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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

Modal inverse problems require regularizing finitely observed eigenmode expansions corrupted by truncation noise, raising the question of how the diagonal penalty shape should be chosen. The paper derives a prior-determined power law, explains its robustness through Berry decorrelation and Weyl mode counts, and finds that diagonal learning rarely improves on it while cross-mode coupling does.

  • Problem

    The paper asks what shape the diagonal Tikhonov regularizer should have when finite observations are contaminated by truncation noise across modal inverse problems.

  • Method

    The paper combines Bayesian analysis with Berry’s random-wave conjecture and Weyl’s eigenvalue-counting law to derive a power-law shape from the prior spectrum.

  • Results

    Across 187 rooms, the closed form stays within 5.82% median relative cost of per-room oracle tuning, while diagonal architectures match it within 1 pp and cross-mode coupling improves over the diagonal oracle by 1.9–13.5 pp.

  • Takeaways & Limitations

    Within the diagonal family, learning has narrow scope because the loss landscape is approximately flat; learning starts to help through non-diagonal cross-mode coupling.

  • Takeaways & Limitations

    The study uses random convex 2D polygon rooms and uniformly random sensors; extension to 3D and validation on structured sensor arrays remain future work.

Abstract

from arXiv · show

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

Modal inverse problems must recover infinitely expandable states from finitely many measurements contaminated by truncation noise. The paper asks whether the diagonal regularizer’s shape is universal across rooms and argues that isotropic truncation noise makes it a prior-determined power law.

  • Motivation: Finite sensors leave modal inverse problems underdetermined, while discarded modes contaminate measurements as truncation noise.In the acoustic instance, K=50 retained modes and M=8 microphones yield 8 equations for 100 unknowns, while typical rooms contain over 300 modes.
  • The shape question: The central question is whether the power-law exponent p should depend on the room or equal the prior’s spectral-decay exponent.The penalty family is Γ_k = λ_k^p, with p=0 treating modes equally and p=2 strongly suppressing high frequencies.
  • Why the noise decides: If truncation noise is isotropic, the optimal exponent is p∗=|s|, determined by the signal prior rather than room-specific structure.Berry’s conjecture motivates decorrelation across modes, while Weyl’s law supplies enough modes for concentration despite empirical violations.
  • Contribution: The paper’s novel claim is that Γ_k ∝ λ_k^|s| is room-independent, allowing one prior-based shape to be reused across rooms.The numerical value of |s| is prior-specified; the claimed contribution is its domain independence.
  • Contribution: Across 187 in-scope rooms, p=|s| stays within 5.82% median relative cost of per-room tuning, while diagonal architectures remain within 1 pp of the closed form.The reported diagonal results include M3 within ±0.3 pp at every T and M1 and M2 within 1 pp.

2 Related Work

Prior work established regularization strength selection and studied shape selection through covariance-dependent estimators, learned regularizers, and domain physics. This paper positions its contribution as explaining why Berry’s conjecture and Weyl’s law reduce the shape question to the prior spectrum in modal inverse problems.

  • Regularization: Classical methods reliably select Tikhonov strength α, but the shape of the diagonal penalty Γ remains a separate problem.The related work distinguishes scalar strength selection from choosing which modes to penalize.
  • Shape selection: Existing optimal-shape results depend on the signal covariance, motivating a physics-based route to determine that covariance analytically.The paper connects power-law excitation priors to a single spectral-decay parameter |s|.
  • Learning: Learned regularizers often rediscover classical variational solutions or reduce to hyperparameter tuning, framing the paper’s diagonal-saturation question.The cited literature includes algorithm unrolling, bilevel optimization, and learned-parameter convergence results.
  • Physics: Berry’s random-wave conjecture and Weyl’s law provide the domain-physics basis for approximate truncation-noise isotropy in generic rooms.Berry supplies approximate modal decorrelation, while Weyl supplies many high-frequency modes.
  • Applications: In room acoustics, the paper studies penalty shape rather than sensor placement, extending related Bayesian, compressive-sensing, array, and physics-informed approaches.The heat-equation literature provides scaling results but not the exact regularizer or its dependence on κ and t.

3 Problem Setup

The problem setup represents acoustic and heat-related modal states on bounded domains using Laplacian eigenpairs and finite microphone observations. It estimates retained modal coefficients with diagonal Tikhonov regularization and evaluates normalized modal reconstruction error.

  • Domain and state: The framework uses Laplacian eigenpairs on a bounded 2D domain and validates the acoustic instance across 187 random convex polygons.Heat diffusion serves as a cross-PDE check rather than the primary simulation setting.
  • Modal representation: Each eigenfunction is a spatial pattern, while modal amplitudes encode its contribution and typically lose energy with eigenvalue according to prior exponent s.The exponent depends on excitation statistics rather than room geometry.
  • Dynamics: Uniform damping makes acoustic modal evolution mode-independent, distinguishing the acoustic model from heat diffusion.The modal variance is defined from random initial conditions for the coefficient pairs.
  • Observation model: The observation model retains K=50 modes, samples M microphones, and separates modeled retained signal from discarded-mode truncation noise.Stacking microphones over T snapshots gives ỹ = Φ̃a_0 + η̃.
  • Estimator and metric: The estimator minimizes penalized least squares with α>0 controlling overall strength and diagonal Γ controlling per-mode penalty.Reconstruction quality is measured by normalized modal MSE P, where P=0 is perfect and P=1 equals zero-guessing performance.

4 Why p∗= s: The Three-Step Argument

The three-step argument links Bayesian optimality under isotropic noise to Berry-based modal decorrelation and Weyl-based concentration. Together, these mechanisms explain why the power-law penalty remains effective even when empirical noise anisotropy is moderate.

  • 4.1 Step 1: If the noise is flat, the answer is immediate: Under isotropic truncation noise and a Gaussian prior with Σ_kk ∝ λ_k^-s, the optimal diagonal penalty has power-law shape Γ_k ∝ λ_k^s.The noise level determines only α, while the prior determines the shape.
  • 4.1 Step 1: If the noise is flat, the answer is immediate: The penalty suppresses low-variance modes more strongly because they have less signal energy to lose, reducing the shape problem to the signal covariance.For wave-chaotic systems, the paper treats that covariance as analytically determined by physics.
  • 4.2 Step 2: Berry’s conjecture: Berry’s conjecture supplies the needed approximation that high-frequency eigenfunctions sampled at generic sensors are decorrelated across modes.The paper uses a weaker sensor-location version of the random-wave conjecture.
  • 4.2 Step 2: Berry’s conjecture: The Herfindahl index H quantifies whether truncation noise is concentrated in one discarded mode or distributed across many modes, with small H supporting isotropy.The operator norm of the anisotropy matrix is bounded by its Frobenius norm.
  • 4.3 Step 3: Weyl’s law: why the conclusion survives Berry violations: Weyl’s law gives a median of approximately 263 truncated modes in the studied rooms, providing enough terms for concentration.This mode count helps the conclusion survive imperfect Berry behavior.
  • 4.3 Step 3: Weyl’s law: why the conclusion survives Berry violations: Across 187 rooms at T=1000, higher noise anisotropy correlates with lower cost, with Spearman ρ=−0.30 and p<10^-4, showing shape insensitivity to moderate anisotropy.The paper identifies eigenvalue dynamic range, rather than noise anisotropy, as the bottleneck and calls this Weyl dominance.

5 Empirical Verification on Acoustic Rooms

Across 187 in-scope acoustic rooms, the population power-law exponent remains near-optimal across observation windows, while the reconstruction landscape becomes increasingly flat.

  • Setup: The excitation exponent is recovered as |ŝ|=1.13±0.05, and the flat-landscape pattern persists under changes to truncation rank, sensor count, and excitation exponent.The estimate has bootstrap 95% CI [1.08, 1.18].
  • Landscape flatness: The median P-range across p ∈[0, 3] compresses ∼14×, from 0.18 at T=1 to 0.013 at T=2100.The per-room oracle p⋆ drifts from approximately 1.2 to 2.0, but the basin widens faster than the optimum shifts.
  • Landscape flatness: At every observation window, a fixed |s|=1.13 remains inside the broad optimum basin.Figure 1 compares the fixed exponent with per-T oracle optima across 187 rooms using K=50 and M=8.
  • Oracle comparison: The median relative cost of using |s| instead of per-room tuning stays below 5.82% at every observation window.Across 187 rooms, 68.4% incur less than 1.1 pp absolute cost, while the worst in-scope cost is 7.61 pp.
  • Isotropy verification: The Berry prediction fits pooled modal samples with KS statistic D=0.037 across N=77,968 observations.The empirical distribution deviates from χ2(1) by at most 3.7%.

6 Can Diagonal Learning Improve Upon |s|?

Diagonal architectures do not robustly improve on the closed-form power law, even when they learn different spectra; cross-mode coupling is the mechanism that surpasses the diagonal oracle.

  • Beyond the diagonal family: M1, M2, and M3 remain on the ridge(|ŝ|) curve, whereas LIR is the only model that falls below the per-room oracle.Figure 2 compares learned spectra and reconstruction error across the diagonal models and LIR.
  • Diagonal saturation: M3 matches the ridge at |s| within 0.31 pp at every T despite qualitatively different learned spectra across seeds.At T=1000, five seeds produce effective exponents from −0.22 to 1.09 while achieving nearly identical reconstruction error.
  • Population versus per-room fitting: Per-room slope fitting does not improve on |s| and is 0.28 pp worse at T=50.Per-room estimation noise σper-room≈0.27 exceeds the population-median standard error σpop≈0.026.
  • Diagonal saturation: Across tested diagonal parameterizations, no learned model achieves robustly lower error than Γk=λ^|s| across operating points.Across three architectures, five training sizes, and three snapshot counts, 212 of 212 valid evaluations have ΔP≥0 relative to the per-room oracle.
  • Beyond the diagonal family: LIR introduces per-layer learned gradient-descent parameters to exploit the full A⊤A structure rather than only per-mode weights.The architecture uses L steps with (ηl, αl, Dl) and 52L total parameters.
  • Beyond the diagonal family: At L=10, LIR improves over the diagonal oracle by 9.4, 13.5, and 1.9 pp at the three reported observation windows.Its errors are P={0.621, 0.459, 0.103}, versus the oracle’s {0.715, 0.594, 0.122}.

7 Extension to Heat Diffusion

Heat diffusion fits the same regularization framework after adding a theoretically determined exponential correction to the power-law prior, with only the spectral exponent estimated from data.

  • Heat regularizer: Heat diffusion requires Γk∝λk^|s|·e^{cλk}, because the Green’s function adds exponential modal decay to the power-law initial spectrum.The exponential factor is inverted in the regularizer through Γ∝Σ−1.
  • Heat regularizer: The correction c=2κt is fixed by observation time and diffusivity rather than fitted from data.Only |s| is estimated, so the two-parameter form introduces no new free parameter.
  • Cross-PDE verification: Across five diagnostic rooms, fitted slopes of ĉ versus ctheory lie in [0.97, 1.00] with R2≥0.998.Per-room intercepts are below 0.10 in absolute value.
  • Cross-PDE verification: For synthetic heat data, |s|≈1.0 rather than 1.13 because the excitation statistics differ, while the framework remains competitive with per-room oracle tuning.At late times, signal decay below measurement noise coincides with a breakdown of the isotropy assumption.

8 Discussion and Conclusion

The paper concludes that isotropy makes the closed-form power law a saturation point for diagonal regularization, while remaining gains require structural extensions beyond per-mode weighting.

  • Main conclusion: Under approximate Berry-Weyl isotropy, Γk=λk^|s| depends on the excitation process rather than the room.The population median stays within 5.82% relative cost of per-room oracle tuning across the evaluated rooms and observation windows.
  • Main conclusion: No diagonal learned architecture robustly improves on the closed form, while LIR improves over the diagonal-family oracle by 1.9–13.5 pp through non-diagonal coupling.The remaining gains are attributed to structural extensions such as cross-mode coupling, temporal dynamics, and trajectory sensing.
  • Scope and limitations: The study is limited to random convex 2D polygonal rooms with uniformly random sensors, while 3D and structured sensor layouts remain extensions.Structured arrays may introduce correlations not captured by the uniform-sensor analysis.
  • Empirical scope: A real-data pilot confirms the flat-landscape prediction but recovers |ŝ|=0.83, below the population value |s|=1.13.The gap is consistent with recording-chain effects that cannot be disentangled from one static array.
  • Future directions: Sequential temporal modeling remains an open frontier because the current treatment regards snapshots as exchangeable rather than exploiting full-recording modal dynamics.Kalman and state-space methods are identified as candidate directions.
  • Theory and evidence: The exact Bayes-optimal diagonal shape is established under isotropic Gaussian noise and a diagonal Gaussian prior; Berry’s conjecture and Weyl’s law motivate approximate isotropy empirically.The empirical verification, rather than the physical heuristics alone, supplies the primary evidence for the approximation.

A.1 Bayesian derivation of Proposition 1

Under isotropic truncation noise and a Gaussian power-law prior, the Bayesian derivation identifies Γ_k ∝ λ_k^|s| as the MMSE-optimal diagonal Tikhonov shape. Empirical analyses show approximate isotropy and a flat diagonal loss landscape, while clarifying that anisotropic noise permits non-diagonal improvements beyond the diagonal family.

  • Bayesian derivation: Γ_k ∝ λ_k^|s| is Bayes-optimal within the diagonal Tikhonov family under exact isotropy and a Gaussian power-law prior.The MAP estimator coincides with the posterior mean and is therefore MMSE-optimal among all estimators under the jointly Gaussian model.
  • Generative model: The model assumes independent modal coefficients with variance c λ_k^-s and isotropic truncation noise with variance σ^2 per measurement.Low-frequency modes carry larger prior variance, while high-frequency modes carry less energy and are penalized more strongly.
  • Scope of optimality: The diagonal result does not imply global MMSE optimality under approximate isotropy, because the true Bayes estimator can couple modes through E.The paper’s per-room oracle is restricted to the diagonal Tikhonov family, and reconstruction cost is governed primarily by eigenvalue dynamic range rather than noise anisotropy.
  • Empirical isotropy: Across 187 rooms, noise anisotropy is moderate, with median ||E||op = 0.58, mean 0.88, and 95th percentile 2.42.The right tail is associated with larger Berry violations, while the median room is close to the Frobenius bound.
  • Empirical isotropy: The closed-form exponent remains inexpensive because signal dynamic range is about 80:1 versus roughly 4:1 noise anisotropy across sensor directions.The population-median relative cost peaks at 5.82% at T=1000, and 68.4% of in-scope rooms remain below 1.1 percentage points.
  • Berry–Weyl mechanism: Berry agreement is close to the χ2(1) prediction with pooled KS D = 0.037, and Weyl’s mode supply preserves robustness despite imperfect decorrelation.The Berry pass rate peaks at 89% for K ∈ [50, 200], while high mode counts become statistically oversensitive despite small absolute deviations.

B.4 Worst-5-rooms tail analysis

The worst Berry-agreement rooms do not exhibit a corresponding reconstruction failure. Their elevated costs are limited and are attributed mainly to long-window oracle dispersion rather than Berry violations themselves.

  • Selection: The five-room analysis selects the largest KS statistics from 168 rooms after excluding 29 rooms with fewer than 50 truncated modes.The accompanying table sorts these five rooms by descending Berry-test statistic D.
  • Worst-room outcomes: Room 00835 reaches δ(T=1000) = 25.2% but stays at δ ≤3% for T ≤100, indicating long-window oracle dispersion rather than Berry failure.The five selected rooms are mid-to-large rooms with Ktotal > 400 and near-median Herfindahl indices H ≈ 0.003–0.004.
  • Worst-room outcomes: Even the worst-Berry rooms have reconstruction costs within the main-text range, so Berry agreement is only weakly predictive of cost.Across 168 eligible rooms, worse Berry agreement tends to increase cost slightly, but the effect remains small.
  • Interpretation: Weyl dominance explains why genuine non-Gaussian cross-correlations do not translate into meaningful reconstruction cost.The conclusion is that imperfect isotropy is sufficient for the diagonal loss landscape to remain flat.

C.1 Estimating |s| in practice

The population exponent is stable across rooms and datasets, while broad loss landscapes make precise per-room tuning largely unnecessary. Learned diagonal models likewise find different spectra without improving reconstruction error.

  • |s| estimates are stable: the median is 1.13, with bootstrap 95% CI [1.08, 1.18] and deconvolved inter-room std 0.05.
  • Using subsets of 50 rooms shifts p by approximately 0.07 and increases reconstruction cost by less than 0.1 percentage points at every T.
  • The worst boundary room has a 10.6 pp recoverable gap, but 78.5% of its total error is the irreducible oracle floor.
  • Across in-scope rooms, the maximum absolute cost is 7.61 pp, while 68.4% incur less than 1.1 pp absolute cost.
  • The loss landscape compresses 20× from T = 1 to T = 2100, and the oracle floor falls from 0.72 to 0.12 before rebounding to 0.15.
  • M3 seeds learn exponents from −0.22 to 1.09 yet achieve essentially identical reconstruction error, indicating no gradient signal selects a unique diagonal spectrum.

D.4 Robustness to dataset parameters

Robustness tests preserve diagonal saturation across truncation rank, sensor count, and excitation regime. Capacity checks show that M3 can recover target power laws, so its failure to improve reflects the loss landscape rather than representational limits.

  • Robustness to dataset parameters: The exponent is regime-specific rather than universal: |s| changes from 1.13 to 1.29, while Γ_k = λ_k^|s| remains the prescribed form.
  • Robustness to dataset parameters: At M = 16, reconstruction error is lower, but the landscape remains flat and M3 still cannot improve on the formula.
  • Robustness to dataset parameters: Five of six varied-parameter cells have ∆P ≥0, so M3 does not beat the formula across Kmax = 100, M ∈{8, 16}, and |s| = 1.29.
  • Per-room variation: Per-room spectral slopes correlate with oracle exponents at T ∈{50, 100} with ρS = 0.40, yet per-room tuning remains worse than the population exponent.
  • Model capacity verification: M3 recovers target exponents {0.35, 1.35, 2.50} for targets {0.5, 1.5, 2.5}, reaching the target exactly at 2.5.

E Heat Equation: Extended Results

Heat diffusion preserves the prior-driven power-law regularizer only after adding the Green’s-function factor e^(-2κλ_kt), which substantially improves spectral fits over a pure power law. The correction is empirically validated, moderately robust to κ error, and supports a one-parameter fallback when diffusivity is uncertain, while late-time noise concentration limits diagonal Bayes optimality.

  • E.1 Why heat diffusion is different from acoustics: Heat requires Γ_k ∝ λ_k^|s| e^(-2κλ_kt) because high-frequency modes decay exponentially faster, a correction determined by the known diffusivity and observation time.The exponential rate is c = 2κt, while |s| is the only parameter estimated from data.
  • E.1 Why heat diffusion is different from acoustics: The exponential-power-law model achieves R2 > 0.99 at all snapshots, whereas the pure power law reaches only R2 = 0.87–0.93 and incurs residuals of about 30 log-units at late times.The exponential-power residuals remain bounded by approximately 0.1 log-units.
  • E.3 Sensitivity to κ uncertainty: A ±20% error in κ changes the fitted rate by ±20% but increases reconstruction cost by less than 0.8 percentage points relative to the two-parameter oracle.The two-parameter regularizer is strictly better when κ is known within ±10% and approximately equivalent to the one-parameter oracle at ±20%.
  • E.4 One-parameter fallback and two-parameter gain: At T = 2100, the two-parameter heat oracle gives P = 0.334, compared with P = 0.347 for the one-parameter heat oracle and P = 0.523 for the acoustic exponent.The identity baseline gives P = 1.210 in the same setting.
  • E.4 One-parameter fallback and two-parameter gain: The two-parameter gain is 0–2% at the population level but 3–4% per room, reflecting mild variation in optimal c rather than a systematically learnable shape advantage.The one-parameter fallback sacrifices 3–4% per-room improvement but requires no material-property knowledge.
  • E.6 Herfindahl index degradation under heat diffusion: At late observation times, heat-noise concentration can invalidate the isotropy condition required for diagonal Bayes optimality, although early snapshots dominate the average regime.The effective contributor count falls sharply over time, and a non-diagonal estimator could in principle outperform the diagonal regularizer late.

F.3 Interpretation

The closed-form power law remains effective under prior misspecification because the noise-isotropy mechanism and spectral flatness are largely separate from the prior. Heavy-tailed and correlated priors alter the exact optimum but leave the diagonal landscape sufficiently flat for the formula to remain a practical approximation, including when Berry’s conjecture fails.

  • Prior robustness: Under the heavy-tailed prior, using |s| costs 11.0% at T = 100, corresponding to P = 0.412 versus the oracle’s P = 0.371, or a 4.1 percentage-point difference.This is the study’s worst prior-robustness case, but the landscape flatness is still only 1.536.
  • Prior robustness: The correlated prior behaves nearly like the Gaussian case: δ = 5.3% and flatness = 1.452 at T = 100, despite mild off-diagonal covariance.The diagonal power law remains a good approximation even though the exact optimum is no longer diagonal.
  • Prior robustness: Across Gaussian, heavy-tailed, and correlated priors, the diagonal landscape remains flat, so Γ_k = λ_k^|s| stays a useful approximation despite prior mismatch.The non-Gaussian cases are not exactly Bayes-optimal for any quadratic Γ, but their approximation error remains limited.
  • Berry violations and Weyl dominance: Berry’s conjecture fails sharply in rectangular rooms, with GOE rejected at p < 10^-5, yet Weyl dominance preserves the regularizer’s robustness by supplying enough truncated modes to flatten the landscape.The rectangular stress test uses analytical eigenpairs and removes FEM discretization error.
  • Berry violations and Weyl dominance: The mechanism is limited to settings with many truncated modes: very small rooms where Ktotal − K is not much larger than one fall outside the framework’s intended regime.Weyl dominance relies on concentration from a sufficiently large truncated-mode count.

G.2 Eigenvalue spacing: Berry fails

Rectangular rooms provide a stringent test because their integrable eigenfunctions violate Berry’s random-wave premise and strongly reject GOE statistics. Nevertheless, adding truncated modes rapidly flattens the regularization landscape, supporting Weyl dominance as the broader explanation for robustness and preserving the diagonal shape under additional isotropic sensor noise.

  • Eigenvalue spacing results: Rectangular rooms reject GOE with D > 0.19 and p < 10^-5, while generic convex rooms match GOE with D = 0.010 and p = 0.17.The square room is the strongest violation, with D = 0.58, due to eigenvalue degeneracies.
  • Control experiment: Adding 25 truncated modes collapses the 3 × 6 room’s landscape ratio from 3.49× to 1.31×, showing that reconstruction robustness does not require Berry statistics.By Ktotal = 100, the ratio reaches the generic convex reference of 1.28×.
  • The Weyl dominance principle: Weyl dominance works because many bounded, non-random rank-one contributions concentrate toward isotropy even when individual rectangular eigenfunctions are not Gaussian or independent.The mechanism is a generalized law-of-large-numbers effect rather than universal validity of Berry’s conjecture.
  • The Weyl dominance principle: The resulting anisotropy is moderate, with empirical median ||E||op = 0.58 across 187 rooms, but the signal dynamic range dominates the noise eigenvalue ratio and leaves the landscape shallow.The reported comparison is approximately 80:1 for signal dynamic range versus 4:1 for noise eigenvalue ratio.
  • Additional noise sources: When electronic sensor noise is added, the optimal response is to increase the scalar strength α while leaving the shape Γ_k ∝ λ_k^|s| unchanged.The truncation and electronic noise components are both approximately isotropic in the analyzed setting.

H.2 Mild sensor noise anisotropy

Mild sensor calibration mismatch perturbs truncation-noise covariance only modestly, while the framework’s broader validity remains constrained by damping, aperture, and truncated-mode regimes.

  • < 3 dB microphone mismatch perturbs ∥E∥op by O(0.1), smaller than the empirical truncation anisotropy ∥E∥op ≈0.58, so Γk = λk^|s| remains robust.
  • Frequency-dependent damping can invalidate uniform-damping assumptions; its exponential correction matters mainly at higher frequencies or under strong absorption.
  • As K approaches Ktotal, the truncation band shrinks, the Herfindahl index rises, isotropy weakens, and the closed-form regularizer requires empirical validation.
  • Within a fixed compact aperture, increasing microphone count or recording length cannot overcome near-rank deficiency; recovery requires a larger aperture or spatially distinct sampling.
  • Across room scales, the K=50 aperture analysis reports retained-mode wavelengths, aperture ratios, and maximum spatial variation for the miniDSP array.

I.3 Empirical real-data validation

Real and matched synthetic recordings support the framework’s flat-landscape prediction, but real-data recovery of the prior exponent is substantially weaker than in the controlled synthetic case. The compact aperture creates an effective rank constraint, motivating spatial or temporal measurement diversity while leaving trajectory optimization open.

  • Results: maxp P/minp P falls from 1.06 at T=10 ms to 1.00 for T ≥50 ms, while δ(|ˆs|) remains ≤1.92% across tested windows.The optimal exponent p⋆ is zero at every T on both real and synthetic data.
  • Results: Real recordings recover |ˆs| = 0.83 with 95% CI [0.75, 0.96], stable across RT60 ∈[0.5, 1.5] s.The synthetic comparison instead recovers |ˆs| = 1.06 with R2 = 0.73, within the population prior |s|=1.13.
  • Results: The real-data slope underestimates the synthetic-controlled estimate by ∆|ˆs| = 0.23 and loses ∆R2 = 0.63, attributing the residual to recording-chain effects and prior idealization.The matched synthetic dataset controls room, microphones, source positions, modes, RT60, and the analysis pipeline.
  • Aperture diagnostics: At D/ℓmin ≈12.8%, the geometry-only and recorded matrices have top-three energy shares of 99.99% and 99.76%, respectively, indicating effective rank ∼3 across K=50 modes.These diagnostics support the predicted aperture-bounded rank deficiency while the flat-landscape prediction survives compactness.
  • Measurement diversity: The aperture constraint can be addressed spatially with a larger array or temporally by moving a compact array, yielding M · N effective measurement points over N positions.The temporal option preserves the compact hardware footprint and acquires spatial diversity through motion.
  • Measurement diversity: A sufficiently mixing trajectory can restore eigenfunction orthogonality in expectation, but trajectory design under path-length or duration budgets remains future work.The trajectory formulation and its theoretical analysis are left open.
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