passing","state evolution","Gaussian coupling","dimension-free bounds","non-asymptotic analysis","Gaussian concentration","Wasserstein distance"],"falsifier":"Simulate the exact coupling of Definition 2 for a non-separable, full-memory Lipschitz recursion with a numerically matched pair $(m,\\Sigma)$, across $n = 10^3, 10^4, 10^5$ at fixed $T$: if $\\max_{t \\in [T]}\\|x_t - y_t\\|$ grows with $n$ (for instance like $\\sqrt{n}$), or if the empirical probability $\\mathbb{P}[\\max_t \\|x_t - y_t\\| \\ge C\\sqrt{r}]$ exceeds $2e^{-r}$, Theorem 1 is refuted. The same setup with $T = c\\log n$ chosen so that $T + \\ln\\kappa(\\Sigma)$ exceeds $C\\log n$ tests the predicted breakdown of the scaling condition (12), namely that $n^{-1/2}\\|X - Y\\|$ should then fail to vanish.","tokens_in":23761,"feed_emoji":"🎲","tokens_out":20829,"duration_ms":180844,"temperature":0.7,"pith_summary":"This paper sets out to show that the state-evolution picture behind approximate message passing (AMP) — the replacement of a complicated iterative algorithm by a Gaussian process whose covariance is updated by a deterministic recursion — holds at finite dimension, uniformly across all $n$ coordinates at once, not only in an asymptotic or coordinate-by-coordinate sense. The main claim, Theorem 1, couples the iterates $x_t$ of a generalized first-order method (a recursion $x_t = A f_t(x_{<t}) + g_t(x_{<t})$ covering gradient-based optimization and AMP) to a conditionally Gaussian comparison process $y_t = m_t(y_{<t}) + w_t$, and bounds the maximum distance by a constant depending only on the number of steps $T$, the Lipschitz constant $L$, and the eigenvalues of the covariance $\\Sigma$, with probability at least $1 - 2e^{-r}$. If the construction is correct, the Gaussian comparison becomes a rigorous finite-sample tool for non-separable, full-memory nonlinearities, with explicit tail bounds instead of asymptotic $o(\\sqrt{n})$ rates. A complementary lower bound shows the error cannot be driven to zero, so the $\\mathrm{O}(1)$ scale is intrinsic, and the paper improves the allowable iteration count from $T = o(\\log n / \\log \\log n)$ to $T + \\ln \\kappa(\\Sigma) = O(\\log n)$.","feed_headline":"Gaussian coupling makes AMP error independent of dimension","feed_subtitle":"The gap between iterates and their Gaussian proxy depends on iteration count and smoothness, not on dimension.","key_machinery":"The load-bearing object is the explicit coupling of Definition 2. At each step the nonlinear response $f_t(y_{<t})$ is turned into a nearly orthonormal vector $q_t$ by Gram–Schmidt against the previous $q_s$ (normalized to length $\\sqrt{n}$), and the noise entering the comparison process is $z_t = A q_t - \\frac{1}{n}\\sum_{s \\le t}\\langle z_s, q_t\\rangle q_s + \\sum_{s \\le t} A'_{st} q_s$, where $A'$ is an independent copy of $A$; the correction terms are Onsager-like debiasing terms that make the $z_t$ exactly i.i.d. standard Gaussian, so $w_t = \\sum_{s \\le t}\\Omega_{st} z_s$ has covariance $\\Sigma = \\Omega^\\top\\Omega$. The proof that the increments are independent is a Gaussian-conditioning argument (Lemma 9, generalizing the lemma in [2]): conditional on the past, a GOE matrix has Gaussian law with computable mean and covariance, and orthogonality of the row spaces forces the conditional means to zero and conditional variances to the identity. Three quantitative tools complete the argument: a discrete-time stability bound (Lemma 14) that inflates the one-step mismatch by $(1 + L_f + L_g)^{T-1}$; Gaussian concentration of measure (Lemma 10) that controls the mismatch terms $\\psi_1, \\psi_2$ at the $\\sqrt{r}$ scale; and a Cholesky perturbation inequality (Lemma 12, from [57]) that turns errors in the empirical second-moment matrix into errors in the Cholesky factors at the cost of a $\\log(4T)$ factor.","core_discovery":"The central claim is Theorem 1: for $A \\sim \\mathrm{GOE}(n)$ and $L$-Lipschitz, possibly non-separable, full-memory nonlinearities $f_t, g_t$, if the mean functions $(m_t)$ and covariance $\\Sigma$ of the comparison process are matched to the original recursion by the recursive state-evolution construction in Definition 1 and $\\Sigma$ is positive definite, then there is a coupling of the two processes under which $\\mathbb{P}\\big[\\max_{t \\in [T]} \\|x_t - y_t\\| \\ge C\\sqrt{r}\\big] \\le 2e^{-r}$ for all $0 \\le r \\le n$, with $C$ depending only on $T$, $L$, and the extreme eigenvalues of $\\Sigma$. In plain terms, the whole $n$-dimensional trajectory of the algorithm stays within a dimension-free distance of a Gaussian process whose law is known, up to an exponentially small failure probability. The proof is one-shot rather than inductive: an independent copy of the Gaussian matrix supplies a stochastic correction so that the comparison process has exactly its target law, and then Gaussian concentration converts the mismatch between $(f,g)$ and $(m,\\Sigma)$ into the tail bound. Theorem 6 complements this with an exact formula for the quadratic Wasserstein distance in a constant/linear setting, showing the coupling distance cannot vanish with $n$ and that for autoregressive updates with $|\\lambda| > 1$ the optimal error grows exponentially in $T$.","pith_inferences":["The construction doubles as a simulation scheme and a diagnostic: because the coupling needs only two independent GOE draws and the Gram–Schmidt basis built from the algorithm's own responses, one can generate the comparison process exactly; conversely, if the empirical law of the increments $z_t$ departs from i.i.d. Gaussian, the algorithm is outside the regime where state evolution describes it.","The bound's dependence on $\\kappa(\\Sigma)$ suggests the guarantee degrades exactly where AMP analyses usually operate: close to a state-evolution fixed point the conditional variance of the increments collapses, so $\\Sigma$ becomes near-singular and the constant $C$ blows up before the asymptotic description takes over. A numerical check of the error versus $\\kappa(\\Sigma)$ along the iteration pat","Because the argument is a one-shot comparison rather than an induction over steps, it is plausible that the same coupling strategy transfers to continuous-time limits and to stochastic-gradient dynamics, where inductive decorrelation proofs are awkward; the paper gestures at dynamical mean-field theory but does not make this transfer."],"forward_implications":["State evolution becomes a finite-sample theorem: for fixed $T$, the full trajectory $\\{x_1,\\dots,x_T\\}$ of an AMP or gradient-based method is within $\\mathrm{O}(1)$ of a Gaussian process with deterministically evolving covariance, with exponential tails, for non-separable nonlinearities that earlier non-asymptotic results could not handle.","The iteration count may grow with dimension: the condition $T + \\ln\\kappa(\\Sigma) = O(\\log n)$ suffices for $n^{-1/2}\\|X - Y\\|$ to vanish, improving on the previous $T = o(\\log n/\\log\\log n)$ regime for finite-sample AMP.","Because the GOE result transfers to asymmetric Gaussian matrices with correlated rows and columns via the arguments of [17,18], the dimension-free bound applies beyond the symmetric ensemble in which it is proved.","The lower bound shows the $\\mathrm{O}(1)$ scale is not an artifact of the coupling: in the exactly solvable linear setting the quadratic Wasserstein distance between $x_t$ and $y_t$ is bounded below by $(\\sqrt{2}-1)^2$ times the state-evolution variance $\\alpha_t^2$ for every $n$, and with $|\\lambda| > 1$ autoregressive updates the optimal error grows exponentially in $T$."],"supporting_citations":[{"why":"Supplies the Gaussian-conditioning lemma that the coupling proof generalizes as its Lemma 9; this is the mechanism that makes the increments $z_t$ independent standard Gaussians.","marker":"[2]"},{"why":"Defines generalized first-order methods, the algorithm class (1) whose iterates are the object of the dimension-free bound.","marker":"[23]"},{"why":"Establishes state evolution for non-separable functions and the debiasing-coefficient formula (5); this is the asymptotic theory whose separability and limit assumptions the paper removes.","marker":"[28]"},{"why":"Provides the earlier finite-sample AMP analysis with separable nonlinearities and the iteration scaling $T = o(\\log n/\\log\\log n)$ that the paper improves on.","marker":"[30]"},{"why":"Gives entrywise non-asymptotic guarantees for general first-order methods under row-separable nonlinearities; it is the core comparison showing what the paper's added generality costs.","marker":"[34]"},{"why":"Supplies the theorem on Gaussian conditional distributions (its Theorem 13.2) that Lemma 8 and the conditioning argument rest on.","marker":"[51]"},{"why":"Provides the closed-form quadratic Wasserstein distance between Gaussian measures used to compute the exact lower bound in Theorem 6.","marker":"[54]"},{"why":"Supplies the Gaussian concentration inequality (Lemma 10) that converts Lipschitz dependence on the driving noise into the exponential tail bound.","marker":"[55]"},{"why":"Provides the Cholesky perturbation inequality used in Lemma 12 to control the empirical second-moment mismatch $\\Delta_2$.","marker":"[57]"}],"fun_headline_variants":["AMP error independent of dimension via Gaussian coupling","Gaussian coupling yields dimension-free AMP bounds","Gaussian coupling removes dimension from AMP error bounds","Sharp dimension-free bounds for AMP via Gaussian coupling","One-shot Gaussian coupling proves dimension-free AMP analysis"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument presupposes that the state-evolution recursion of Definition 1 has a solution — matched mean functions $m_t$ and a positive-definite covariance $\\Sigma$, with each $m_t$ Lipschitz — and the paper proves no existence or uniqueness theorem for such a matched pair for general Lipschitz $(f,g)$; Theorem 1 as stated does not even list the Lipschitz condition on $m_t$ that the proof of Theorem 5, from which it follows, requires.","fun_headline_variants_meta":{"raw":{"variants":["AMP error independent of dimension via Gaussian coupling","Gaussian coupling yields dimension-free AMP bounds","Gaussian coupling removes dimension from AMP error bounds","Sharp dimension-free bounds for AMP via Gaussian coupling","One-shot Gaussian coupling proves dimension-free AMP analysis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2894,"prompt_tokens":994,"completion_tokens":1900,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":1831}},"tokens_in":610,"tokens_out":1900,"duration_ms":15203,"temperature":1.0,"reasoning_tokens":1831,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:31:43.216648+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the exact coupling of Definition 2 for a non-separable, full-memory Lipschitz recursion with a numerically matched pair $(m,\\Sigma)$, across $n = 10^3, 10^4, 10^5$ at fixed $T$: if $\\max_{t \\in [T]}\\|x_t - y_t\\|$ grows with $n$ (for instance like $\\sqrt{n}$), or if the empirical probability $\\mathbb{P}[\\max_t \\|x_t - y_t\\| \\ge C\\sqrt{r}]$ exceeds $2e^{-r}$, Theorem 1 is refuted. The same setup with $T = c\\log n$ chosen so that $T + \\ln\\kappa(\\Sigma)$ exceeds $C\\log n$ tests the predicted breakdown of the scaling condition (12), namely that $n^{-1/2}\\|X - Y\\|$ should then fail to vanish.","supporting_citations":[{"cited_title":"The estimation error of general first order methods,","cited_arxiv_id":null,"evidence_quote":"Defines generalized first-order methods, the algorithm class (1) whose iterates are the object of the dimension-free bound."},{"cited_title":"State evolution for approximate message passing with non-separable functions,","cited_arxiv_id":null,"evidence_quote":"Establishes state evolution for non-separable functions and the debiasing-coefficient formula (5); this is the asymptotic theory whose separability and limit assumptions the paper removes."},{"cited_title":"Finite sample analysis of approximate message passing algorithms,","cited_arxiv_id":null,"evidence_quote":"Provides the earlier finite-sample AMP analysis with separable nonlinearities and the iteration scaling $T = o(\\log n/\\log\\log n)$ that the paper improves on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem on Gaussian conditional distributions (its Theorem 13.2) that Lemma 8 and the conditioning argument rest on."},{"cited_title":"The Fr´ echet distance between multivariate normal distributions,","cited_arxiv_id":null,"evidence_quote":"Provides the closed-form quadratic Wasserstein distance between Gaussian measures used to compute the exact lower bound in Theorem 6."},{"cited_title":"On Parlett’s matrix norm inequality for the Cholesky decomposi- tion,","cited_arxiv_id":null,"evidence_quote":"Provides the Cholesky perturbation inequality used in Lemma 12 to control the empirical second-moment mismatch $\\Delta_2$."}],"review_version":1}