{"id":"c13d6067-8f9d-4702-a609-c1f452f17a96","arxiv_id":"1908.09959","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A sharp information-theoretic threshold for approximate recovery of a planted sparse submatrix is derived, and an overlap gap property is proved that blocks local MCMC algorithms in a conjecturally hard phase.","lead":"The authors analyze how hard it is to find a planted block of elevated values buried in a large random matrix. They prove a sharp signal threshold for statistical recovery and show that in a broad intermediate-signal regime, local search algorithms provably get stuck on suboptimal solutions.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4 rests on the imported zero-temperature Γ-limit of Theorem 2.1, which is under-verified; §4.1's displayed functionals have a factor-2 mismatch in the linear term and the two-species atomic case is not checked.","rationale":"The reader identified the variational formula as the weakest assumption, and I agree. The proof of Theorem 1.4 is a clean derivative analysis once Theorem 2.1 is granted; the risky step is precisely the zero-temperature Γ-limit, which is not derived in the paper but imported from the authors' [50]. The visible factor-2 mismatch between (4.3), Theorem 2.1, and the displayed Eβ/E in Section 4.1 is a concrete symptom that this passage is not fully pinned down. I also noted a smaller issue in Theorem 3.2: the reflected chain as defined (re-normalizing Q on the boundary without a self-loop) is not reversible with respect to ν(·|A), so the claimed stationarity used in the proof is not immediate; this is likely fixable by including the self-loop at ∂A and does not affect the OGP itself. The main concern remains Theorem 4.2. Because the imported theorem is central, a fair verdict is conditional: the paper should be accepted once the authors supply a self-contained proof of the Γ-convergence step or explicitly verify the hypotheses of [50] for this two-species constrained functional, including the atomic case and the factor 2. I do not think the argument is wrong; the risk is that the central variational formula is currently asserted rather than fully demonstrated.","tokens_in":38659,"tokens_out":46411,"duration_ms":457855,"concrete_test":"Independently re-derive Theorem 4.2: prove that the functionals (4.3), with ν=βµ([0,s])ds, Γ-converge as β→∞ to P(ν,Λ)=ρu^1_ν(0,0)+(1−ρ)u^2_ν(0,0)−Λ1q−Λ2(ρ−q)−2∫_0^ρ s dν(s), including measures ν with an atom at ρ and a recovery sequence valid for both Λ1 and Λ2 simultaneously. Specifically, check whether the factor of 2 in the linear term survives the Γ-limit; if the limit is instead −∫s dν, recompute the bound (2.9) and the sign of ∂qE(ρ^{2−δ}) in Theorem 1.4. A cheaper numerical cross-check: evaluate E(q;ρ,λ) from Theorem 2.1 for ρ=10^-2, λ=5√(log(1/ρ)/ρ) at q=ρ² and q=ρ^{1.9} and compare with direct maximization of (1.5) for N=2000,4000 to see whether the predicted derivative sign change appears.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim Theorem 1.4 is derived by differentiating the constrained energy E(q;ρ,λ), which is given by Theorem 2.1. Theorem 2.1 is obtained from Theorem 4.2, the zero-temperature limit of the finite-temperature variational problem. The proof of Theorem 4.2 is not self-contained: it invokes [50, Theorem 3.2] for the Γ-convergence of the Parisi functionals and then asserts the extension to the two-species constrained functional in Lemma 4.4 in a few lines. Two concrete risks are visible. First, the functionals Eβ and E defined in Section 4.1 contain the linear term −∫ s dν(s), whereas the finite-temperature functional (4.3) and the target P in Theorem 2.1 contain −2∫ s dν(s); if this factor is not correctly transferred by the Γ-limit, the formula for E(q) changes, and the derivative estimate (2.9)–(2.10) that produces the OGP gap is no longer valid. Second, the imported Γ-convergence theorem is for a single-species Parisi functional; here the same measure ν must simultaneously recover the limits for the two species with different terminal data Λ1 and Λ2, and ν may carry an atom at ρ after the c=0 reduction. The paper does not verify that the recovery sequence from [50, Lemma 3.4] works for both species or that atoms are handled. If Γ-convergence fails for any q in [ρ²,ρ^{1+ε}], the signs of ∂qE used to create the gap collapse, and with them the OGP and the MCMC barrier.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies approximate support recovery in the planted principal submatrix model A = (λ/N)vv^T + W, where W is a GOE matrix, v has density ρ, and λ is a constant. The main results are: (i) an information-theoretic threshold at λ ≈ sqrt((1/ρ) log(1/ρ)) for approximate recovery, with MLE optimal up to constants; (ii) an ε-overlap-gap property (OGP) for the constrained maximum likelihood energy E(q;ρ,λ) in the window C1 sqrt((1/ρ) log(1/ρ)) < λ < ρ^{-α} for α < 2 - sqrt(2) and ρ small; (iii) a consequence that a natural family of local reversible MCMC samplers requires exponential time to leave the low-overlap region; and (iv) a spectral algorithm achieving approximate recovery when λ > 1/ρ. The central technical engine is a Parisi-type variational formula for E(q;ρ,λ), obtained as a zero-temperature limit of a two-species spin glass free energy, followed by a delicate analysis of the derivative of E with respect to q using SDE representations and quantile bounds.","tokens_in":39024,"tokens_out":15155,"duration_ms":135603,"significance":"If the main theorem is fully established, the paper would be a significant contribution to the statistical-computational gap literature: it gives structural landscape evidence for hardness in sparse PCA / planted submatrix recovery, connects to the OGP framework, and provides a clean algorithmic consequence for local MCMC. The information-theoretic threshold and the spectral result are also useful. The proof strategy is sophisticated: it combines Guerra interpolation, Aizenman-Sims-Starr bounds, Ghirlanda-Guerra identities, ultrametricity, and a zero-temperature Γ-convergence result imported from the authors' prior work. The paper is honest about relying on this machinery, but the missing verification of the two-species zero-temperature Γ-convergence and some visible factor inconsistencies mean the central claim is currently not fully supported as written. The result is plausible and the high-level strategy is credible, but a careful repair of the variational proof is needed before the paper can be accepted.","major_comments":[{"comment":"There is a factor-of-two mismatch in the functionals used for the zero-temperature limit. The finite-temperature functional Pβ in (4.3) and the target functional P in Theorem 2.1 both contain the linear term −2∫ s dν(s), whereas the functionals Eβ and E defined immediately before Lemma 4.4 contain −∫ s dν(s). As written, Lemma 4.4 proves Γ-convergence to a functional that is not the one appearing in Theorem 4.2 or Theorem 2.1. Since Theorem 2.1 is the foundation for the derivative estimates in (2.9)–(2.10) that produce the overlap gap, this discrepancy is load-bearing. The authors should either correct the coefficient and re-check the subsequent compactness and convergence arguments, or explicitly explain a change of variables that removes the factor of two.","section":"Section 4.1, Lemma 4.4 and Theorem 4.2"},{"comment":"The extension of the zero-temperature Γ-convergence result to the two-species constrained functional is asserted rather than proved. The imported result [50, Theorem 3.2] is for a single Parisi functional, while here the same measure ν must simultaneously recover the limits for two species with different terminal data Λ1 and Λ2, and the limiting measure may require treatment of atoms after the c=0 reduction in Section 2. The claim in the proof of Lemma 4.4 that the recovery sequence 'does not depend on the functional itself' is not sufficient, because the minimizer of the sum is not automatically a recovery sequence for both summands. This is not a cosmetic gap: if the Γ-convergence fails for some q in the critical interval, the sign changes of ∂_q E that create the gap collapse. Please provide a detailed proof of Lemma 4.4 or a precise citation to a two-species version of the Γ-convergence theorem.","section":"Section 4.1, Lemma 4.4 and Section 6.2"},{"comment":"The displayed chain in the proof of Lemma 4.6 appears algebraically inconsistent with the derivative formula in Lemma 5.2. Lemma 5.2 states ∂βF = 2β∫(ρ²−q²)dµ(q), but the proof of Lemma 4.6 writes β∫(ρ−q)dµ(q) ≤ (1/(4ρ))β∫(ρ²−q²)dµ(q) = (1/(4ρ))∂βF. Combining these identities gives a right-hand side of (1/(8ρ))∂βF after the displayed inequality, not (1/(4ρ))∂βF, and the inequality itself does not follow from (ρ−q) ≤ (ρ²−q²)/(ρ+q). Since Lemma 2.2's constant (1/2)√(ρ log(1/ρ)) enters the quantile bounds (2.9) and hence the OGP window, this factor must be reconciled. Please re-derive the constants in Lemmas 5.2 and 4.6 carefully.","section":"Section 5, Lemmas 5.2 and 4.6"}],"minor_comments":[{"comment":"The phrase 'with minor modification' is too terse for a step that carries the entire zero-temperature limit. Even if the factor issue is resolved, the lemma deserves a full proof or a precise statement of the modified assumptions under which [50] applies.","section":"Section 4.1, Lemma 4.4"},{"comment":"The Ghirlanda-Guerra perturbation argument is only sketched. The paper correctly notes that the self-overlap R11 is constant on Σ_N(ρ,q), which removes the usual self-overlap terms, but the construction of the parameters (x_p) and the verification of the identities should be written out or quoted more explicitly.","section":"Section 6.2.2, Lemma 6.4"},{"comment":"The notation q = ρ²−δ is easy to misread as ρ² − δ; it should be typeset consistently as ρ^{2−δ}. The displayed derivative bound surrounding (2.10) also contains an unmatched parenthesis that should be fixed.","section":"Section 2, proof of Theorem 1.4"},{"comment":"The condition y = o_ρ(ρ) in Theorem 3.3 is consistent with Definition 1.3 only after taking ρ → 0 with ε fixed; this quantifier should be stated explicitly to avoid apparent conflict with the condition y < ρ^{1+ε}.","section":"Section 3, Theorem 3.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in its overall strategy and the OGP/MCMC landscape picture is compelling, but the variational formula on which the main theorem rests is not currently demonstrated as written. The factor-of-two issue in Section 4.1 and the unproved two-species Γ-convergence assertion are central, not merely expository. I recommend major revision with a request for a complete proof of Theorem 2.1's zero-temperature limit, rather than rejection. The information-theoretic and spectral parts appear sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper makes a genuine advance: it proves the first rigorous overlap gap property for the planted submatrix model in the constant-ρ, ρ→0 regime, and it extracts a clean MCMC barrier from that OGP. The information-theoretic threshold for the MLE and the spectral rounding for λ > 1/ρ are also solid. The two-species Parisi-type variational formula is a substantial piece of work, not a routine import.\n\nThe soft spot is real and it sits right at the load-bearing joint. In Section 4.1, the functionals Eβ and E are displayed with the linear term −∫ s dν(s), while the finite-temperature functional (4.3) and the target P in Theorem 2.1 both carry −2∫ s dν(s). The proof of Lemma 4.4 never addresses this term, so if the Γ-convergence statement is taken at face value, the limit is the wrong functional. That would change the derivative of E(q) and the sign of ∂q E at ρ²−δ, which is exactly what creates the OGP gap. It may be a typo, but it is not a harmless one; the authors need to correct the display and verify that the Γ-limit recovers the factor 2.\n\nThe second concern is related: the Γ-convergence result is imported from [50], which is single-species, and the paper asserts the extension to two species with different terminal data in a few lines. The atom at ρ is handled by a linear transformation, but the recovery sequence from [50, Lemma 3.4] is not shown to work for both species simultaneously. Lemma 6.4 (the Ghirlanda–Guerra perturbation) is also only sketched, and the embedding of the constrained two-species model into the ultrametric framework is not fully detailed.\n\nIf those gaps are repairable, this is an important paper. As written, I would not rely on Theorem 1.4 without a careful check of Section 4.1. The information-theoretic and spectral parts stand on their own.\n\nRecommendation: yes, send it to peer review — this is exactly the kind of paper a serious referee should spend time on — but the referee's first job is to resolve the factor-2 mismatch. I wouldn't cite the OGP result in my own work until that is settled.","headline":"Rigorous OGP for planted submatrix recovery is a real step forward, but the zero-temperature variational proof has a visible factor-2 mismatch that must be fixed before the main theorem holds up.","tokens_in":39495,"tokens_out":10017,"would_cite":false,"duration_ms":89986,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q87","60C05","82B44","68Q25","62H25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The planted submatrix model has an overlap gap at moderate signal strengths, making the likelihood landscape disconnected in overlap and forcing reversible local MCMC samplers to take exponential time to reach high-overlap configurations.","keywords":["overlap gap property","principal submatrix recovery","spin glasses","Parisi formula","statistical-computational gap","Markov chain Monte Carlo","approximate support recovery","spectral method"],"falsifier":"Compute $E(q;\\rho,\\lambda)$ from the variational formula for a concrete pair inside the claimed window, say $\\rho=10^{-3}$ and $\\lambda=10\\sqrt{(1/\\rho)\\log(1/\\rho)}$, and check the sign of $\\partial_q E$ at $q=\\rho^2$ and at $q=\\rho^{2-\\delta}$ with $\\delta\\approx 0.2$; if the derivative is nonnegative at $q=\\rho^{2-\\delta}$, or if the energy curve has no interior local maximum between $\\rho^2$ and $\\rho^{1+\\varepsilon}$, then the overlap gap property, and with it the exponential hitting-time bound, fails exactly where Theorem 1.4 claims it holds.","tokens_in":38506,"feed_emoji":"🧩","tokens_out":11825,"duration_ms":111920,"temperature":0.7,"pith_summary":"This paper studies a planted principal submatrix problem: an $N\\times N$ Gaussian matrix contains a hidden $k\\times k$ submatrix ($k=\\rho N$) with elevated mean $\\lambda/N$, and the goal is to recover a constant fraction of its support. The authors establish that approximate recovery by the maximum likelihood estimator is possible exactly when $\\lambda$ is at least a constant times $\\sqrt{(1/\\rho)\\log(1/\\rho)}$, and impossible below that scale. Their main discovery, however, is about the likelihood landscape: in the high-sparsity, moderate-signal window $\\sqrt{(1/\\rho)\\log(1/\\rho)} \\ll \\lambda \\ll \\rho^{-\\alpha}$ for $\\alpha<2-\\sqrt{2}$, the constrained energy has a local maximum at a suboptimal overlap, so near-optimal configurations split into well-separated low- and high-overlap clusters. They prove this overlap gap property implies that local reversible Markov chains, started at a random overlap, require exponential time to reach the good cluster. A straightforward rounding of the top eigenvector succeeds once $\\lambda>1/\\rho$, showing the barrier sits between the information-theoretic and spectral regimes.","feed_headline":"Overlap gap blocks local samplers in submatrix recovery","feed_subtitle":"At moderate signals, the likelihood splits into separated overlap clusters, so MCMC needs exponential time to recover the support.","key_machinery":"The load-bearing object is the constrained ground state energy $E(q;\\rho,\\lambda)=\\lim_{N\\to\\infty}\\frac{1}{N}\\max\\{(x,Wx)+\\lambda q^2 : x\\in\\Sigma_N(\\rho N),\\,(x,v)=Nq\\}$. Theorem 2.1 evaluates this limit as $\\lambda q^2+\\min_{\\nu,\\Lambda}P(\\nu,\\Lambda)$, a Parisi-type variational formula over monotone measures $\\nu$ and Lagrange multipliers $\\Lambda$; the PDE in the formula is solved through a family of diffusions. The proof of OGP comes from differentiating this functional: at $q=\\rho^2$ the derivative is $2\\lambda\\rho^2>0$, while at $q=\\rho^{2-\\delta}$ the derivative is negative for the claimed parameter range, forcing an interior local maximum and hence the gap. The SDE representation and the bound $\\int_0^\\rho m(t)\\,dt \\le \\frac{1}{2}\\sqrt{\\rho\\log(1/\\rho)}$ control the derivative and complete the sign-change argument.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.4: for any $\\alpha<2-\\sqrt{2}$, any $C_1>2$, all sufficiently small $\\rho$, and all $\\lambda$ with $C_1\\sqrt{(1/\\rho)\\log(1/\\rho)}<\\lambda<\\rho^{-\\alpha}$, the planted submatrix model has the $\\varepsilon$-overlap gap property. That means there are overlaps $w<\\rho^2<x<y<\\rho^{1+\\varepsilon}$ at which the constrained ground state energy $E(q;\\rho,\\lambda)$ satisfies $E(w)<E(x)$ and $E(y)<E(x)$, with the supremum over the interval $(w,\\varepsilon\\rho]$ strictly below the supremum over $[\\varepsilon\\rho,\\rho]$. The likelihood landscape is therefore disconnected along the overlap axis: any near-maximizer is either nearly uncorrelated with the planted vector or strongly correlated with it, never in between. Combining this structural statement with the free-energy-well argument gives Theorem 1.5: a reversible nearest-neighbor Markov chain on the Hamming graph, initialized from the Gibbs measure conditioned on the low-overlap interval, has exponentially small probability of escaping that interval before time $\\exp(cN)$.","pith_inferences":["The same derivative-sign analysis should extend the overlap gap to the full sub-spectral window $\\sqrt{(1/\\rho)\\log(1/\\rho)} \\ll \\lambda \\ll 1/\\rho$; the paper stops at $\\rho^{-\\alpha}$ for $\\alpha<2-\\sqrt{2}$ only because the current scaling estimates degrade there, and such an extension would make the hard phase coincide with the conjectured computational threshold.","The free-energy-well argument never uses the detailed-balance reversibility beyond Theorem 3.2; only the overlap's Lipschitz property and the exponential concentration of the Gibbs weights are needed, so the barrier should apply to any local search whose stationary distribution is approximately Gibbs, including simulated tempering or parallel tempering variants.","A numerical evaluation of the variational formula for finite $\\rho$ (for example $\\rho=10^{-3}$) could chart the finite-size precursor of the gap and may show the OGP appears already at moderate $N$, providing a direct test of the proof's scaling predictions.","For the $k=o(N)$ sparse PCA analogue, the same two-species Lagrange-multiplier decoupling is a natural starting point for an overlap-gap proof, suggesting the barrier found here is not an artifact of the constant-density constraint."],"forward_implications":["Approximate support recovery is information-theoretically possible in the whole OGP window, yet no reversible local MCMC algorithm can achieve it in polynomial time from a random start; the gap between statistical possibility and local algorithmic success is rigorous, not conjectural.","The threshold for the MLE is sharp up to a constant factor: recovery succeeds for $\\lambda>(2+\\varepsilon)\\sqrt{(1/\\rho)\\log(1/\\rho)}$ and fails when $\\lambda=o(\\sqrt{(1/\\rho)\\log(1/\\rho)})$.","For $\\lambda>1/\\rho$, rounding the leading eigenvector of $A$ recovers a constant fraction of the support, so the hard phase sits strictly between the information-theoretic threshold and the spectral threshold.","As a by-product, the first-order asymptotics of the largest sum of entries over all $N\\rho\\times N\\rho$ principal submatrices of an i.i.d. Gaussian matrix is determined by the value $E(\\rho^2;\\rho,0)/\\sqrt{2}$.","The barrier becomes more pronounced as $\\rho\\to 0$: the difficult window in $\\lambda$ widens with sparsity."],"supporting_citations":[{"why":"It supplies the cavity-method lower bound for the free energy of the constrained two-species model.","marker":"[4]"},{"why":"It provides the zero-temperature $\\Gamma$-convergence result that turns the finite-temperature Parisi functional into the ground-state energy formula.","marker":"[50]"},{"why":"It supplies the Ruelle probability cascade expectation formulas and their correspondence with Parisi-type PDEs, used throughout the interpolation argument.","marker":"[7]"},{"why":"It supplies the ultrametricity theorem that reduces the limiting overlap distribution to Ruelle cascades in the lower bound.","marker":"[67]"},{"why":"It provides the Ghirlanda-Guerra identities and the general mean-field framework for overlap distributions.","marker":"[68]"},{"why":"It supplies the Slepian comparison inequality and Gaussian concentration used to control the maximum of the noise Hamiltonian.","marker":"[22]"},{"why":"It supplies the largest-eigenvalue phase transition used by the spectral rounding algorithm for $\\lambda>1/\\rho$.","marker":"[12]"},{"why":"It supplies the data-processing and Gaussian channel capacity inequalities used to prove the information-theoretic impossibility part of Theorem 1.2.","marker":"[32]"}],"fun_headline_variants":["Overlap gap traps local samplers in submatrix recovery","Landscape split pins MCMC in submatrix recovery","Moderate signals create hard landscape for MCMC","OGP proves MCMC fails near optimal recovery","Hidden matrix recovery: OGP blocks Markov chains"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the claim that the rescaled maximum-likelihood value at every fixed overlap is exactly described by a tractable mean-field variational formula; if that formula is wrong for any overlap between the trivial solution and the good solution, the disconnectedness of the landscape and the sampling barrier both vanish.","fun_headline_variants_meta":{"raw":{"variants":["Overlap gap traps local samplers in submatrix recovery","Landscape split pins MCMC in submatrix recovery","Moderate signals create hard landscape for MCMC","OGP proves MCMC fails near optimal recovery","Hidden matrix recovery: OGP blocks Markov chains"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000373,"raw_usage":{"total_tokens":2075,"prompt_tokens":1110,"completion_tokens":965,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":726,"completion_tokens_details":{"reasoning_tokens":890}},"tokens_in":726,"tokens_out":965,"duration_ms":7710,"temperature":1.0,"reasoning_tokens":890,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:57:50.290098+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $E(q;\\rho,\\lambda)$ from the variational formula for a concrete pair inside the claimed window, say $\\rho=10^{-3}$ and $\\lambda=10\\sqrt{(1/\\rho)\\log(1/\\rho)}$, and check the sign of $\\partial_q E$ at $q=\\rho^2$ and at $q=\\rho^{2-\\delta}$ with $\\delta\\approx 0.2$; if the derivative is nonnegative at $q=\\rho^{2-\\delta}$, or if the energy curve has no interior local maximum between $\\rho^2$ and $\\rho^{1+\\varepsilon}$, then the overlap gap property, and with it the exponential hitting-time bound, fails exactly where Theorem 1.4 claims it holds.","supporting_citations":[{"cited_title":"Extended variational principle for the sherrington- kirkpatrick spin-glass model","cited_arxiv_id":null,"evidence_quote":"It supplies the cavity-method lower bound for the free energy of the constrained two-species model."},{"cited_title":"On the unbalanced cut problem and the generalized sherrington- kirkpatrick model","cited_arxiv_id":null,"evidence_quote":"It provides the zero-temperature $\\Gamma$-convergence result that turns the finite-temperature Parisi functional into the ground-state energy formula."},{"cited_title":"Spin glass computations and Ruelle’s probability cascades","cited_arxiv_id":null,"evidence_quote":"It supplies the Ruelle probability cascade expectation formulas and their correspondence with Parisi-type PDEs, used throughout the interpolation argument."},{"cited_title":"The Parisi ultrametricity conjecture","cited_arxiv_id":null,"evidence_quote":"It supplies the ultrametricity theorem that reduces the limiting overlap distribution to Ruelle cascades in the lower bound."},{"cited_title":"The Sherrington-Kirkpatrick model","cited_arxiv_id":null,"evidence_quote":"It provides the Ghirlanda-Guerra identities and the general mean-field framework for overlap distributions."},{"cited_title":"Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices","cited_arxiv_id":null,"evidence_quote":"It supplies the largest-eigenvalue phase transition used by the spectral rounding algorithm for $\\lambda>1/\\rho$."},{"cited_title":"Elements of information theory john wiley & sons","cited_arxiv_id":null,"evidence_quote":"It supplies the data-processing and Gaussian channel capacity inequalities used to prove the information-theoretic impossibility part of Theorem 1.2."}],"review_version":1}