{"id":"92b8a323-87bd-4842-aa80-6089faaf7386","arxiv_id":"2507.04983","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A self-normalized detector based on the largest eigenvalues of deformed Wigner matrices detects online, with controlled false alarm rate, the moment a latent signal crosses the detectability threshold.","lead":"This paper builds a statistical test that watches a stream of noisy, matrix-valued data and sounds an alarm when a hidden signal becomes strong enough to be visible. The test uses the leading eigenvalue of each matrix, a self-normalized detector, and a theory proving that the alarm threshold does not depend on unknown noise levels.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.2's proof requires the support of s/σ to be bounded away from 1, an assumption absent from the model and violated by the Uniform(0,1) simulations; this unproven rigidity/Tracy-Widom step underpins Theorem 3.8.","rationale":"The reader's weakest assumption coincides with the most load-bearing gap I find. The central claim is Theorem 3.8, whose proof cascades from the rigidity and Tracy-Widom fluctuation of the largest eigenvalue under the null (Lemma 3.2). The proof of that lemma is the only place where the distribution of the signal strength enters, and it does so through a compact-support condition that is neither stated in Assumption 3.1 nor implied by the model's 'support in [0,σ)'. The simulations use exactly the kind of P(1) (Uniform(0,1), Beta(2,4)) for which the support's supremum equals σ, making the Lee-Schnelli condition fail. I do not claim the theorem is false; the near-critical mass may be negligible and the result may be salvageable via other RMT tools. But as written, the proof does not cover the claimed assumptions, and every later step (truncation, Gaussian approximation, tail bound) depends on Lemma 3.2. This warrants a conditional verdict: the paper should either add the missing assumption to the model (and avoid claiming the uniform case), or provide a proof of Lemma 3.2 that handles support approaching the edge. A secondary concern is that equation (4.1) for the critical-value simulation appears to differ from the correct Monte Carlo expression derived from (3.4)-(3.5) (the term −(m+k)/m W(m) should be −m W(m)); this affects reproducibility of the reported quantiles, though Table 1 suggests the printed formula may be a typo. This does not change the verdict but should be corrected. Overall, I agree with the reader's CONDITIONAL assessment: the central idea is sound and the paper is valuable, but the proof has a concrete, identified gap that must be resolved before Theorem 3.8 can be accepted as stated.","tokens_in":32326,"tokens_out":29081,"duration_ms":263947,"concrete_test":"Run a focused Monte Carlo check of Lemma 3.2(b) in the design of Section 4.1: set σ=1, n=10^4, generate s ~ Uniform(0,1) and x uniform on the sphere, form M = sxx^T + W/√n with GOE W, and record n^{2/3}(λ_max - 2) over 10^5 replications; compare the empirical distribution with Tracy-Widom via a KS test. If the KS statistic exceeds the 0.95 threshold, the missing compact-support assumption is substantive for the theorem's range of applicability; if it matches within Monte Carlo error, the gap is purely in the proof and can be closed by replacing the Lee-Schnelli citation with an edge-universality theorem valid for spikes approaching the edge, which would also make the stated Assumptions sufficient.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Lemma 3.2 (Section 5.5), the authors verify Lee-Schnelli's condition (5.33) by fixing λ_l, λ_r with λ_r < 1 such that the support of λ = s/σ is contained in [λ_l, λ_r]. They then use 1/x^2 ≥ 1/λ_r to get ∫(v-x)^{-2} dF^V(v) ≥ (n-1)/(n λ_r) ≥ 1+ε for large n. The model only assumes P(1) has support in [0,σ), which does not ensure any λ_r < 1 exists. For the simulation design in Section 4.1 with P(1) = Uniform(0,1) and σ=1, the support of s/σ equals [0,1); for every λ_r < 1, P(s/σ > λ_r) > 0. Then inf_{x∈[λ_l,λ_r]} ∫(v-x)^{-2} dF^V(v) ≤ (n-1)/n + o(1) ≈ 1, so (5.33) cannot hold for any fixed ε>0. Consequently, the cited edge universality and rigidity results [48, 44] do not apply. Lemma 3.2(a)-(b) are the basis for the truncation events Λ_t (3.8), the early replacement property (Remark 5.1), Lemma 5.3, and hence the Gaussian approximation (Theorem 3.6) and tail bound (Lemma 3.7) that prove Theorem 3.8. Without an additional argument handling signals with support approaching σ, the central claim is unproven for the settings actually simulated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a monitoring procedure for a time series of deformed Wigner matrices, aiming to detect in real time the transition from a subcritical to a supercritical signal. The proposed detector is a self-normalized functional of cumulative sums of the largest eigenvalues, designed so that all nuisance parameters cancel and the limit is a pivotal Brownian functional. The main theoretical result, Theorem 3.8, claims weak convergence of the detector under H0, and Corollary 3.1 claims asymptotic level control and power consistency under H1. The proof strategy combines eigenvalue rigidity and Tracy-Widom fluctuations for subcritical deformed Wigner matrices with Gaussian approximations for weakly dependent triangular arrays. The paper also reports simulations and two applications.","tokens_in":32655,"tokens_out":24974,"duration_ms":284275,"significance":"If the gaps identified below are repaired, this is a valuable and original contribution: it is, to my knowledge, the first monitoring procedure for an eigenvalue phase transition in a stream of random matrices, and the self-normalization construction is genuinely parameter-free. The Gaussian approximation for the partial-sum process of leading eigenvalues is innovative, and the empirical sections demonstrate that the method can work in realistic finite samples. The main theoretical mechanism is credible and uses appropriate external results, but the manuscript is not currently acceptable because the central theorem is not proved for the settings that are actually simulated, and the power claim is overbroad.","major_comments":[{"comment":"The proof of Lemma 3.2 chooses 0 ≤ λ_l ≤ λ_r < 1 independent of n such that the support of λ = s/σ is contained in [λ_l, λ_r], and uses λ_r < 1 to verify condition (5.33) of Lee and Schnelli. This uniform support-gap condition is not stated in Assumption 3.1 or in the model specification before (1.3), and it fails for the Uniform(0,1) design of Section 4.1 when σ = 1, since that distribution has mass arbitrarily close to 1. Near-critical values of s can change the rigidity and fluctuation scales of the largest eigenvalue, so the cited results [48, 44] do not justify Lemma 3.2(a)-(b) for such P(1). Because Lemma 3.2 drives the truncation events Λ_t in (3.8), Remark 5.1, Lemma 5.3, and hence Theorems 3.6, 3.7, and 3.8, the central claim is unproved for the settings actually simulated. The authors should either add a uniform gap assumption to the model and adjust the simulations accordingly, or provide a separate argument covering supports that accumulate at σ.","section":"Section 5.5, Lemma 3.2"},{"comment":"The normalizing definitions are internally inconsistent. In (3.11), Y_t is defined without the factor n^{2/3} that appears in the same Y_t in (5.2), and the denominator in (3.9) as printed is √m τ_n rather than √(mτ_n). Theorem 3.6 states a coupling of P_m to √τ_n B_m, while the proof in Section 5.1 concludes with a coupling to a standard Brownian motion B_m, and the proof of Theorem 3.8 also substitutes a standard Brownian motion. As written, Theorem 3.6 cannot be true for all three formulations simultaneously, and the reader cannot verify the pivotal cancellation that leads to (3.6). These displays need to be reconciled.","section":"Section 3.3, Eqs. (3.9)-(3.11), Theorem 3.6"},{"comment":"The power proof analyzes only the case k* ≍ m and asserts that all other cases are analogous. This is not correct for small k*. For fixed k*, the dominant term R_3 in the decomposition of D_m(2k*) is of order n^{2/3} m^{-1/2}, while (5.24) shows that V_m/√τ_n is tight. Under Assumption 3.4 with θ > 4/3, this ratio tends to 0, so the claimed power 1 for every D ≥ 0 in (3.13) is false. The statement of Corollary 3.1(b) must either restrict k* to be sufficiently large relative to θ, or the proof must contain a separate argument for early change points.","section":"Section 5.4, Corollary 3.1(b)"}],"minor_comments":[{"comment":"The phrase \"Exiting approaches\" should be \"Existing approaches\".","section":"Section 2"},{"comment":"In the display defining the φ-mixing coefficients, the expression contains a duplicated \"sup_n sup_n\" that should be cleaned up.","section":"Section 3.1"},{"comment":"The phrase \"null sequence null-sequence\" is a duplicate and should be reduced to a single description of the sequence (d_m).","section":"Section 5.1"},{"comment":"The proof of Lemma 3.7 refers to \"Lemma C.1 of [46]\" without stating the version used; since [46] is a functional-data paper, the relevant lemma and its hypotheses should be stated explicitly.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The support-gap issue is the most important correctness problem: it affects the paper's own simulation design, not merely an edge case. If the authors add a support-gap assumption, they must also revisit the simulation section, because Uniform(0,1) with σ=1 would have to be replaced by a distribution whose support is bounded away from 1. The near-critical regime may be of practical interest, so I would not advise rejecting outright, but the current version overclaims the range of validity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a careful referee, but the referee will need to sit with the proof of Lemma 3.2. The paper develops a sequential detector for the emergence of a supercritical signal in a stream of deformed Wigner matrices, based on partial sums of the leading eigenvalue, and self-normalizes so the limit is pivotal. The Gaussian approximation for the partial sum process (Theorem 3.6) and the tail bound for the detector at large horizons (Lemma 3.7) are substantial pieces of work, and the self-normalization strategy that avoids assuming a long-run variance exists is a real extension of the method. This is the first monitoring procedure for phase transitions in this matrix model, and it does not hide a fitted nuisance parameter.\n\nThe soft spot is real and load-bearing. In the proof of Lemma 3.2 (Section 5.5), the authors verify the Lee-Schnelli condition (5.33) by fixing λ_r < 1 with the support of s/σ inside [λ_l, λ_r]. The stated model only requires support in [0, σ). For the Uniform(0,1) and Beta(2,4) designs with σ=1, the support reaches 1, so no such λ_r exists, and the integral in (5.33) is at most ~1, so the condition fails for any fixed ε. Without Lemma 3.2, the eigenvalue rigidity and Tracy-Widom fluctuations that drive the truncation events, the early replacement property, and eventually the Gaussian approximation and Theorem 3.8, are not justified. In short, the paper proves its headline result only under an assumption its own simulations violate.\n\nTwo smaller points: the definition of Y_t is inconsistent between (3.11) and (5.2), and the critical values are simulated on a finite horizon T=500; the latter looks stable in the table, so I would not hold it against them.\n\nThe paper is for researchers in change-point monitoring and random matrix theory. The idea is fresh and the execution is careful. I would send it to referees with an explicit request to work through Section 5.5 and the simulations. It needs a major revision, not a desk rejection.","headline":"Genuinely new monitoring method for streams of deformed Wigner matrices; the main theorem has a gap that excludes the paper's own simulations, so it needs a careful referee.","tokens_in":33200,"tokens_out":4262,"would_cite":false,"duration_ms":44912,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H15","62L10","62M10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A monitoring test watches the largest eigenvalue of a stream of deformed Wigner matrices and declares a supercritical phase transition the moment a latent rank-one signal becomes detectable; self-normalization makes the limiting null…","keywords":["change point analysis","high-dimensional matrices","self-normalization","sequential test","time series","deformed Wigner matrices","BBP phase transition","leading eigenvalue"],"falsifier":"Generate the model with $P^{(1)}$ supported on $[0, \\sigma - n^{-\\gamma}]$ for a slowly decaying gap and compare the empirical false-alarm rate of the test to the nominal level; if the rate drifts away from $\\alpha$ as $n, m$ grow, or if $n^{2/3}(\\lambda - 2\\sigma)$ under this design visibly departs from the Tracy-Widom distribution, the theorem's stated generality is false.","tokens_in":32073,"feed_emoji":"📈","tokens_out":12627,"duration_ms":119473,"temperature":0.7,"pith_summary":"This paper develops an online test for the moment a hidden rank-one signal in a noisy matrix stream becomes strong enough to be seen. The model is a time series of deformed Wigner matrices $M_t = s_t x_t x_t^\\top + W_t/\\sqrt{n}$; while the signal strength $s_t$ is below the noise scale $\\sigma$, the largest eigenvalue stays near the noise edge $2\\sigma$ with universal Tracy-Widom fluctuations, and only when $s_t > \\sigma$ does it separate to $s_t + \\sigma^2/s_t$. The proposed detector aggregates the largest eigenvalues of the monitoring windows after a training period and divides by a training-sample normalizer, so the test statistic is self-normalized and needs no unknown parameters. The main result shows that under the null hypothesis the supremum of this detector converges to a pivotal Brownian functional, giving asymptotic level control, and under the alternative it diverges, giving power one. The practical payoff is a procedure that can flag the transition in real time as new matrix-valued observations arrive.","feed_headline":"Top eigenvalue of a noisy matrix stream reveals when a signal appears","feed_subtitle":"Self-normalized detector reaches a pivotal limit, so no unknown variances need estimating.","key_machinery":"The carrying object is the self-normalized detector $\\Gamma_{n,m}(k) = D_m(k)/V_m$, where $D_m(k) = \\frac{n^{2/3}\\sqrt{m}}{m+k}\\left(\\sum_{t=m+1}^{m+k}\\lambda_t - \\frac{k}{m}\\sum_{t=1}^m \\lambda_t\\right)$ compares the post-training average largest eigenvalue with the training average, and $V_m = \\frac{n^{2/3}}{m^{3/2}}\\sum_{t=1}^m \\left|\\sum_{s=1}^t \\lambda_s - \\frac{t}{m}\\sum_{s=1}^m \\lambda_s\\right|$ is the training-sample normalizer that cancels the variance scale. The supporting machinery is a Gaussian approximation (Theorem 3.6) that couples the eigenvalue partial-sum process $P_m(x) = \\frac{1}{\\sqrt{m}\\tau_n}\\sum_{t=1}^{\\lfloor mx \\rfloor} n^{2/3}[\\lambda_t - b(n)]$ to a Brownian motion on intervals of length $T_m \\asymp m^{1/2-\\rho}$, proved by truncating eigenvalues to the event $\\Lambda_t = \\{|\\lambda_t - 2\\sigma| < n^{\\varepsilon - 2/3}\\}$, bounding the truncated variables, concentrating the finite-sample variance $\\tau_n$, and applying a functional CLT for $\\phi$-mixing arrays; a tail bound (Lemma 3.7) controls all later $k$. The truncation is what turns the difficult eigenvalue process into a sum of bounded, weakly dependent, centered variables.","core_discovery":"The central claim is that the emergence of a supercritical signal in a time series of spiked Wigner matrices can be detected online from the extremal eigenvalues alone. Under the null hypothesis that all signal strengths stay subcritical, the self-normalized detector $\\Gamma_{n,m}(k) = D_m(k)/V_m$ converges in distribution, as $m, n \\to \\infty$ with $m \\asymp n^\\theta$, to $\\sup_{0 \\le x < \\infty} [B(1+x) - B(1) - xB(1)] / [(1+x)\\int_0^1 |B(s) - sB(1)|\\,ds]$, where $B$ is standard Brownian motion. Because the limit is pivotal, rejecting when $\\Gamma_{n,m}(k)$ exceeds the upper-$\\alpha$ quantile gives an asymptotically exact test of level $\\alpha$. Under the alternative, once a signal with strength in the supercritical range appears at some unknown time, the detector tends to infinity and the test is consistent. The proof combines a Gaussian approximation for the eigenvalue partial-sum process with eigenvalue truncation and tail bounds, and the self-normalization cancels the unknown variance scale in finite samples rather than estimating a long-run variance.","pith_inferences":["The self-normalization strategy should transfer to other high-dimensional spectral statistics with Tracy-Widom-type edge fluctuations, such as leading singular values of spiked covariance matrices, where moment bounds for the edge are likewise unavailable.","A natural extension is to monitor several leading eigenvalues or the eigenvector overlap in addition to the top eigenvalue; the pivotal limit would change, but the finite-sample variance cancellation could be preserved.","The simulations place $P^{(1)}$ on $[0,1]$ while the proof needs the subcritical support to stay bounded away from $\\sigma = 1$; a reader can test whether the empirical size degrades as the support approaches $1$, which would mark the practical boundary of the theorem.","Because the limiting functional involves only Brownian motion, the same critical values apply to any model admitting a comparable Gaussian approximation, making this construction a template for other matrix-stream monitoring problems."],"forward_implications":["The test has asymptotic size $\\alpha$ under the null hypothesis, so a stream with no detectable signal produces false alarms only at the nominal rate, with no unknown variance or signal distribution to estimate.","Under the alternative, once a supercritical signal appears, the probability that the detector eventually exceeds the threshold tends to one, so the transition is identified in the limit.","The critical values depend only on the Brownian functional; the paper reports $q_{0.90}=4.57$ and $q_{0.95}=5.85$ from simulation, making implementation a matter of comparing a single path to a fixed threshold.","Weak dependence (exponentially decaying $\\phi$-mixing coefficients) and a polynomial relation $m \\asymp n^\\theta$ between training length and dimension are allowed, covering realistic streaming regimes.","In the paper's applications, the detector flags the Cameron Peak Fire from PM2.5 data on August 19, 2020, a few days after the first report, and remains below threshold for the baboon social-interaction data until a synthetic signal is added, which is then detected quickly."],"supporting_citations":[{"why":"Supplies the eigenvalue rigidity and Tracy-Widom fluctuations of the largest eigenvalue in the subcritical case (Lemma 3.2).","marker":"[48]"},{"why":"Provides the universality result that extends the diagonal-deformation results to general rank-one random perturbations.","marker":"[44]"},{"why":"Gives the supercritical eigenvalue limit used in Lemma 3.3 and in the power proof.","marker":"[51]"},{"why":"Supplies the form of the cumulative-sum detector on which $D_m(k)$ is modeled.","marker":"[34]"},{"why":"Provides the functional CLT for $\\alpha$-mixing triangular arrays used in the Gaussian approximation (Theorem 3.6).","marker":"[32]"},{"why":"Gives the moment inequalities for mixing sequences used to bound the partial-sum process in the Gaussian approximation and tail arguments.","marker":"[65]"},{"why":"Supplies the sup-norm bound for tail sums invoked in the proof of Lemma 3.7.","marker":"[46]"},{"why":"Provides the fast decay of supercritical eigenvalue deviations used to prove power consistency in Corollary 3.1.","marker":"[43]"},{"why":"Supplies the covariance bound for bounded mixing variables used to show variance concentration in Lemma 5.5.","marker":"[17]"}],"fun_headline_variants":["Spiked Wigner streams: detect supercritical onset from top eigenvalues","Self-normalized detector for online matrix phase transitions","Real-time detection of latent signals in random matrix time series","Pivotal limit test spots signal birth in deforming matrices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that subcritical signal strengths stay strictly below the critical threshold $\\sigma$, because the proof of Lemma 3.2 requires the support of $s/\\sigma$ to lie in $[\\lambda_l, \\lambda_r]$ with $\\lambda_r < 1$; Assumption 3.1 only states support in $[0,\\sigma)$, so the argument fails if that support approaches $\\sigma$.","fun_headline_variants_meta":{"raw":{"variants":["Spiked Wigner streams: detect supercritical onset from top eigenvalues","Self-normalized detector for online matrix phase transitions","Real-time detection of latent signals in random matrix time series","Pivotal limit test spots signal birth in deforming matrices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00064,"raw_usage":{"total_tokens":2955,"prompt_tokens":961,"completion_tokens":1994,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":1926}},"tokens_in":577,"tokens_out":1994,"duration_ms":17303,"temperature":1.0,"reasoning_tokens":1926,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:36:06.480868+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate the model with $P^{(1)}$ supported on $[0, \\sigma - n^{-\\gamma}]$ for a slowly decaying gap and compare the empirical false-alarm rate of the test to the nominal level; if the rate drifts away from $\\alpha$ as $n, m$ grow, or if $n^{2/3}(\\lambda - 2\\sigma)$ under this design visibly departs from the Tracy-Widom distribution, the theorem's stated generality is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the eigenvalue rigidity and Tracy-Widom fluctuations of the largest eigenvalue in the subcritical case (Lemma 3.2)."},{"cited_title":"and Y IN, J","cited_arxiv_id":null,"evidence_quote":"Provides the universality result that extends the diagonal-deformation results to general rank-one random perturbations."},{"cited_title":"and S OSHNIKOV , A","cited_arxiv_id":null,"evidence_quote":"Gives the supercritical eigenvalue limit used in Lemma 3.3 and in the power proof."},{"cited_title":"and STEINEBACH , J","cited_arxiv_id":null,"evidence_quote":"Supplies the form of the cumulative-sum detector on which $D_m(k)$ is modeled."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the functional CLT for $\\alpha$-mixing triangular arrays used in the Gaussian approximation (Theorem 3.6)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the moment inequalities for mixing sequences used to bound the partial-sum process in the Gaussian approximation and tail arguments."},{"cited_title":"and K OKOSZKA , P","cited_arxiv_id":null,"evidence_quote":"Supplies the sup-norm bound for tail sums invoked in the proof of Lemma 3.7."},{"cited_title":"and Y IN, J","cited_arxiv_id":null,"evidence_quote":"Provides the fast decay of supercritical eigenvalue deviations used to prove power consistency in Corollary 3.1."},{"cited_title":"and S ØRENSEN , M","cited_arxiv_id":null,"evidence_quote":"Supplies the covariance bound for bounded mixing variables used to show variance concentration in Lemma 5.5."}],"review_version":1}