{"id":"48c10d1c-c87c-485c-9057-a16455e35a12","arxiv_id":"2501.07975","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Unobservable symmetries in Bayesian inference require quotienting the error metric, and for extensive-rank matrix factorization this leads to a three-level optimization problem that may be inaccessible to local message-passing methods.","lead":"This note shows when symmetries help or hurt Bayesian inference, and that unobservable invariances force an orbit-aware error metric whose optimal estimator is hard to compute. It matters for the extensive-rank matrix factorization problem, where the standard statistical mechanics toolkit appears to fail.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim that the extensive-rank quotiented estimator cannot be obtained from local posterior marginals rests on an unproven assumption that the extensive symmetry group Sr does not spontaneously break; the paper itself flags this in Sec. V.","rationale":"The reader's verdict is CONDITIONAL, with the weakest assumption being exactly the open question of spontaneous breaking of an extensive symmetry group. My stress-test identifies the same concern as the most load-bearing point: the paper's strong methodological conclusion ('not expressible in terms of local marginal laws', Sec. IV) goes beyond what is proven. The elementary derivations in the paper are correct: Eq. (24) follows from the definition of the quotiented distance, Eq. (26) is the correct Bayes-optimality condition, and the examples in Sec. IV are internally consistent. The paper also honestly flags both limitations, the non-locality of the estimator and the unclear spontaneous-breaking behavior, in Sec. V. However, the argument that the replica/cavity/I-MMSE machinery is inadequate rests on the assumption that the extensive symmetry group cannot be broken into pure states. The paper does not prove this; it only states that the situation is 'much less clear.' Since the finite-group SBM example works precisely because spontaneous breaking occurs, the extensive-rank case is not settled by the paper's analysis. This does not require changing the reader's CONDITIONAL verdict, but it does confirm that the conditionality is justified. The concrete test I propose, comparing symmetric and symmetry-broken replica free energies in a specific model, would settle whether the premise holds or fails, and therefore whether the methodological conclusion is valid or overreaching.","tokens_in":18460,"tokens_out":3116,"duration_ms":35947,"concrete_test":"Choose a tractable extensive-rank model, e.g., X with i.i.d. standard Gaussian entries, n rows and r = alpha n columns, with observations Y = XX^T + Z for Gaussian noise Z. Compute the replica free energy at fixed alpha and signal-to-noise ratio under two ansatze: (i) the fully symmetric ansatz, and (ii) a permutational symmetry-breaking ansatz where the replica order parameter is a permutation matrix or a convex combination of permutation matrices. If, for some alpha and SNR, the symmetry-broken free energy is strictly lower than the symmetric one, then the posterior does spontaneously break Sr, and the claim that local marginal-based replica/cavity methods are inadequate for the quotiented distance is invalidated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central methodological conclusion is that, for extensive-rank matrix factorization with column i.i.d. prior, the Bayes-optimal estimator for the quotiented distance dG in Eq. (24) must be computed from the three-level problem of Eq. (26), and that the usual replica/cavity machinery based on the graphical model (1) is inadequate because the estimator is not expressible in terms of local posterior marginals. This conclusion is load-bearing for the paper's main message, but it depends on an unproven premise: that the posterior measure cannot spontaneously break the extensive symmetry group Sr into pure states in the thermodynamic limit. The paper explicitly says 'It is much less clear that such a phenomenon can occur when the symmetry group grows' (Sec. V), and it offers no argument, such as a free-energy barrier estimate or a cluster-size bound, that would rule out spontaneous breaking. In the finite-symmetry SBM case, the same difficulty is resolved precisely by spontaneous breaking: symmetry-breaking solutions of the cavity equations select pure states, and the quotiented-distance estimator is recovered from local marginals of one pure state. If an analogous phenomenon occurs for Sr with r proportional to n, then the replica/cavity approach would remain viable and the statement that Eq. (26) 'is not expressible in terms of local marginal laws of the posterior probability' (Sec. IV) would be false. The paper provides observations and a clear formulation of the three-level problem, but not a proof of the impossibility that its methodological conclusion requires.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This note examines the role of symmetries in Bayesian inference, distinguishing beneficial symmetries that reduce the complexity of low-degree polynomial estimators from detrimental unobservable symmetries that require quotiented distances. The paper derives the Bayes-optimal estimator for the quotiented distance dG in a generic setting and specializes to the extensive-rank matrix factorization problem, where the column-permutation symmetry Sr leads to a three-level optimization problem (Eq. (26)) involving a bipartite matching inner problem. It argues that this estimator is not expressible in terms of local posterior marginals and therefore challenges the standard replica/cavity approach, while leaving the possibility of spontaneous symmetry breaking as an open question.","tokens_in":18662,"tokens_out":8684,"duration_ms":82916,"significance":"If the conclusions hold, the paper clarifies why the extensive-rank matrix factorization problem has resisted standard statistical mechanics treatments and provides a concrete starting point for future work. The derivations in Sec. IV are clean and correct, the examples are instructive, and the paper is honest about the open issues. The main value is conceptual; the paper does not prove the impossibility of local-marginal formulations, but it formulates the correct objective and highlights the role of the growing symmetry group.","major_comments":[{"comment":"The central methodological claim that the Bayes-optimal estimator for the quotiented distance is 'not expressible in terms of local marginal laws of the posterior probability PX|Y' is stated categorically in Sec. IV but depends on an unproven premise: the absence of spontaneous breaking of the extensive symmetry group Sr. The paper itself acknowledges in Sec. V that 'It is much less clear that such a phenomenon can occur when the symmetry group grows.' In the SBM case, spontaneous breaking of the finite symmetry group is precisely what allows the quotiented-distance estimator to be recovered from local marginals of a pure state. If an analogous phenomenon occurs for Sr with r proportional to n, the statement in Sec. IV would be false. The author should either provide an argument (e.g., a free-energy barrier estimate or a cluster-size bound) that spontaneous breaking cannot occur, or explicitly reformulate the claim as conditional on the symmetric posterior and discuss how symmetry-broken solutions of the replica/cavity equations would alter the conclusion.","section":"Sec. IV (Eq. (26)) and Sec. V"}],"minor_comments":[{"comment":"The phrase 'local marginal laws of the posterior probability PX|Y' is ambiguous: it could mean the marginals of the full (symmetric) posterior or the marginals of a pure state. Clarifying this distinction would prevent a misreading, since the SBM example shows that marginals of a pure state can suffice.","section":"Sec. IV, last paragraph"},{"comment":"The phrase 'the consequences this implies for the statistical mechanics treatment' is stronger than the exploratory tone of the note supports; consider softening to 'the potential consequences' or 'the challenges this poses.'","section":"Abstract"},{"comment":"The observation model in Eq. (21), Y_p = \\sum_i (S_i)^p, is artificial; a brief remark that it is chosen for illustrative purposes would help the reader calibrate the scope of the example.","section":"Sec. IV, example with power sums"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a well-written note with correct illustrative calculations and an honest discussion of open issues. The main weakness is the categorical statement about local marginals, which is not supported by a proof of absence of spontaneous symmetry breaking for extensive groups. If the author addresses this with a clear conditional statement and a more detailed discussion of the spontaneous-breaking scenario, the paper would be suitable for publication. The fit with the journal is appropriate for a conceptual note."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: this is a clear, honest note, and the main observation is right. When a symmetry acts trivially on the observations, the error metric must be quotiented by that symmetry, and for extensive-rank matrix factorization that yields a three-level optimization with an assignment-problem inner max, not a local posterior-marginal computation. The paper's stronger methodological conclusion—that the replica/cavity/AMP machinery is inadequate—is not proven. It rests on the assumption that the extensive symmetry group Sr does not spontaneously break into pure states in the thermodynamic limit. The paper itself flags this as open in Sec. V, so the reader's conditional verdict is fair.\n\nWhat is genuinely useful: the explicit formulation of Eq. (26) and its connection to the linear assignment problem is a good reformulation, and the order-statistics/permutation example is a neat illustration of a quotiented distance that still admits a closed-form estimator. The beneficial-symmetry discussion follows Hunt-Stein/equivariance arguments and is correctly credited. The paper does not oversell: Eq. (24) is attributed to [32], and the examples are mostly pedagogical. No parameters are fitted and no circularity burdens the argument.\n\nSoft spots, in proportion: the statement that the optimal estimator is \"not expressible in terms of local marginal laws\" is asserted, not demonstrated. As the paper itself notes, in the finite-symmetry SBM case the same difficulty is resolved by spontaneous breaking: local marginals in one pure state do give the quotiented-distance estimator. Whether that can happen when r is proportional to n is genuinely open. No free-energy barrier or cluster-size bound is given. So the central claim is conditional on a negative answer to that question. That does not make the paper wrong; it makes it a programmatic note rather than a theorem.\n\nWho should read it: anyone working on extensive-rank matrix factorization or on statistical mechanics of inference with unobservable symmetries. It deserves a serious referee, because the reformulation is useful and the open problem is well-posed. I would recommend acceptance as a note after the author either explicitly frames the local-marginal claim as conditional or adds a discussion of what a spontaneous-breaking scenario would imply.\n\nBest,\n[Your name]","headline":"A clear, honest note that correctly identifies the quotient-distance issue in extensive-rank matrix factorization, but its strong claim that local statistical mechanics tools fail rests on an unproven no-spontaneous-breaking assumption.","tokens_in":19246,"tokens_out":2687,"would_cite":true,"duration_ms":27502,"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":"In Bayesian inference, unobservable signal symmetries force a quotiented error metric; in extensive-rank matrix factorization, Bayes-optimal estimation becomes a three-level problem with a linear-assignment inner step.","keywords":["Bayesian inference","symmetries in inference problems","quotiented distance","extensive-rank matrix factorization","statistical mechanics of disordered systems","linear assignment problem","equivariant estimators","low-degree polynomial method"],"falsifier":"Take $Y=XX^T+Z$ with $r=\\alpha n$ and i.i.d. prior entries, and compare, at growing $n$, the risk of the estimator from Eq. (26) (computed by solving the inner assignment problem numerically) with the risk predicted by the symmetric replica, cavity, or approximate-message-passing solution. If the local-marginal solution already achieves the same quotiented risk, the paper's claim that local posterior marginals are inadequate is refuted; alternatively, show that the posterior overlap distribution conditioned on one observation is a single peak rather than a mixture over column permutations, and the assumed absence of spontaneous symmetry breaking would be false.","tokens_in":18187,"feed_emoji":"🔄","tokens_out":9407,"duration_ms":89689,"temperature":0.7,"pith_summary":"This paper argues that symmetries in Bayesian inference cut both ways. A symmetry acting strongly on the observations can be exploited: restricting estimators to equivariant functions reduces the computational dimension without hurting accuracy. A symmetry acting only on the signal is detrimental: all points on the same orbit are indistinguishable, so the error measure must be quotiented by the group. The paper shows that for extensive-rank matrix factorization this quotiented error turns Bayes-optimal estimation into a three-level problem that is not expressible through local posterior marginals. The upshot is that the standard replica, cavity, and I-MMSE machinery, which is built on local marginal probabilities and the mutual information, is not the right tool for this regime.","feed_headline":"Hidden symmetries turn Bayes-optimal estimation into a matching problem","feed_subtitle":"For matrix factorization, unobservable column permutations force a quotiented distance and a three-level estimator.","key_machinery":"The load-bearing object is the quotiented distance $d_G(S,\\hat S)=\\inf_{g\\in G} d(g\\cdot S,\\hat S)$, which identifies all signals that differ by an unobservable symmetry. For extensive-rank matrix factorization with $G=\\mathrm{S}_r$ acting on the columns of $X$, this distance becomes the squared Frobenius norms minus twice the maximum over column permutations of the trace of a matching matrix; that inner maximum is the bipartite matching (linear assignment) problem. It is this inner optimization that turns estimation into a 'three-level problem': minimize over the estimator, average over the posterior, and maximize over the group element. The paper also uses the rearrangement inequality in the $\\mathrm{S}_N$ example to show that for some groups the quotiented distance collapses to an ordinary distance on ordered coordinates, which is why that case remains tractable.","core_discovery":"The paper's central claim is that unobservable invariances must be gauged away before defining the distance between signal and estimator, and that doing so changes the structure of the Bayes-optimal estimator. For a signal $S$ with symmetry group $G$ leaving observations invariant, the correct distance is $d_G(S,\\hat S)=\\inf_{g\\in G} d(g\\cdot S,\\hat S)$, and the optimal estimator minimizes $\\mathbb{E}[d_G(S,x)\\mid Y]$. In the extensive-rank matrix factorization problem $Y=XX^T+Z$ with column-i.i.d. prior, this gives the three-level formula of Eq. (26): minimize over $x$ the quantity $\\|x\\|^2-2\\,\\mathbb{E}[\\max_{\\pi\\in\\mathrm{S}_r}\\sum_{i,\\mu} X_{i,\\mu}x_{i,\\pi(\\mu)}\\mid Y]$. The inner maximization over permutations is a bipartite matching problem, so the estimator cannot be reduced to posterior expectations of simple functions of $X$. The paper concludes that the mutual information $I(S;Y)$ no longer controls the average risk through an I-MMSE-type relation, and that the graphical-model view of the posterior is an inadequate starting point in the extensive-rank regime.","pith_inferences":["This analysis implies that current approximate-message-passing state evolutions for extensive-rank matrix factorization, which track only low-dimensional order parameters, may be missing a genuinely matching-like order parameter; a testable consequence is that algorithms initialized with a permutation-aligned guess will outperform symmetric AMP at the same noise level.","The rearrangement-inequality example suggests a general design rule: when the symmetry group is a sorting group with totally ordered orbits, the quotiented distance becomes explicit and the optimal estimator is a posterior mean of an orbit representative; for groups like $\\mathrm{S}_r$ acting on the columns of a matrix with $n>1$, no such ordering exists, which is a structural reason for the diffi","The same quotiented-distance prescription should apply to supervised learning with label-invariant symmetries: if a loss function compares predictions with unquotiented distances, the symmetry of the data-generation process makes the learning target inconsistent; quotienting the loss by the group is a direct, testable modification of equivariant-network training.","The three-level structure is likely to appear also in dictionary learning and blind source separation, where column permutation and sign ambiguities are unobservable; this suggests the difficulty is not specific to matrix factorization but generic to extensive-rank inverse problems."],"forward_implications":["For any inference model with an unobservable symmetry, using the naive square-error or Hamming distance makes the Bayes-optimal estimator trivial (for example, the posterior mean vanishes), so the quotiented distance is mandatory to extract any useful estimate.","In the symmetric stochastic block model, finite symmetry groups allow the computation through symmetry-broken pure states; the paper's framework explains why this route is available there but doubtful when the symmetry group grows with system size.","The mutual information $I(S;Y)$ and the minimum mean-square error are no longer linked by the I-MMSE relation once the distance is quotiented, so information-theoretic thresholds do not directly give the quotiented-distance risk.","The Bayes-optimal estimator for extensive-rank matrix factorization must solve a bipartite matching problem inside a posterior average; the paper suggests a message-passing solution of the assignment problem as the inner step of a multi-level statistical-mechanics treatment.","Symmetries acting on the observations remain an asset: restricting to equivariant low-degree polynomials reduces the dimension of the variational space without increasing the risk, as in matrix denoising."],"supporting_citations":[{"why":"Introduced the statistical-mechanics treatment of matrix factorization that the paper revisits; its later failure motivates the search for a correct extensive-rank theory.","marker":"[43]"},{"why":"Extended the replica treatment of Bayes-optimal matrix factorization; together with [43] it is the object of the paper's critique.","marker":"[44]"},{"why":"The work that unveiled the incorrectness of the initial proposal, leaving the extensive-rank regime without a satisfactory statistical-mechanics treatment.","marker":"[45]"},{"why":"The stochastic block model analysis that first used a quotiented Hamming distance and symmetry-broken cavity solutions; the finite-symmetry benchmark against which the extensive-rank case is contrasted.","marker":"[23]"},{"why":"Supplies the rearrangement inequality used to compute the quotiented distance exactly in the $\\mathrm{S}_N$ example.","marker":"[67]"},{"why":"Provides the polynomial-time algorithm for the bipartite matching and assignment problem that is the inner step of Eq. (26), showing it is computationally tractable but analytically nontrivial.","marker":"[68]"},{"why":"Recent extensive-rank matrix denoising work that already discussed the quotiented distance and used a greedy approximation of the assignment; the paper positions its observations relative to this effort.","marker":"[32]"},{"why":"The I-MMSE identity connecting mutual information and mean-square error in Gaussian channels, which the paper argues ceases to hold for quotiented distances.","marker":"[69]"},{"why":"Establishes the symmetrization argument that equivariant low-degree polynomials are sufficient for optimal estimation, the basis of the 'symmetries as an asset' section.","marker":"[42]"},{"why":"Message-passing solution of the assignment problem, proposed as the inner-level solver for the three-level statistical-mechanics treatment.","marker":"[72]"}],"fun_headline_variants":["Unobservable symmetries turn Bayes-optimal estimation into matching","Bayes-optimal estimator becomes bipartite matching under gauged symmetries","Symmetries can help or hinder Bayesian inference — gauge them away","Invisible symmetries force a matching problem in optimal inference","For matrix factorization, unobservable permutations demand quotiented distance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusion that local statistical-mechanics tools fail for extensive rank rests on the assumption that the posterior distribution does not spontaneously split into components related by column permutations in the thermodynamic limit; if it did, symmetry-broken replica or cavity solutions could still compute the quotiented-distance estimator.","fun_headline_variants_meta":{"raw":{"variants":["Unobservable symmetries turn Bayes-optimal estimation into matching","Bayes-optimal estimator becomes bipartite matching under gauged symmetries","Symmetries can help or hinder Bayesian inference — gauge them away","Invisible symmetries force a matching problem in optimal inference","For matrix factorization, unobservable permutations demand quotiented distance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000724,"raw_usage":{"total_tokens":3206,"prompt_tokens":867,"completion_tokens":2339,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":2265}},"tokens_in":483,"tokens_out":2339,"duration_ms":15272,"temperature":1.0,"reasoning_tokens":2265,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:29:13.185186+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $Y=XX^T+Z$ with $r=\\alpha n$ and i.i.d. prior entries, and compare, at growing $n$, the risk of the estimator from Eq. (26) (computed by solving the inner assignment problem numerically) with the risk predicted by the symmetric replica, cavity, or approximate-message-passing solution. If the local-marginal solution already achieves the same quotiented risk, the paper's claim that local posterior marginals are inadequate is refuted; alternatively, show that the posterior overlap distribution conditioned on one observation is a single peak rather than a mixture over column permutations, and the assumed absence of spontaneous symmetry breaking would be false.","supporting_citations":[{"cited_title":"Barbier and N","cited_arxiv_id":null,"evidence_quote":"Introduced the statistical-mechanics treatment of matrix factorization that the paper revisits; its later failure motivates the search for a correct extensive-rank theory."},{"cited_title":"Equivalence of Approximate Message Passing and Low-Degree Polynomials in Rank-One Matrix Estimation","cited_arxiv_id":"2212.06996","evidence_quote":"Extended the replica treatment of Bayes-optimal matrix factorization; together with [43] it is the object of the paper's critique."},{"cited_title":"Sakata and Y","cited_arxiv_id":null,"evidence_quote":"The work that unveiled the incorrectness of the initial proposal, leaving the extensive-rank regime without a satisfactory statistical-mechanics treatment."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The stochastic block model analysis that first used a quotiented Hamming distance and symmetry-broken cavity solutions; the finite-symmetry benchmark against which the extensive-rank case is contrasted."},{"cited_title":"Singer and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the rearrangement inequality used to compute the quotiented distance exactly in the $\\mathrm{S}_N$ example."},{"cited_title":"Sigworth","cited_arxiv_id":null,"evidence_quote":"Provides the polynomial-time algorithm for the bipartite matching and assignment problem that is the inner step of Eq. (26), showing it is computationally tractable but analytically nontrivial."},{"cited_title":"Matrix Inference in Growing Rank Regimes","cited_arxiv_id":"2306.01412","evidence_quote":"Recent extensive-rank matrix denoising work that already discussed the quotiented distance and used a greedy approximation of the assignment; the paper positions its observations relative to this effort."},{"cited_title":"Hardy, J","cited_arxiv_id":null,"evidence_quote":"The I-MMSE identity connecting mutual information and mean-square error in Gaussian channels, which the paper argues ceases to hold for quotiented distances."},{"cited_title":"Lelarge and L","cited_arxiv_id":null,"evidence_quote":"Establishes the symmetrization argument that equivariant low-degree polynomials are sufficient for optimal estimation, the basis of the 'symmetries as an asset' section."},{"cited_title":"Altarelli, A","cited_arxiv_id":null,"evidence_quote":"Message-passing solution of the assignment problem, proposed as the inner-level solver for the three-level statistical-mechanics treatment."}],"review_version":1}