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Why Learning Rediscovers the Closed-Form Diagonal Regularizer

Jeahn Han, Pyojin Kim

arXiv:2609.09656v1stat.MLcs.LGcs.ROeess.AS

TL;DR

Choosing a robust diagonal regularizer for modal inverse problems is difficult because finite measurements leave truncation noise from discarded modes. This paper combines isotropy theory with Weyl-law analysis and finds that a prior-determined closed-form power law is near-optimal across rooms, while gains require cross-mode coupling.

  • Problem

    Finite measurements leave truncation noise from discarded modes, raising the question of which diagonal penalty shape is best and whether it depends on the room.

    Source: §1 · p. 1

  • Method

    The paper derives power-law regularization under isotropic truncation noise and uses Berry’s conjecture and Weyl’s law to assess robustness to anisotropy.

    Source: §4.1 · p. 4, §4.3 · p. 5

  • Results

    5.82% relative cost is the population-median gap from per-room oracle tuning, while no diagonal architecture improves over the closed form and non-diagonal coupling improves by 1.9–13.5 pp.

    Source: §1 · p. 2, §8 · p. 9

  • Takeaways & Limitations

    Within the diagonal family, the closed form leaves little room for robust learning gains; the remaining opportunity is cross-mode coupling.

    Source: §8 · p. 9

    The evaluation uses random convex 2D polygonal rooms and uniformly random sensors; structured sensor layouts and 3D extensions remain to be validated.

    Source: §8 · p. 9

Abstract

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We identify a diagonal saturation principle in modal inverse problems: when truncation noise is isotropic, the Bayes-optimal Tikhonov shape is a closed-form power law Gamma_k proportional to lambda_k^|s| set by the prior alone, independent of the domain. Berry's random-wave conjecture decorrelates the truncation noise across modes, and Weyl's eigenvalue counting law supplies enough modes for the conclusion to survive empirical Berry violations. Together they predict an approximately flat loss landscape across the per-mode family, leaving narrow scope for a diagonal regularizer to robustly beat the closed form. On FEM-simulated acoustic rooms, the closed form is near-optimal relative to per-room oracle tuning across observation windows, and three diagonal architectures trained on the same data match its reconstruction error within 1 pp despite learning qualitatively different spectra. The framework extends to heat diffusion via a known exponential Green's function correction with no new free parameters. Saturation is restricted to the diagonal family: Learned Iterative Ridge crosses the boundary by exploiting cross-mode coupling, locating where learning starts to help.

1 Introduction

The paper argues that isotropic truncation noise makes the Bayes-optimal diagonal regularizer a room-independent power law Γ_k ∝ λ_k^|s|. Theory, room-acoustic experiments, neural diagonal models, and heat-diffusion analysis support diagonal saturation while identifying cross-mode coupling as the route beyond it.

Problem setup

  • The motivating inverse problem reconstructs modal weights from few microphone measurements, where finite sensing leaves discarded modes as truncation noise that can dominate the error budget.With K = 50 retained modes and M = 8 microphones, one snapshot supplies 8 equations for 100 unknowns, while typical rooms contain over 300 modes.Source: §1 · p. 1

Theory

  • Berry’s random-wave conjecture [Berry, 1977] and Weyl’s law [Weyl, 1912, Ivrii, 2016] motivate approximately isotropic truncation noise, yielding p* = |s| and little robust scope for diagonal adaptation.Weyl’s law supplies enough discarded modes for truncation noise to concentrate around its isotropic average.Source: §1 · p. 2

Core claim

  • The central claim is that the optimal diagonal shape is room-independent: Γ_k ∝ λ_k^|s|, with |s| fixed by the prior rather than learned from each domain.This claim persists under changes in K, M, and excitation exponent.Source: §1 · p. 2

Verification

  • The closed form’s median relative cost stays below 5.82% at every observation window, while 68.4% of rooms remain within 1.1 pp absolute of per-room tuning.These results cover 187 in-scope rooms; 10 boundary-regime rooms with K_total ≤ K are reported separately.Source: §1 · p. 2

Diagonal saturation

  • Three neural diagonal architectures match the closed form’s P-vs-T curve despite learning qualitatively different spectra, with M3 within ±0.3 pp and M1/M2 within 1 pp at every T.This flat landscape indicates diagonal saturation, including for learned deep-unrolling regularizers [Gregor and LeCun, 2010, Adler and Öktem, 2018, Aggarwal et al., 2018].Source: §1 · p. 2

Cross-PDE consistency

  • For heat diffusion, the Green’s-function correction predicts Γ_k ∝ λ_k^s · e^(2κtλ_k), and per-room fits recover its rate within [0.97, 1.00].The paper presents this as a cross-PDE consistency check rather than physical validation.Source: §1 · p. 2

2 Related Work

Prior work resolves Tikhonov strength selection more reliably than penalty shape, while existing shape methods depend on data or assumed smoothness. This paper connects Berry’s conjecture and Weyl’s law to isotropic truncation noise, explaining why a covariance- and prior-determined closed-form shape recurs across rooms and related inverse problems.

Tikhonov regularization

  • Tikhonov strength α has established selection rules, but the shape matrix Γ remains harder: existing approaches use minimax shrinkage, hierarchical Bayes, or spectral Bayes under data or smoothness assumptions.The cited methods include the L-curve, generalized cross-validation, discrepancy principle, Pinsker’s estimator, hierarchical Bayesian weights, and manifold spectral Bayesian structures.Source: §2 · p. 2

Core contribution

  • Berry’s conjecture and Weyl’s law reduce the regularizer-shape question to the signal covariance, so a power-law prior requires only the scalar |s| and yields no robust room-adaptive gain.Alberti et al. [2021] established covariance dependence, while Berry’s conjecture and Weyl’s law supply the isotropy and mode abundance used here.Source: §2 · p. 2

Learned regularization

  • Learned regularizers often rediscover classical solutions, with learned parameters converging to variational solutions and bilevel optimization reducing to hyperparameter tuning.Related examples include untrained CNN denoisers and plug-and-play penalties collapsing to denoiser spectral penalties [Hurault et al., 2022].Source: §2 · pp. 2–3

Spectral physics

  • Berry’s conjecture models high-frequency room eigenmodes as random waves, while Weyl’s law supplies linearly growing 2D mode counts; their regularization-design connection was previously absent.Berry’s conjecture is supported by quantum ergodicity but can fail for scarred states and symmetric geometries.Source: §2 · p. 3

Room acoustics

  • Room-acoustics research spans Bayesian, compressive-sensing, array, neural, and Gaussian-process methods, whereas this work asks whether the penalty shape is shared across rooms.The Matérn-kernel GP regularity parameter is analogous to |s|, distinguishing this shape question from sensor-placement studies.Source: §2 · p. 3

Heat equation

  • For backward heat diffusion, exponential modal decay e^−κλ_kt causes exponential noise amplification; existing minimax and variational theories provide scaling but not the exact regularizer or its κ,t dependence.The cited heat-equation literature includes Beck et al. [1985], Kaipio and Fox [2011], Knapik et al. [2013], and Hohage and Weidling [2017].Source: §2 · p. 3

3 Problem Setup

The paper formulates modal inverse problems on bounded two-dimensional domains using Laplacian eigenmodes, with acoustics as the main instance and heat diffusion as a cross-PDE check. It models truncated microphone observations with a diagonal Tikhonov estimator and evaluates normalized modal reconstruction error.

Modal formulation

  • The framework represents the state on a bounded 2D domain through Laplacian eigenpairs, with acoustic validation across 187 random convex polygons and a heat-equation cross-check.Each eigenfunction is treated as a spatial pattern, while modal amplitudes encode its time-dependent contribution.Source: §3 · p. 3
  • Modal-amplitude variance decays as a power law in eigenvalue, with exponent s determined by excitation statistics rather than room geometry.Higher modes therefore generally carry less energy in physical settings.Source: §3 · p. 3

Dynamics and assumptions

  • The acoustic model uses mode-independent damping γ, whereas heat diffusion differs because its damping is not uniform across modes.The initial-condition variance σ2_a,k is defined for the random coefficients (c_k, β_k).Source: §3 · p. 3

Truncation and observations

  • The observation model retains K = 50 modes and records M microphones over T snapshots, separating the retained signal from discarded higher-mode truncation noise.After stacking observations, the model is written as ỹ = Φ̃ a0 + η̃, where Φ̃ combines spatial measurements and temporal basis functions.Source: §3 · p. 3

Estimation and metric

  • The estimator recovers initial modal amplitudes by penalized least squares with strength α > 0 and diagonal penalty matrix Γ ≻ 0, using normalized modal MSE P as the quality metric.P = 0 denotes perfect reconstruction, while P = 1 is no better than guessing zero.Source: §3 · p. 4

4 Why p∗= s: The Three-Step Argument

The argument derives a prior-determined power-law regularizer by combining Bayesian isotropy, Berry’s eigenmode decorrelation, and Weyl’s abundance of truncated modes. It concludes that moderate noise anisotropy does not materially change the regularizer shape, with eigenvalue dynamic range as the limiting factor.

4 Why p∗= s: The Three-Step Argument

  • The three-step argument identifies the optimal regularizer shape as a power law set by the prior spectral decay rate s.The noise level determines only the overall regularization strength, not the per-mode shape.Source: §4 · p. 4, §4.1 · p. 4

4.1 Step 1: If the noise is flat, the answer is immediate

  • Under isotropic truncation noise, Bayesian MAP estimation yields a regularizer whose shape is determined entirely by the prior.Modes carrying less signal energy are penalized more, while the noise level controls only overall strength.Source: §4.1 · p. 4

4.2 Step 2: Berry’s conjecture: the idealized isotropy mechanism

  • Berry’s random-wave conjecture [Berry 1977] supplies approximate sensor-level decorrelation between high-frequency eigenmodes, supporting isotropic truncation noise.The resulting anisotropy depends on how concentrated the discarded-mode noise power is: many comparable contributors promote isotropy, while one dominant mode undermines it.Source: §4.2 · pp. 4–5

4.3 Step 3: Weyl’s law: why the conclusion survives Berry violations

  • Weyl’s eigenvalue counting law [Weyl 1912, Ivrii 2016] provides roughly 263 truncated modes in the rooms, enough for concentration despite Berry violations.Across 187 rooms at T=1000, higher anisotropy correlates with lower cost (Spearman ρ = −0.30, p < 10−4), indicating that eigenvalue dynamic range—not noise anisotropy—is the bottleneck.Source: §4.3 · p. 5

5 Empirical Verification on Acoustic Rooms

Across 187 in-scope acoustic rooms, the reconstruction landscape over diagonal exponents is broad and shallow, while the fixed |s|=1.13 remains within the optimum basin across snapshot counts. Its measured cost stays small in most rooms, supporting the closed-form choice despite per-room oracle tuning.

Robustness

  • The flat-landscape pattern persists when truncation rank, sensor count, excitation exponent, prior tail behavior, and correlation are varied.The baseline uses K=50 modes and M=8 microphones; robustness checks include K=100, M∈{8, 16}, |s|=1.29, Student-t ν=3, and correlated-Gaussian ρ=0.3 alternatives.Source: §5 · p. 5

Landscape flatness

  • Berry isotropy and Weyl eigenvalue spacing limit curvature, concentrating per-room optima and explaining why diagonal tuning has little robust advantage.At T=50, the standard deviation of per-room p∗ is 0.31; the framework’s remaining diagonal structure becomes effectively ill-defined as data increase.Source: §5 · pp. 5–6
  • The median P-range across p ∈[0, 3] compresses ∼14× from 0.18 at T=1 to 0.013 at T=2100, while the oracle drifts from ≈1.2 to ≈2.0.The basin widens faster than the optimum shifts, keeping |s|=1.13 inside every basin.Source: §5 · p. 5, Figure 1 · p. 6

Empirical verification of isotropy

  • Across 77,968 pooled sensor samples, the Berry-isotropy test gives KS statistic D = 0.037 against χ2(1), with deviations no larger than 3.7%.The test evaluates whether |Ω| · φk(xm)^2 follows the predicted χ2(1) law for uniformly random sensor positions.Source: §5 · p. 6

Cost of using |s|

  • 68.4% of 187 in-scope rooms incur less than 1.1 pp absolute cost from using |s|=1.13, while the worst in-scope cost is 7.61 pp in the prior-dominated regime.The relative cost is measured against per-room oracle tuning across all snapshot counts; the worst-case median cost is 5.82% in the data-rich regime.Source: §5 · p. 6, Table 1 · p. 6

6 Can Diagonal Learning Improve Upon |s|?

Learning does not robustly improve the physics-derived diagonal regularizer Γ_k = λ_k^|s|: different learned spectra achieve essentially the same error, while cross-mode coupling via LIR beats the diagonal oracle.

6 Can Diagonal Learning Improve Upon |s|?

  • M3’s learned spectra vary substantially across seeds yet yield identical reconstruction error, with effective exponents ranging from −0.22 to 1.09 at T=1000.This indicates a flat loss landscape across qualitatively different reachable diagonal regularizers.Source: §6 · pp. 6–7
  • M3 recovers the theoretical exponent in the median, while M2 learns a shallower power law whose population-median error differs by at most 0.009.At T=1000, p̂_M3 = 1.13 matches theory’s |s| = 1.13, whereas p̂_M2 = 0.62; M2’s power-law fit has R2 > 0.96.Source: Figure 2 · p. 7, §6 · p. 7
  • Across diagonal architectures, M3 remains within 0.31 pp of Γ_k = λ_k^|s| at every observation window, so learning does not recover the per-room oracle gap.The gaps P_M3 − P_ridge are −0.19 pp at T=1, +0.16 pp at T=100, and +0.31 pp at T=1000.Source: §6 · p. 7, Table 2 · p. 7
  • Per-room slope fitting fails to improve on |s| because estimation noise σ_per-room ≈ 0.27 overwhelms the population-median standard error σ_pop ≈ 0.026.The s_room reconstruction is 0.28 pp worse at T=50, consistent with shrinkage toward the population mean outperforming per-room plug-in estimation [James et al., 1961, Stein, 1956].Source: §6 · p. 7

6.1 Beyond the Tikhonov family

  • At L=10, LIR improves over the diagonal oracle by 9.4, 13.5, and 1.9 pp at T=10, 100, and 500, respectively.LIR achieves P = {0.621, 0.459, 0.103} versus the diagonal oracle’s {0.715, 0.594, 0.122}.Source: §6.1 · p. 8

7 Extension to Heat Diffusion

Heat diffusion fits the same diagonal-regularization framework after adding a theory-determined exponential correction to the power-law prior. The correction is competitive with per-room oracle tuning, while its validity is limited when late-time signal falls below measurement noise.

Physical scope

  • Heat differs from acoustics in temporal structure and excitation statistics, yielding |s| ≈1.0 rather than 1.13, while retaining a power-law initial component plus PDE-specific theoretical corrections.Recovering the initial state itself is exponentially ill-posed and is not attempted; the method estimates current-state amplitudes at terminal snapshots.Source: §7 · pp. 8–9

Theory-derived correction

  • The optimal heat regularizer multiplies the acoustic power law by e^{c λ_k}, inverting the Green’s-function exponential through Γ ∝ Σ^-1.The correction uses c = 2κt, so the exponential factor is determined by theory rather than learned freely.Source: §7 · p. 8

Parameter estimation

  • R2 > 0.98 for t > 100 ms, while five-room regressions recover c with slopes [0.97, 1.00], R2 ≥0.998, and intercepts below 0.10.Only |s| is estimated from data; c = 2κt is fixed by the chosen observation time and material diffusivity κ.Source: §7 · p. 8

Cross-PDE consistency

  • The heat formula is competitive with per-room oracle tuning while truncation noise dominates, but late-time prior decay causes isotropy to break down.At late observation times, exponential prior decay drives signal energy below measurement noise and the performance margin grows; this is a cross-PDE consistency check, not independent physical validation.Source: §7 · p. 9

8 Discussion and Conclusion · A Exact Theorem and Approximate Isotropy Argument

The paper concludes that approximate isotropy yields a domain-independent closed-form diagonal regularizer that nearly matches per-room oracle tuning, while improvements require non-diagonal or temporal structure. Its theorem, Berry–Weyl argument, empirical results, and limitations define both the scope and remaining frontier of this conclusion.

8 Discussion and Conclusion

  • Under approximate isotropy, the Bayes-optimal diagonal shape is Γ_k = λ_k^|s|, with |s| determined by the excitation process rather than the room (Prop. 1).The isotropy approximation is supported by Berry’s conjecture, Weyl’s law, and empirical verification.Source: §8 · p. 9, A Exact Theorem and Approximate… · p. 13
  • The closed form stays within 5.82% relative cost of per-room oracle tuning, while Learned Iterative Ridge improves over the diagonal-family oracle by 1.9–13.5 pp.The closed form remains near-optimal across truncation levels and rooms, whereas LIR gains by exploiting non-diagonal coupling.Source: §8 · p. 9
  • No diagonal learned architecture improves over the closed form because approximate isotropy makes the per-mode loss landscape empirically flat.The remaining gains are therefore structural rather than improvements to diagonal spectral tuning.Source: §8 · p. 9
  • Per-room fitting is disfavored: σ_per-room ≈ 0.27 exceeds σ_pop ≈ 0.026 by roughly an order of magnitude, placing the problem in a James–Stein shrinkage regime.The deconvolved inter-room signal is effectively zero, and population-mean shrinkage empirically outperforms per-unit plug-in.Source: §8 · p. 9
  • The framework extends to heat diffusion without new free parameters, while electronic noise changes only α and frequency-dependent damping composes with |s| through the heat framework.These extensions preserve the diagonal regularizer’s form while providing cross-PDE consistency and noise robustness.Source: §8 · p. 9
  • The conclusion is limited to the tested diagonal setting: non-diagonal estimators, trajectory sensing, and sequential temporal models remain the principal avenues beyond its ceiling.The paper specifically identifies cross-mode coupling, temporal dynamics, and full-recording state-space methods as remaining frontiers.Source: §8 · p. 9
  • A real-data pilot confirms aperture-bounded rank-3 sampling and a flat landscape but estimates |ŝ| = 0.83 below the population |s| = 1.13.The gap is consistent with recording-chain effects that cannot be disentangled from one static array, motivating trajectory-based sensing.Source: §8 · p. 9

A Exact Theorem and Approximate Isotropy Argument

  • The appendix formalizes the result as an exact isotropic-Gaussian theorem and motivates approximate isotropy through Berry’s conjecture and Weyl’s law.Berry–Weyl reasoning is physically motivated, while empirical verification in §5 and Appendix B supplies the primary adequacy evidence.Source: A Exact Theorem and Approximate… · p. 13

A.1 Bayesian derivation of Proposition 1 … B.3 Boundary-stratified KS statistics

Under isotropic truncation noise and a Gaussian power-law prior, the Bayes-optimal diagonal Tikhonov shape is Γ_k ∝ λ_k^|s|, while Berry/Weyl analyses show approximate isotropy is sufficient in practice. Extended validation finds modest boundary and room-level deviations that do not materially disrupt the regularizer’s robustness.

A.1 Bayesian derivation of Proposition 1

  • Γ_k ∝ λ_k^|s| is the globally MMSE-optimal diagonal Tikhonov shape under exact isotropy and a Gaussian power-law prior, with noise and operator structure affecting only scalar strength.With approximate anisotropy, the unconstrained Bayes estimator is non-diagonal, while the per-room oracle remains restricted to the diagonal family.Source: §A.1 · p. 14

A.2 Anisotropy bound derivation

  • Across 187 rooms, median ||E||op is 0.58 versus the 0.60 Frobenius estimate, but the mean is 0.88 and the 95th percentile is 2.42 because of Berry-violating right-tail rooms.Signal dynamic range exceeds noise anisotropy in 91% of rooms; the population-median cost of staying at |s| peaks at 5.82%, with 68.4% below 1.1 percentage points.Source: §A.2 · p. 16

A.3 Herfindahl index: numerical values

  • The Herfindahl index has median 0.005, ranges from 0.002 to 0.03, and remains finite asymptotically, so its small operating-point values are a finite-sample property rather than a guarantee.For scene_00850, H = 0.0038 corresponds to 260 effective contributors among 433 truncated modes.Source: §A.3 · pp. 16–17

A.2 Anisotropy bound derivation

  • Per-room reconstruction cost is governed by eigenvalue dynamic range rather than noise anisotropy, with Figure 4 reporting ρ = −0.30 between relative cost and ||E||op.Large rooms combine low anisotropy with wider eigenvalue spectra and higher oracle-exploitable cost.Source: §A.2 · p. 16, Figure 4 · p. 17

A.4 Temporal stacking

  • Temporal stacking introduces rank-two temporal covariance that shifts p*(T) away from |s|, but its median reconstruction cost peaks at 5.6% for T = 1000 and decreases by T = 2100.The residual anisotropy is temporal because spatial factors still concentrate toward I_M.Source: §A.4 · pp. 17–18

B.1 Eigenfunction value distribution

  • Berry validation finds pooled KS D = 0.037 against χ²(1), tightening to D = 0.029 for k ≥ 10, while pass rate peaks at 89% for K ∈ [50, 200].Weyl’s law supplies enough modes for robust concentration despite imperfect individual-mode decorrelation.Source: §B.1 · p. 18

B.2 Per-room KS distribution

  • Per-room KS statistics have median D = 0.042, IQR [0.031, 0.058], and maximum D = 0.12; even the worst rooms retain flat reconstruction-cost landscapes.The Berry pass rate reaches 88.5% for Ktotal ∈ [50, 200), and KS variation does not predict reconstruction outcome because Weyl dominance provides a large margin.Source: §B.2 · pp. 18–19

B.3 Boundary-stratified KS statistics

  • Boundary effects are modest: near-boundary sensors have KS D = 0.048 versus 0.039 for interior sensors, and interior sensors dilute this deviation in the aggregate covariance.The table reports the near-boundary statistic as 23% above the interior value, while both remain within the regime where regularizer shape is robust.Source: Table 3 · p. 19, §B.3 · pp. 19–20

B.4 Worst-5-rooms tail analysis · C Acoustic Experiments: Extended Results

Worst-Berry rooms exhibit real but weakly consequential deviations from isotropy: even the largest observed cost is low at short windows, while Weyl dominance preserves a flat landscape. The extended acoustic results provide the methodological and per-room context behind the paper’s headline claims.

B.4 Worst-5-rooms tail analysis

  • The tail analysis selects the five largest KS statistics D from 168 of 197 validation rooms after excluding 29 rooms with fewer than 50 truncated modes.Table 4 presents these five rooms sorted by D descending.Source: §B.4 · p. 20, Table 4 · p. 20
  • Even room 00835, with D = 0.098 and δ(T=1000) = 25.2%, has δ ≤3% for T ≤100; its long-window cost reflects oracle dispersion, not Berry failure.The five worst-Berry rooms have near-median Herfindahl indices, H ≈0.003–0.004, rather than elevated concentration.Source: §B.4 · p. 20
  • The selected worst-Berry rooms are mid-to-large, with 6–10 vertices and Ktotal > 400, while their fixed KS sample size is K·M = 400.The observed association between D and mode count therefore reflects geometry rather than statistical power.Source: §B.4 · p. 20
  • Across 168 eligible rooms, worse Berry agreement predicts slightly higher reconstruction cost, but the statistically significant correlation is weak.The worst-Berry rooms remain well within the cost range reported in the main text.Source: §B.4 · p. 20
  • Berry violations do not translate into meaningful reconstruction cost, because Weyl dominance keeps the loss landscape effectively flat despite imperfect isotropy.The analysis finds that some rooms have genuinely non-Gaussian cross-correlations, but their reconstruction costs remain within the main-text range.Source: §B.4 · p. 20

C Acoustic Experiments: Extended Results

  • The appendix supplies methodological details, per-room breakdowns, and extended analyses supporting the headline claims that the landscape is flat, Berry’s KS statistic is 0.037, and using |s| has small cost.These claims correspond to Figure 1 and Table 1 in the main text.Source: C Acoustic Experiments: Extended Results · p. 20

C.1 Estimating |s| in practice … D.3 Could a better model do more?

This section evaluates practical estimation of the population exponent and the scope for improving the closed-form diagonal regularizer. Across models and operating conditions, variation in per-room oracle exponents is limited in predictive value, while differing learned spectra need not improve reconstruction error.

C.1 Estimating |s| in practice

  • Median |ŝ| = 1.13 across 187 rooms, with bootstrap 95% CI [1.08, 1.18], and the estimate stabilizes by N = 50 rooms.Room-level OLS fits log modal variance against log eigenvalue using K = 50 retained modes; the method assumes power-law continuation into the truncation band.Source: §C.1 · p. 21, Figure 7 · p. 22
  • A 6% worst-case exponent-estimation deviation shifts p by approximately 0.07 and increases reconstruction cost by less than 0.1 percentage points at every T.This robustness follows from the flat loss landscape near its minimum.Source: §C.1 · p. 21

C.3 Per-room variation: the worst cases

  • Even the worst rooms are dominated by irreducible error: scene 00806 has 78.5% oracle-floor error, while the worst in-scope gap is 7.61 pp at T = 5.Across all rooms at T = 1000, 66.5% have absolute cost below 1.1 pp, rising to 68.4% among in-scope rooms.Source: §C.3 · pp. 23–24

C.2 Full cost table

  • Across rooms, relative cost rises from 0.5% at T = 1 to 5.6% at T = 1000, then declines to 3.3% at T = 2100.The non-monotonicity reflects differing extrema of the adaptation gap and oracle floor, not signal decay; absolute gaps remain below 10.6 pp at all T.Source: §C.2 · pp. 22–23, Figure 8 · p. 23, §C.4 · p. 24

C.5 Adaptation gap and oracle floor

  • The adaptation gap peaks at 1.04 pp at T = 500, while the oracle floor falls from 0.72 at T = 1 to 0.12 at T = 1000.The landscape range compresses 20× by T = 2100 as posterior concentration makes the regularizer nearly irrelevant; the 61-point grid reliably finds the global minimum because the basin is much wider than Δp = 0.1.Source: §C.5 · p. 24, Table 6 · p. 24, §C.6 · p. 25

C.7 Per-room landscape gallery

  • Across six rooms, population p = 1.13 remains inside the broad P/P0 ≤ 1.05 basin despite per-room p⋆ ranging from approximately 0.4 to 1.4.The rooms span 3–10 vertices, supporting the persistence of the flat basin across geometries.Source: Figure 9 · p. 25, §C.7 · p. 26

D.1 Architecture specifications

D.2 Training curves for all models

  • At T = 1000, M3 seeds learn exponents from −0.22 to 1.09 yet achieve the same reconstruction error, while inference-accessible features explain less than 2.3% of oracle-exponent variance.Room geometry collapses to a room-size factor explaining only 4% of variance, and the random forest reaches R2 = 0.14 at T = 1000 but below zero at T = 50.Source: §D.2 · p. 27, Table 7 · p. 28, §D.3 · p. 28

D.3 Could a better model do more?

  • A random forest using all features achieves R2 = 0.14 at T=1000 and drops below zero at T=50.Thus, 86% of p⋆ variance remains unexplained by the feature-based model.Source: §D.3 · p. 28, Figure 11 · p. 29
  • No accessible feature predicts p⋆ reliably: nine features have maximum absolute Spearman correlation 0.15 among inference-accessible features.The strongest accessible relationship is weak, while moderate geometry-dependent correlations are dominated by room-area proxies.Source: Figure 11 · p. 29
  • The Ktotal-adjusted estimator wins by 0.39 pp at T=1000 but loses by up to 3.4 pp elsewhere, leaving |s| strictly safer across operating conditions.This comparison covers 187 in-scope rooms, with OLS fitting and evaluation restricted to the in-scope subset.Source: §D.3 · p. 28, Table 9 · p. 29

D.4 Robustness to dataset parameters … D.8 Learned Iterative Ridge (LIR)

Across varied datasets, per-room exponent fitting, capacity checks, and seed sweeps, diagonal learning does not robustly improve the closed-form power law. Learned Iterative Ridge (LIR) escapes this diagonal saturation through cross-mode coupling, outperforming oracle Tikhonov at sufficient depth.

D.4 Robustness to dataset parameters

  • The exponent is regime-specific but plugs universally into Γ_k = λ_k^|s|; at M = 16, more sensors lower overall P while the flat landscape persists.The two settings use |s| = 1.13 and |s| = 1.29, and exact equality across M is not expected because the observation matrix changes the noise projection.Source: §D.4 · pp. 29–30
  • All six varied-parameter cells except one show ∆P ≥0, so M3 cannot beat the formula; the lone ∆P = −0.004 cell reflects exponent mismatch, not shape advantage.The varied dataset uses Kmax = 100, M ∈{8, 16}, and |s| = 1.29; Table 10 evaluates 197 validation rooms across five seeds.Source: §D.4 · p. 29, Table 10 · p. 30

D.5 Per-room vs population spectral exponent

  • Per-room exponents do not improve reconstruction: at T = 50, δ(sroom) = 0.88% versus δ(|s|) = 0.60%, and at T = 1000 only 45% of rooms benefit.Although sroom correlates with oracle p⋆ at T ∈{50, 100} with ρS = 0.40, noisy estimation and oracle drift undermine per-room tuning.Source: §D.5 · pp. 30–31, Table 11 · p. 30

D.6 Per-seed sweep heatmap

  • Across 225 training configurations, 13 small-training M3 runs at T = 1000 failed numerically, whereas every n = 800 main-text seed succeeded.The failures arose from ill-conditioned ˜Φ⊤˜Φ; Figure 12 marks failed configurations with grey cells.Source: §D.6 · p. 31, Figure 12 · p. 31

D.7 Model capacity verification

  • M3 recovers target power-law exponents monotonically up to ptarget = 2.5, showing its failure against |s| reflects loss-landscape flatness rather than insufficient capacity.M3 obtains precovered = {0.35, 1.35, 2.50} for ptarget = {0.5, 1.5, 2.5}; M2 saturates at ˆp ≤0.86, while M1 is non-monotonic.Source: §D.7 · pp. 31–32, Figure 13 · p. 32

D.8 Learned Iterative Ridge (LIR)

  • At depth L ≥5, LIR beats oracle Tikhonov across all observation windows, with the largest improvement at T = 100; excessive depth overfits.At T = 1, L = 20 has lower training loss (0.580 vs. 0.590) but higher test P (0.630 vs. 0.621) than L = 10.Source: §D.8 · p. 32, Table 12 · p. 32
  • LIR’s advantage comes from cross-mode coupling: removing off-diagonal estimator entries is catastrophic, while composed maps distribute regularization more richly than Tikhonov’s two-parameter rational function.At T = 100, stripping off-diagonals changes Tikhonov P from 0.435 to 82.5 and LIR P from 0.459 to 6.6; coupling originates from A⊤A.Source: §D.8 · p. 33, Table 13 · p. 33

E Heat Equation: Extended Results … F.2 Results table

This section extends the diagonal regularizer analysis to heat diffusion, examining the predicted exponential correction, its robustness, and the limits of diagonal optimality under late-time anisotropy. It also evaluates prior-exponent selection through oracle regret and loss-landscape flatness across contrasting times and prior families.

E.1 Why heat diffusion is different from acoustics

  • Heat modes decay as e^-κλ_kt, so the regularizer combines a power law with e^-2κtλ_k rather than using a pure power law.The exponential factor reflects PDE evolution, while the power-law term reflects initial conditions; predictions use κ > 0, with κ = 1 only in the synthetic experiments.Source: §E.1 · pp. 34–35

E.2 The three-step fitting procedure

  • R2 > 0.99 for the two-parameter model versus 0.87–0.93 for the pure power law, whose late-snapshot residuals reach ∼30 log-units.The fit uses OLS predictors log λ_k and λ_k; only |s| is estimated because c = 2κt is known before data collection.Source: §E.2 · pp. 35, 37
  • All five fitted slopes lie in [0.97, 1.00] with R2 ≥0.998, confirming that the estimated exponential rate tracks c_theory = 2κt.The pooled slope is 0.987 with 95% CI [0.953, 1.022], which contains 1.0; this is a consistency check because the synthetic forward model and modal basis share eigenvalues.Source: §E.2 · pp. 36–37, Figure 19 · p. 37

E.3 Sensitivity to κ uncertainty

  • A ±20% error in κ changes P by less than 0.8 percentage points relative to the oracle two-parameter regularizer, and a one-parameter fallback remains practical when κ is poorly known.At ±20% knowledge, the two-parameter regularizer is approximately equivalent to the one-parameter oracle; the one-parameter regularizer sacrifices 3–4% per-room improvement.Source: §E.3 · pp. 37–38, Table 15 · p. 38

E.4 One-parameter fallback and two-parameter gain

  • The acoustic exponent gives P = 0.523 (2.3× improvement), the one-parameter heat oracle gives P = 0.347 (3.5×), and the two-parameter oracle gives P = 0.334 (3.6×).The population gain is 0–2%, while the per-room median improvement is 3–4%; this gap reflects mild room-to-room variation in optimal c, not a systematically exploitable shape advantage.Source: §E.4 · p. 38, Table 16 · p. 38

E.5 Per-room spectral fits

  • Across five rooms, the representative-time fitted c/c_theory ratio stays within 2.2% of unity and every c-vs-t regression has R2 > 0.998.Room 00826 has a calibration intercept α0 = −0.072, but its single-time fit exactly matches c_theory and the offset does not affect the slope or cross-PDE conclusion.Source: §E.5 · pp. 39–40

E.6 Herfindahl index degradation under heat diffusion

  • By t_obs ≈250 ms, effective contributor count 1/H falls from ∼20–80 at 12 ms to ∼1–6; at 493 ms, H reaches 0.99 in the smallest diagnostic room.Proposition 1’s diagonal Bayes-optimality is guaranteed strictly at early t_obs under isotropic noise; late concentration could allow a non-diagonal estimator to outperform the diagonal regularizer.Source: §E.6 · p. 40

F Prior Robustness / F.1 Experimental setup

  • Prior-robustness experiments replace the Gaussian independent power-law prior with Student-t ν = 3 and adjacent-mode-correlated Gaussian alternatives having ρ = 0.3.The alternatives preserve marginal variance while testing heavy tails and broken independence; ν = 3 has finite variance and infinite kurtosis.Source: F Prior Robustness · p. 41, §F.1 · p. 41

F.1 Experimental setup

  • For 20 rooms and each prior, the experiment sweeps p on a 61-point grid at T ∈ {1, 100}, without neural networks or GPU computation.It computes the oracle p∗, δ(|s|), and the loss-landscape ratio max P/min P over p ∈ [0, 3]. Testing excludes T = 1000 because P → 0 without truncation noise, making δ meaningless.Source: §F.1 · p. 41

F.2 Results table

  • Table 19 reports δ(|s|) as the cost of using the population exponent rather than the per-prior oracle, with Flatness = max P/min P and 1.0 denoting a perfectly flat landscape.The table’s metrics quantify exponent misspecification and loss-landscape flatness over p ∈ [0, 3].Source: §F.2 · p. 41

F.3 Interpretation · G Rectangular Control Experiment · G.1 Setup

Prior misspecification leaves the diagonal power-law regularizer effective because noise isotropy and landscape flatness depend mainly on the noise and eigenvalue spectrum. Rectangular-room controls further show that Weyl dominance preserves this robustness even when Berry’s random-wave premise fails.

F.3 Interpretation

  • Gaussian and correlated priors preserve near-baseline behavior, with the correlated prior’s mild adjacent-mode dependence leaving the diagonal power law a good approximation.Gaussian results are δ = 1.2% at T = 1 and 4.9% at T = 100; correlated-prior results are δ = 0.9% and 5.3%.Source: §F.3 · pp. 41–42
  • Under the adversarial heavy-tailed prior, the formula incurs at most 11.0% relative cost, while the loss landscape remains moderately flat.At T = 1, δ = 3.8% and flatness = 1.179; at T = 100, δ = 11.0% and flatness = 1.536.Source: §F.3 · p. 42
  • The heavy-tailed prior shifts the optimal exponent toward zero because aggressive high-mode penalization can discard rare, informative large amplitudes.Even at T = 100, P = 0.412 versus the oracle’s P = 0.371, a 4.1 pp difference on a reconstruction error already at 37%.Source: §F.3 · p. 42
  • Across all three priors, Γ_k = λ_k^|s| remains useful because noise isotropy is prior-independent and Weyl eigenvalue spacing limits landscape curvature.For non-Gaussian priors, no quadratic Γ is exactly Bayes-optimal, but the flat landscape makes the power-law approximation error small.Source: §F.3 · p. 42

G Rectangular Control Experiment

  • Rectangular Dirichlet rooms provide a sharp Berry-isotropy stress test because their analytically known product-of-sines eigenfunctions have regular nodal lines and non-Gaussian cross-correlations.Analytical eigenpairs also remove FEM discretization error, isolating the combined integrability and boundary-condition departure.Source: G Rectangular Control Experiment · p. 42, §G.1 · p. 43
  • The rectangular control shows that the formula survives simultaneous Berry failure and boundary-condition departure through Weyl dominance, which supplies enough truncated modes to flatten the landscape.The resulting margin is described as robust enough to absorb both departures.Source: G Rectangular Control Experiment · p. 43

G.1 Setup

  • The setup spans four rooms—3 × 6, 2 × 8, 4 × 4, and 3 × 5 m—with 239–287 total modes, retaining K = 50 modes and using M = 8 random sensors.The square room tests common eigenvalue degeneracies, while the 2 × 8 room tests a 4:1 aspect ratio; P(p) is evaluated at T = 100.Source: §G.1 · p. 43

G.2 Eigenvalue spacing: Berry fails … H.1 Electronic sensor noise

Berry’s conjecture fails for rectangular rooms, but Weyl’s mode-count dominance makes those violations largely irrelevant to reconstruction: adding enough truncated modes flattens the regularizer landscape. Electronic sensor noise changes only the optimal strength, not the closed-form regularizer shape.

G.2 Eigenvalue spacing: Berry fails

  • Generic convex rooms match GOE statistics, whereas rectangular rooms reject GOE and are closer to Poisson, with the 4 × 4 room reaching D = 0.58.The convex-room result uses 50 pooled rooms and ∼13,000 spacings; the rectangular result uses four rooms and ∼2,000 spacings.Source: §G.2 · p. 43

G.3 Landscape ratio: but it doesn’t matter

  • At Ktotal = 75, adding 25 truncated modes collapses the landscape ratio from 3.49× to 1.31×, reaching 1.28× by Ktotal = 100 and 1.25× at Ktotal = 287.Figure 21b shows the landscape becomes indistinguishable from the generic convex reference after sufficient truncation modes.Source: Figure 21 · p. 44, §G.3 · p. 44
  • Despite non-random rectangular eigenfunctions, hundreds of bounded rank-one contributions concentrate toward isotropy, producing a generalized law-of-large-numbers effect.The rectangular control has 237 contributing terms, so bounded correlations with limited variance are overwhelmed by averaging.Source: §G.3 · p. 44, §G.4 · p. 45

G.4 The Weyl dominance principle

  • Weyl dominance makes Berry violations irrelevant when the signal dynamic range (λK/λ1)|s| ≈80:1 exceeds noise anisotropy of ∼4:1.The empirical median anisotropy is ∥E∥op = 0.58 across 187 rooms, yet it is insufficient to curve P(p).Source: §G.4 · p. 45
  • The framework requires Ktotal −K ≫1 and can fail in very small rooms with too few supported modes for meaningful acoustic reconstruction.Such rooms have wavelengths exceeding their dimensions, so the framework is explicitly not intended for them.Source: §G.4 · p. 45

H Sensor Noise and Model Mismatch

  • The noise analysis extends beyond truncation noise because electronic sensor noise, calibration errors, and model mismatch also contribute to the real measurement error budget.The appendix analyzes how these additional error sources affect the optimal regularizer.Source: H Sensor Noise and Model Mismatch · p. 45

H.1 Electronic sensor noise

  • Electronic sensor noise leaves Γk ∝ λk^s unchanged while increasing α, so the same formula applies whether truncation, electronics, or both dominate.Electronic noise is exactly isotropic, while truncation noise is approximately isotropic by the Berry/Weyl argument; their combined power changes strength but not shape.Source: §H.1 · pp. 45–46

H.2 Mild sensor noise anisotropy … I.2 Numerical instantiation

The appendix bounds when the diagonal regularizer remains robust and identifies physical sampling constraints that can invalidate modal recovery. Mild anisotropy is benign, while frequency-dependent damping, concentration diagnostics, and compact-aperture limits define when corrections or empirical validation are needed.

H.2 Mild sensor noise anisotropy

  • Realistic microphone mismatch below 3 dB perturbs ∥E∥op by O(0.1), much less than the empirical truncation anisotropy ∥E∥op ≈0.58, so Γk = λ^|s|_k remains robust.Comparable or larger sensor anisotropy could curve the P(p) landscape, but typical calibration mismatch adds only a small covariance perturbation.Source: §H.2 · p. 46

H.3 Frequency-dependent damping

  • Frequency-dependent damping follows γk = γ0 + γ1λk and requires an exponential regularizer correction when γ1 is known, whereas an elevated power-law exponent absorbs it when γ1 is unknown.The correction is structurally identical to the heat-equation factor e^-2γ1λkt.Source: §H.3 · p. 46
  • Below 500 Hz, γ1λKt ≪1 for retained modes, so uniform damping and Γk = λ^|s|_k usually suffice; strong absorption or higher frequencies may require the correction.The heat-equation analysis supplies the complete treatment when frequency-dependent damping is no longer negligible.Source: §H.3 · p. 47

H.4 Practical diagnostic: the Herfindahl check

  • The Herfindahl index provides a data-free diagnostic: H < 0.01 supports Γk = λ^|s|_k, whereas H > 0.1 requires empirical validation because increasing K toward Ktotal weakens isotropy.The impossibility pattern persists at M ∈{8, 16}, confirming the diagnostic’s stated M-sensitivity check.Source: §H.4 · p. 47

I Aperture Constraint on Physical Validation

  • The aperture appendix formalizes the spatial-sampling constraint, derives the aperture-to-wavelength bound, instantiates it for typical rooms, and reports a pilot measurement motivating follow-up work.These analyses distinguish structural spatial undersampling from ordinary signal-to-noise limitations.Source: I Aperture Constraint on Physical… · p. 47, §I.1 · p. 47

I.1 The aperture-to-wavelength bound

  • A compact array with D/ℓmin ≪1 produces nearly parallel sampling columns and noise amplification, so increasing microphones or recording length cannot repair the structural rank deficiency.Recovery requires enlarging the aperture until D/ℓmin is O(1) or sampling spatially distinct positions over time.Source: §I.1 · p. 47

I.2 Numerical instantiation

  • No room size supports a static compact array for K=50 recovery while preserving the framework’s assumptions, because smaller rooms create near-field effects and too few total modes.The closet has only 141 modes below the simulation ceiling, barely above the 50 retained modes.Source: §I.2 · p. 47, Table 20 · p. 48
  • For T ≥50 ms, the real-data Tikhonov landscape is flat with maxp P/ minp P = 1.00 and p⋆=0, while real |ŝ|=0.83 differs from synthetic |ŝ|=1.06.The slope gap Δ|ŝ|=0.23 signals prior mismatch, despite the flat regularizer landscape.Source: Figure 22 · p. 48

I.3 Empirical real-data validation · I.4 Resolution via distributed temporal sampling · NeurIPS Paper Checklist

This section validates the predicted regularizer behavior on real-room measurements, examines distributed temporal sampling as a hardware-preserving resolution strategy, and documents the paper’s scope, reproducibility, limitations, and release plans.

I.3 Empirical real-data validation

  • At T=10 ms, maxp P/minp P = 1.06, collapsing to 1.00 for T ≥50 ms, while δ(|ˆs|) ≤1.92% across tested T and p⋆=0 throughout real and synthetic data.The results support the flat-landscape prediction and identify ridge as optimal for the aperture-bounded spatial Gram.Source: §I.3 · pp. 48–49
  • |ˆs| = 0.83 with 95% CI [0.75, 0.96] on real recordings, versus |ˆs| = 1.06 (R2 = 0.73) synthetically; the real-data fit loses ∆R2 = 0.63.The controlled comparison attributes the gap to recording-chain effects, including prior mismatch, frequency-dependent damping, truncation, loudspeaker response, near-field violations, and Nsrc=5.Source: §I.3 · p. 49
  • The compact aperture makes Φ effectively rank ∼3 across K=50 modes: its top three singular directions capture 99.99% of energy, compared with 99.76% for recorded Y.This confirms aperture-bounded rank deficiency while preserving the flat-landscape prediction but undermining slope-recovery quality.Source: §I.3 · p. 49

I.4 Resolution via distributed temporal sampling

  • Temporal distribution preserves a compact hardware footprint by converting M static microphones into M · N effective measurement points along a moving trajectory.A sufficiently mixing trajectory can restore the eigenfunction-sampling isotropy premise, but trajectory optimization and empirical validation remain future work.Source: §I.4 · p. 49

NeurIPS Paper Checklist

  • The checklist reports that claims and limitations are documented, including approximate isotropy, 2D-only scope, the Gaussian prior, Berry failure in rectangles, 3D as future work, and the open non-diagonal question.It also states that Proposition 1 is proved, while Berry is presented as a conjecture verified in §5 and Appendix B.Source: NeurIPS Paper Checklist · pp. 50–51
  • All experimental settings and uncertainty procedures are specified, but open code and data are unavailable at submission and code is promised upon acceptance.The checklist gives K=50, M=8, train/validation splits, optimization details, seeds, confidence intervals, compute resources, and planned release assets.Source: NeurIPS Paper Checklist · pp. 51, 52, 55
  • The study involved no human subjects or crowdsourcing, so IRB approval was not applicable.The checklist marks these activities as not applicable.Source: NeurIPS Paper Checklist · p. 55
  • LLMs were not part of the core methodology; they were used only for writing and editing, not experiments.The checklist therefore marks LLM usage as not applicable.Source: NeurIPS Paper Checklist · p. 56

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