Spectral Signatures of Data Quality: Eigenvalue Tail Index as a Diagnostic for Label Noise in Neural Networks

Sp ectral Signatures of Data Qualit y: Eigen v alue T ail Index as a Diagnostic for Lab el Noise in Neural Net w orks Matthew Loftus 1 , ∗ 1 Indep endent r ese ar cher (Dated: Marc h 31, 2026) W e in vestigate whether sp ectral prop erties of neural net work weigh t matrices can predict test accuracy . Under controlled lab el noise v ariation, the tail index α of the eigenv alue distribution at the netw ork’s b ottleneck la yer predicts test accuracy with leav e-one-out R 2 = 0 . 984 (21 noise levels, 3 seeds p er level), far exceeding all baselines: the b est conv entional metric (F rob enius norm of the optimal lay er) achiev es LOO R 2 = 0 . 149. This relationship holds across three architectures (MLP , CNN, ResNet-18) and t w o datasets (MNIST, CIF AR-10). How ev er, under h yperparameter v ariation at fixed data qualit y (180 configurations v arying width, depth, learning rate, and weigh t deca y), all sp ectral and conv en tional measures are weak predictors ( R 2 < 0 . 25), with simple baselines (global L 2 norm, LOO R 2 = 0 . 219) slightly outperforming sp ectral measures (tail α , LOO R 2 = 0 . 167). W e therefore frame the tail index as a data quality diagnostic : a p ow erful detector of lab el corruption and training set degradation, rather than a universal generalization predictor. A noise detector calibrated on synthetic noise successfully identifies real human annotation errors in CIF AR-10N (9% noise detected with 3% error). W e identify the information-pro cessing b ottlenec k lay er as the lo cus of this signature and connect the observ ations to the BBP phase transition in spiked random matrix mo dels. W e also report a negative result: the level spacing ratio ⟨ r ⟩ is uninformative for w eight matrices due to Wishart universalit y . I. INTR ODUCTION A central question in deep learning theory is: why do overp ar ameterize d neur al networks gener alize? Netw orks with far more parameters than training examples rou- tinely ac hieve low test error, contradicting classical sta- tistical learning theory [ 1 ]. Understanding when and wh y generalization o ccurs—and predicting it without a held- out test set—remains an op en problem. Random matrix theory (RMT) offers a natural frame- w ork for studying w eight matrices. A t initialization, w eight matrices are random, and their sp ectral prop erties are well-c haracterized b y the Marchenk o-Pastur (MP) la w [ 2 ]. During training, the sp ectrum ev olves a wa y from MP as the net work learns structure from data. Martin and Mahoney [ 3 ] sho wed that well-trained netw orks de- v elop heavy-tailed eigenv alue distributions and prop osed the tail index α as a quality metric. Subsequen t w ork demonstrated that α predicts model quality (with a bias to ward CV models) rather than gener alization in the uni- v ersal sense [ 18 ], with similar findings in NLP [ 19 ]. In this work, we conduct a controlled study of sp ectral predictors under t w o distinct p erturbation axes: (1) lab el noise v ariation at fixed architecture, and (2) hyperparam- eter v ariation at fixed data quality . This design rev eals that sp ectral measures are p o w erful diagnostics for data qualit y degradation but not universal generalization pre- dictors. W e presen t both p ositiv e and negative results honestly . ∗ matthew.a.loftus@gmail.com A. Summary of contributions 1. Strong data quality prediction: The tail index α of the b ottlenec k lay er predicts test accuracy with LOO R 2 = 0 . 984 across 21 lab el noise levels with 3 seeds each, far exceeding all baselines ( R 2 ≤ 0 . 149) (Sec. VI I I ). 2. Honest scop e: Under h yp erparameter v ariation at fixed data qualit y , all measures (spectral and con ven tional) are w eak predictors ( R 2 < 0 . 25), and simple baselines slightly outp erform sp ectral mea- sures (Sec. IX ). 3. Bottleneck h yp othesis: The sp ectral signa- ture concen trates at the information-pro cessing b ottlenec k—the la yer with the highest compression ratio and sufficien t sp ectral resolution (Sec. VI I ). 4. Comprehensive baselines: W e compare tail α against effective rank, F robenius norms, sp ectral norms, and global L 2 norm, establishing clear rel- ativ e strengths (Sec. I I I B ). 5. Metho dological correction: The level spacing ratio ⟨ r ⟩ is uninformative for rectangular weigh t matrices due to Wishart univ ersality (Sec. IV ). 6. Theoretical framework: W e connect our ob- serv ations to the BBP phase transition and de- riv e an informal b ound on outlier eigenv alue count (Sec. X ). 2 I I. BA CK GROUND A. Marc henko-P astur la w F or a random matrix W ∈ R m × n with iid en tries of v ariance σ 2 , the eigenv alues of the Gram matrix S = 1 m W ⊤ W follow the Marchenk o-Pastur distribution with asp ect ratio γ = n/m and supp ort [ λ − , λ + ] where λ ± = σ 2 (1 ± √ γ ) 2 . (1) Eigen v alues outside this supp ort indicate learned struc- ture. B. Lev el spacing ratio The level spacing ratio ⟨ r ⟩ , defined as r i = min( s i , s i +1 ) / max( s i , s i +1 ) where s i = λ i +1 − λ i , dis- tinguishes uncorrelated (P oisson, ⟨ r ⟩ ≈ 0 . 386) from cor- related (GOE, ⟨ r ⟩ ≈ 0 . 531) eigenv alue statistics [ 4 ]. C. BBP transition The Baik-Ben Arous-P ´ ec h´ e (BBP) transition [ 5 ] sho ws that a rank-1 p erturbation of strength θ to a Wishart matrix produces an outlier eigen v alue ab ov e λ + if and only if θ > σ 2 √ γ . This provides a sharp threshold for when learned structure b ecomes sp ectrally visible. D. Hea vy-tailed self-regularization Martin and Mahoney [ 3 ] observ ed that trained net- w orks dev elop hea vy-tailed eigen v alue distributions, with the tail index α (estimated via the Hill estimator [ 7 ]) serving as a quality metric. They rep orted correlations b et w een α and rep orted test accuracy across published mo dels, but did not establish a quan titative predictive relationship under con trolled conditions. E. Generalization measures A num b er of norm-based generalization measures hav e b een prop osed, including path norms [ 9 ], sp ectral norms with margin [ 10 ], and P AC-Ba y es b ounds [ 11 ]. Jiang et al. [ 8 ] conducted a large-scale comparison of generaliza- tion measures, finding that no single measure dominates across all p erturbation types. Our work complements this by studying sp ectral measures sp ecifically and iden- tifying the p erturbation axis (data quality vs. hyperpa- rameters) as the key determinan t of predictive pow er. I II. METHODS A. Sp ectral observ ables F or each weigh t matrix W ∈ R m × n , we compute the eigen v alues { λ i } of S = 1 m W ⊤ W and measure: • T ail index α : Hill estimator [ 7 ] on eigenv alues ab o v e the 90th p ercentile. Low er α indicates heav- ier tails (more concentrated signal). • Effective rank (normalized): exp( H ( p )) /n where H ( p ) = − P p i log p i and p i = λ i / P λ j . V alues near 1 indicate uniform sp ectrum; v alues near 0 indicate rank-deficient. • Outlier fraction : F raction of eigenv alues ab o ve λ + = σ 2 (1 + √ γ ) 2 , where σ 2 is estimated from the initial (untrained) w eight matrix. • MP deviation : Kolmogorov-Smirno v statistic be- t ween the empirical eigen v alue distribution and the b est-fit MP distribution. B. Baseline measures W e compare sp ectral observ ables against con ven tional norm-based measures: • Global L 2 norm : p P l ∥ W l ∥ 2 F , the total param- eter norm. • Sum of F rob enius norms : P l ∥ W l ∥ F . • Best-lay er F rob enius norm : max l | corr( ∥ W l ∥ F , test acc) | , c ho osing the sin- gle la yer whose F rob enius norm best correlates with test accuracy . • Max sp ectral norm : max l σ 1 ( W l ), the largest singular v alue across all lay ers. • Pro duct of sp ectral norms : Q l σ 1 ( W l ). C. Arc hitectures W e test three architectures: • MLP: [784 , 512 , 256 , 128 , 10] on MNIST, SGD with momen tum 0.9, lr=0.01, 30 ep o c hs. • CNN: 4 conv lay ers (64, 64, 128, 128 c hannels) + 2 F C lay ers (256, 10) on CIF AR-10, SGD, lr=0.01, 50 ep o c hs, no weigh t decay or data augmen tation. • ResNet-18: Mo dified for 32 × 32 input (3 × 3 first con v, no maxp o ol) on CIF AR-10, SGD, lr=0.01, 50 ep ochs. 3 D. Lab el noise proto col W e use tw o exp erimental designs: EXP-010 (label noise gradien t): F or the MLP/MNIST arc hitecture, we train with 21 evenly spaced lab el noise fractions η ∈ [0 , 1] (i.e., η ∈ { 0 , 0 . 05 , 0 . 10 , . . . , 1 . 0 } ), with 3 indep endent random seeds p er noise level (63 total training runs). A fraction η of training labels are replaced with uniformly random la- b els. T est lab els are alwa ys clean. This pro duces a fine- grained gradient from full generalization ( η = 0) to pure memorization ( η = 1). All metrics are ev aluated using lea ve-one-out (LOO) cross-v alidation across the 21 noise lev els, with seed-av eraged v alues and error bars. EXP-011 (hyperparameter v ariation): F or MLP/MNIST with cle an lab els ( η = 0), we train 180 configurations (3 seeds each, 540 total runs) v ary- ing width ∈ { 64 , 128 , 256 , 512 } , depth ∈ { 2 , 3 , 4 } , learning rate ∈ { 0 . 001 , 0 . 01 , 0 . 1 } , and weigh t deca y ∈ { 0 , 10 − 4 , 10 − 3 , 10 − 2 , 10 − 1 } . E. Ev aluation metric All predictiv e comparisons use leav e-one-out (LOO) cross-v alidated R 2 , whic h p enalizes ov erfitting and pro- vides an honest estimate of out-of-sample predictive p o w er. This is a delib erately conserv ativ e metric: each prediction is made without access to the corresp onding data p oint. Our earlier exp eriments with 6 noise levels ga ve inflated R 2 v alues ( > 0 . 97) due to the small sample size; the 21-lev el LOO R 2 rep orted here is the definitive result. F. Null mo del As a control, w e compare sp ectral properties of net- w orks trained on real labels v ersus fully sh uffled labels ( η = 1), matc hed for training duration and conv ergence lev el. IV. LEVEL SP A CING IS UNINFORMA TIVE F OR WEIGHT MA TRICES Our initial hypothesis was that ⟨ r ⟩ would transition from Poisson (at initialization) to GOE (after training), with the transition sp eed or final v alue distinguishing generalization from memorization. This h yp othesis is incorrect. W e found that ⟨ r ⟩ ≈ 0 . 53 (GOE) at initialization and remains at GOE throughout training, for b oth real and shuffled lab els (Fig. 1 , b ottom-righ t panel). The explanation is Wishart universalit y: for any ran- dom matrix W with iid en tries (regardless of distribu- tion), the Gram matrix 1 m W ⊤ W follows the Wishart- Laguerre ensemble, which has GOE-lik e level spacing b y construction. Since weigh t matrices are initialized with iid en tries and SGD p erturbations are small relativ e to the total matrix norm, ⟨ r ⟩ remains at GOE throughout training. Lesson: The level spacing ratio, designed for Hamilto- nian matrices in quantum chaos, is the wrong observ able for rectangular weigh t matrices. Bulk distribution prop- erties (tail index, outlier fraction, effective rank) are the correct to ols. V. BULK OBSER V ABLES DISTINGUISH GENERALIZA TION FR OM MEMORIZA TION Ha ving identified the correct observ ables, we compare net works trained on real lab els versus sh uffled lab els (MLP/MNIST). A t conv ergence (b oth achieving > 99% training ac- curacy), the sp ectral signatures differ dramatically (T a- ble I ). T ABLE I. Sp ectral prop erties of MLP input la yer (784 → 512) at conv ergence. Real labels: 98.3% test accuracy . Shuffled lab els: 9.9% test accuracy . Observ able Init Real Shuffled Outlier fraction 0% 6.4% 76% T ail α 6.5 2.1 3.5 Effectiv e rank (norm) 0.72 0.61 0.47 MP KS 0.006 0.07 0.23 ⟨ r ⟩ 0.53 0.53 0.53 In terpretation: A generalizing netw ork concentrates learned structure in to a sparse set of large signal eigen- v alues (few outliers, heavy tail, MP bulk preserved). A memorizing net w ork spreads w eight changes across man y sp ectral directions (many outliers, lighter tail, MP bulk destro yed). The level spacing ratio ⟨ r ⟩ is identical for b oth—confirming its uninformativeness. These results are robust to the training-duration con- found: when compared at matched training loss (not matc hed ep o c h), the separation p ersists across the en- tire training tra jectory (Fig. 1 ). VI. AR CHITECTURE-DEPENDENT SPECTRAL SIGNA TURES F or CNNs, the sp ectral signature is layer-typ e dep en- dent . In a CNN trained without regularization to 100% training accuracy on b oth real and shuffled lab els: • Conv lay ers: Both real and shuffled lab els develop hea vy tails and outliers. The deep est conv la yer sho ws 84% outliers for memorization vs 49% for generalization. • FC lay er (classifier.0, 8192 → 256 ): The clean- est discriminator. Generalization compresses the 4 FIG. 1. Sp ectral evolution during training, plotted against training loss (right = untrained, left = conv erged). Real la- b els (blue) and shuffled lab els (red) sho w clearly separated tra jectories for all bulk observ ables but iden tical ⟨ r ⟩ . rank by 30% (0 . 985 → 0 . 691) with heavy tails ( α = 2 . 1). Memorization barely c hanges the rank (0 . 984 → 0 . 933) with light tails ( α = 6 . 1). This reveals wher e learning happ ens: generalization concen trates structure in the FC lay er (decision bound- ary), while memorization concen trates in con v lay ers (p er-example feature enco ding). VI I. THE BOTTLENECK HYPOTHESIS W e prop ose that sp ectral signatures concentrate at the information-pr o c essing b ottlene ck : the la yer with the highest compression ratio (max dimension / min dimen- sion) that has sufficient sp ectral resolution (min(dim) ≳ 50). T ABLE I I. Best sp ectral predictor of test accuracy across arc hitectures under lab el noise v ariation. R 2 v alues are from linear fits to seed-a veraged data at 6 noise levels (see Sec. VII I for the definitive 21-lev el LOO results). Arc hitecture Best Lay er Observ able R 2 MLP/MNIST net.2 (512 → 256) tail α 0.976 CNN/CIF AR-10 classifier.0 (8192 → 256) tail α 0.987 ResNet-18/CIF AR-10 la yer2.1.con v2 eff. rank 0.989 In each case, the b est predictor is a sp ectral prop ert y of a la yer in the netw ork’s information-pro cessing core: • MLP: the middle hidden lay er (distributed com- pression across all lay ers) • CNN: the F C lay er (32 × compression, the explicit b ottlenec k) • ResNet: mid-depth residual blo cks (skip connec- tions distribute compression) The ResNet result is notable: the FC la y er has the highest compression (51 × ) but only 10 eigenv alues, mak- ing sp ectral estimation unreliable. The mid-depth resid- ual blo c ks (128 eigen v alues) serve as the effective b ottle- nec k with sufficient resolution. VI II. QUANTIT A TIVE PREDICTION UNDER LABEL NOISE A. Exp erimen tal design Using the fine-grained noise gradient proto col (EXP- 010, Sec. I I I D ), we train MLP/MNIST at 21 noise levels with 3 seeds each. F or each noise lev el, we av erage sp ec- tral observ ables and test accuracy across seeds and com- pute error bars. W e then ev aluate each predictor using lea ve-one-out cross-v alidated R 2 . B. Results T ABLE I II. Leav e-one-out R 2 for predicting test accuracy from sp ectral and baseline measures under lab el noise v ari- ation (EXP-010: 21 noise lev els, 3 seeds). The tail index α dominates all alternatives. Measure LOO R 2 T ail α (b ottlenec k la yer) 0.984 Effectiv e rank (b ottleneck la yer) 0.750 Best-la yer F rob enius norm 0.149 Global L 2 norm 0.044 Sum of F rob enius norms 0.036 Max sp ectral norm < 0 Pro duct of spectral norms < 0 The tail index α achiev es LOO R 2 = 0 . 984, far ex- ceeding every baseline. The best con v entional measure (F robenius norm of the optimal lay er) explains only 14.9% of v ariance, while α explains 98.4%. Sp ectral norms ac hieve negative LOO R 2 , meaning they predict w orse than a constant (the mean). The effective rank is a distant second at R 2 = 0 . 750, confirming that the tail shap e (not just the rank struc- ture) carries the dominant signal. Figure 2 sho ws the linear relationship. C. Cross-arc hitecture consistency The original 6-p oin t fits across architectures (T able I I ) all ac hieve R 2 > 0 . 97, consisten t with the high-resolution 21-p oin t LOO result. The linear form test acc = a · α bottleneck + b (2) holds across MLP , CNN, and ResNet-18 architectures. 5 FIG. 2. MLP/MNIST: T est accuracy vs. tail index α of the b ottlenec k la y er (net.2.w eight). Each point is a different noise lev el (0%–100%), av eraged o ver 3 seeds with error bars. LOO R 2 = 0 . 984. D. Hill estimator stability The Hill estimator for α dep ends on the choice of threshold quan tile q (the fraction of eigenv alues used in the tail fit). Across threshold quantiles q ∈ [0 . 70 , 0 . 95], the estimated α ranges from 1.47 to 1.87 for a typical trained netw ork. This represents mo derate stability: the qualitativ e conclusion (heavy tail) is robust, but the pre- cise v alue of α v aries by ∼ 25% dep ending on the thresh- old. W e use q = 0 . 90 throughout, following Martin and Mahoney [ 3 ]. The high LOO R 2 confirms that, despite this estimator v ariance, the relative ordering of α across noise levels is highly consisten t. IX. F AILURE UNDER HYPERP ARAMETER V ARIA TION A critical question is whether spectral measures pre- dict generalization under perturbation axes other than data quality . W e test this with EXP-011 (Sec. I I I D ): 180 hyperparameter configurations at fixed (clean) data. A. Results T ABLE IV. Leav e-one-out R 2 for predicting test accuracy un- der hyperparameter v ariation (EXP-011: 180 configs, 3 seeds eac h, clean lab els). All measures are weak. Baselines sligh tly outp erform sp ectral measures. Measure LOO R 2 Global L 2 norm 0.219 Sum of F rob enius norms 0.204 T ail α (b ottlenec k la yer) 0.167 Max sp ectral norm 0.103 Effectiv e rank (b ottleneck la yer) 0.094 All measures are w eak predictors ( R 2 < 0 . 25). Con- v entional norm-based measures (global L 2 norm, sum of F robenius norms) slightly outp erform sp ectral measures (tail α , effective rank). The test accuracy range across configurations is narrow (91.1%–98.6% on MNIST), whic h partly explains the difficult y: all configurations pro duce reasonably goo d mo dels, and the remaining v ari- ation is driven b y optimization dynamics that are not cleanly captured by any single weigh t-space summary statistic. B. In terpretation The con trast b et ween T ables I I I and IV is the cen tral finding of this pap er. Lab el noise degrades data qual- it y in a w ay that leav es a clear, architecture-independent sp ectral fingerprint: memorization of random lab els re- quires spreading weigh t changes across many sp ectral di- rections, pro ducing lighter tails. Hyp erparameter v ari- ation, by contrast, pro duces netw orks that all learn the same clean signal but with differen t optimization tra jec- tories, regularization strengths, and capacit y utilization patterns. These differences are not dominated b y a single sp ectral signature. This is consistent with Jiang et al. [ 8 ], who found that no generalization measure dominates across all perturba- tion t yp es. Our con tribution is iden tifying wher e sp ectral measures excel (data quality) and wher e they do not (h y- p erparameter v ariation), with rigorous LOO baselines. X. THEORETICAL FRAMEWORK A. Outlier b ound via BBP transition By the BBP transition, a rank- r p erturbation ∆ W to a random w eight matrix produces at most r outlier eigen- v alues ab o ve λ + . F or a net work learning k linearly sep- arable classes, the optimal b ottleneck representation has rank O ( k ), pro ducing O ( k ) outliers. F or memorization of N random lab els through a lay er of width n , the p er- turbation requires rank O (min( N , n )). This predicts: • Generalization: O ( k ) outliers at the b ottleneck (ob- serv ed: 33 for k = 10, consistent with ∼ 3 k due to m ulti-lay er interactions) • Memorization: O ( n ) outliers (observed: 389 out of 512, or 76%) B. Conjecture: Monotonicity of α in noise fraction Conjecture. L et f θ : R d → R k b e a neur al net- work with sufficient c ap acity to interp olate N tr aining examples. L et D η denote a tr aining set wher e a fr action η ∈ [0 , 1] of lab els ar e r eplac e d uniformly at r andom. L et W η b e the b ottlene ck weight matrix after tr aining to zer o loss on D η , and let α η b e the tail index of the eigenvalue distribution of 1 m W ⊤ η W η . Then α η is monotonic al ly in- cr e asing in η . 6 Pr o of sketch. At η = 0, the net work learns a rank- k classification b oundary . By the BBP transition, W η dev elops O ( k ) outlier eigenv alues ab ov e the MP bulk, concen trating sp ectral mass in few directions (heavy tail, lo w α ). At η = 1, the netw ork m ust memorize N random lab el assignments. Since random lab els ha v e no shared structure, eac h training example con tributes approxi- mately indep enden tly to ∆ W , pro ducing O (min( N , n )) outlier eigenv alues that spread sp ectral mass across many directions (lighter tail, higher α ). A t in termediate η , the net work must enco de b oth a rank- k signal comp onen t (from the (1 − η ) N clean examples) and a rank- O ( η N ) noise comp onen t (from the η N corrupted examples). As η increases, the noise comp onent gro ws monotonically , spreading sp ectral mass and increasing α . □ This conjecture is supported empirically b y the mono- tonic relationship in Fig. 2 and the out-of-sample v alida- tion in the noise detector (mean error 1.5%). Corollary . Given a c alibr ation set { ( α i , η i ) } m i =1 fr om m ≥ 3 known noise levels, the noise fr action of an un- known dataset c an b e estimate d by line ar interp olation of α , with exp e cte d err or O (1 /m ) . C. Wh y the relationship is linear under lab el noise If lab el noise η controls both test accuracy (linearly: test acc ∝ 1 − η ) and the tail index (linearly: α ∝ η , since more noise requires more sp ectral directions), then com- p osing these relationships gives test acc ∝ − α + const. The linearity of α in η follo ws from the additivity of learned structure: with noise fraction η , the netw ork m ust enco de (1 − η ) N clean examples (rank- k structure) plus η N random examples (rank-min( η N , n ) structure). The tail index reflects this mixture. D. Wh y sp ectral measures fail under h yp erparameter v ariation Under hyperparameter v ariation with clean lab els, the rank of the learned signal is alwa ys O ( k )—the data qual- it y is fixed. What v aries is the optimization tra jec- tory (learning rate, momentum), the capacity utilization (width, depth), and the regularization (weigh t decay). These affect the sc ale of weigh ts (captured by norms) more than the shap e of the sp ectrum (captured by α and effective rank). Martin and Mahoney [ 20 ] identi- fied precisely this distinction: shap e metrics (like α ) and scale metrics (like norms) play complemen tary roles, and datasets that v ary h yp erparameters can exhibit a Simp- son’s paradox where shap e and scale metrics disagree. This explains why norm-based measures ha ve a slight edge (T able IV ): they directly measure the scale v aria- tion that dominates in this regime. XI. DISCUSSION A. Relation to prior w ork Martin and Mahoney [ 3 ] established the connection b e- t ween heavy-tailed w eight sp ectra and net w ork quality , prop osing the tail index α as a quality metric and build- ing the WeightWatcher tool. Their evidence comes from cross-mo del correlations across published arc hitectures (e.g., comparing VGG, ResNet, and DenseNet). Sub- sequen t work [ 18 ] show ed that α predicts mo del quality rather than gener alization , with a bias tow ard CV ar- c hitectures, and extended this to NLP [ 19 ]. Martin and Mahoney [ 20 ] further show ed that shap e metrics ( α ) and scale metrics (norms) play complementary roles, with a Simpson’s paradox arising when data, mo dels, and hy- p erparameters v ary simultaneously—exactly the b eha v- ior we observe in our h yp erparameter v ariation exp er- imen ts (T able IV ). Our work differs from the original HTSR framew ork in three wa ys: (1) w e use c ontr ol le d exp erimen ts with a contin uous noise gradient and prop er LOO cross-v alidation, pro ducing quan titative predictive p o w er ( R 2 = 0 . 984) rather than rank correlations; (2) w e identify the b ottlene ck layer as the optimal measure- men t p oint, rather than av eraging across lay ers (their “w eighted alpha”); and (3) w e rep ort the honest nega- tiv e result under hyperparameter v ariation, scoping the claim to data quality diagnostics—consistent with the qualit y-vs-generalization distinction identified in [ 18 ]. Meng and Y ao [ 13 ] is the closest prior work. They study ho w classification difficult y drives sp ectral phases (Ligh t T ail, Bulk T ransition, Heavy T ail) and prop ose sp ectral criteria for early stopping. Our work extends theirs with: (a) a fine-grained 21-lev el noise gradient with LOO ev aluation, (b) head-to-head baselines showing that con ven tional norm measures fail (LOO R 2 < 0 . 15) while tail α succeeds ( R 2 = 0 . 984), (c) v alidation on real h u- man annotation noise (CIF AR-10N), and (d) the negative result under h yp erparameter v ariation. Y unis et al. [ 14 ] show ed that sp ectral dynamics of w eights distinguish memorizing from generalizing net- w orks, with random lab els pro ducing high-rank solutions. Our w ork complemen ts theirs with quan titative predic- tion rather than qualitativ e distinction, and identifies the sp ecific lay er where the signal concen trates. Thamm et al. [ 15 ] conducted a thorough RMT analy- sis of trained w eight matrices, finding that bulk eigen- v alues follow MP predictions and level spacings agree with RMT. Our level spacing negative result (Sec. IV ) is consistent with and extends their finding by explicitly testing whether lev el spacing distinguishes generalization from memorization (it do es not). Jiang et al. [ 8 ] benchmark ed 40+ generalization mea- sures and found that no single me asure dominates across all p erturbation types. Our results are consistent: sp ec- tral measures achiev e near-p erfect prediction on one axis (data qualit y) while failing on another (hyperparame- ters). 7 The information b ottlenec k framew ork of Tishb y et al. [ 6 ] predicts that optimal representations compress task-irrelev ant information. Our sp ectral measurements pro vide a concrete, computable signature of this com- pression: few large eigen v alues (task-relev ant directions) with a suppressed MP bulk (compressed noise). B. Practical applications Sp ectral noise detector (v alidated on real-w orld noise): W e demonstrate a practical label noise detection to ol v alidated on b oth synthetic and r e al noise. A linear mo del calibrated on 5 synthetic noise levels predicts test accuracy on held-out noise lev els with OOS R 2 > 0 . 97 on b oth MLP/MNIST (mean error 1.5%) and CNN/CIF AR- 10 (mean error 2.7%). Critically , the same to ol detects real human annotation noise in CIF AR-10N [ 12 ]: aggre- gate lab els ( ∼ 9% noise) detected with 3% error, worst- annotator lab els ( ∼ 40% noise) detected with 9% error. The detector transfers from synthetic calibration to real- w orld noise without retraining. V alidation on real-w orld noise (CIF AR-10N): T o test transfer from synthetic to real noise, we ap- plied the CNN/CIF AR-10 detector (calibrated on syn- thetic noise) to CIF AR-10N [ 12 ], which contains human re-annotations with known noise rates. The detector correctly identifies clean data as clean (predicted noise = 0%) and detects real annotation noise at every level: aggregate labels ( ∼ 9% noise) detected with 3% error, w orst-annotator lab els ( ∼ 40% noise) detected with 9% error. The underestimation at high noise is exp ected: real annotation noise is class-dep enden t (e.g., cat–dog confusion), while our calibration assumes uniform noise. This is, to our kno wledge, the first demonstration of a sp ectral to ol detecting real human annotation errors, cal- ibrated from syn thetic noise alone. Relationship to sample-level noise detection: Metho ds such as Confident Learning [ 16 ] and small-loss selection [ 17 ] identify which sp e cific samples are misla- b eled, using p er-sample losses or predicted probabilities. Our spectral metho d answers a different question: how noisy is this dataset over al l? It provides a single dataset- lev el estimate of corruption rate from the weigh t sp ec- trum alone, without examining individual samples. The t wo approac hes are complemen tary: a sp ectral diagnostic could serve as a fast, cheap trigger—if α indicates signif- ican t noise, a more exp ensive sample-level metho d can then identify the corrupted examples. Early stopping under noise: When training on po- ten tially corrupted data, monitoring α can detect when the netw ork transitions from learning signal to memoriz- ing noise—the onset of tail lightening indicates o v erfit- ting to corrupted lab els. C. Limitations • The tail index is a strong predictor under lab el noise v ariation but a weak predictor under hyper- parameter v ariation ( R 2 = 0 . 167). It should not b e used as a universal generalization diagnostic. • W e test on small-scale net w orks (MLP , small CNN, ResNet-18) and standard b enchmarks. Scaling to large language mo dels and vision transformers is an imp ortan t op en question. • The Hill estimator for α sho ws mo derate instabil- it y across threshold quantiles ( α ranges 1.47–1.87 for thresholds q ∈ [0 . 70 , 0 . 95]). While the relative ordering is robust (as evidenced by the high LOO R 2 ), absolute v alues of α should be interpreted with caution. • The Hill estimator requires ≳ 50 eigen v alues for sta- bilit y , limiting application to narrow la yers. In fine- tuning scenarios where only a narrow output lay er is trained (e.g., a 10-class FC lay er on a frozen pre- trained bac kb one), the sp ectral signal is to o w eak for reliable detection. The to ol requires training la yers with sufficient spectral resolution. • W e do not pro ve a formal generalization bound; our theoretical framework is heuristic. • The h yp erparameter v ariation negative result holds on b oth MNIST (accuracy range 91.1%–98.6%) and CIF AR-10 (accuracy range 62.7%–78.8%), confirm- ing it is not an artifact of the narrow MNIST range. • The CIF AR-10N detector underestimates noise at high corruption levels (9% error at 40% noise), b ecause real annotation noise has class-dep enden t structure that uniform synthetic calibration do es not capture. XI I. CONCLUSION W e hav e shown that the tail index of the eigen v alue distribution at a neural net work’s bottleneck lay er is a p o w erful diagnostic for data quality . Under controlled lab el noise v ariation, it predicts test accuracy with LOO R 2 = 0 . 984, while all con ven tional baselines fail (LOO R 2 < 0 . 15). More practically , a detector calibrated on syn thetic noise successfully identifies real human anno- tation errors in CIF AR-10N—to our knowledge, the first suc h demonstration using sp ectral metho ds. This predictive p ow er do es not transfer to hyperpa- rameter v ariation at fixed data quality , where all mea- sures are w eak. W e rep ort this honestly: sp ectral analysis rev eals data quality , not gener alization in the universal sense. The tail index is best understoo d as a sensitive diagnostic for lab el corruption, training set degradation, and annotation quality—problems that are p erv asive in 8 real-w orld machine learning but p o orly served by existing to ols. A CKNOWLEDGMENTS Computational exp eriments were p erformed on Apple M1 hardware using PyT orc h with MPS acceleration. All co de is av ailable at https://github.com/MattLoftus/ rmt- neural . [1] C. Zhang, S. Bengio, M. Hardt, B. Rech t, and O. 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