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When Noise Estimation Hides Basis Misspecification in Repeated Bayesian Inverse Problems

Rares Dimitrie Grozavescu, Mark Girolami

arXiv:2609.37762v1stat.MEmath.STstat.MLstat.TH

TL;DR

Basis-restricted priors can lose field coverage when the truth has out-of-basis components, and fitting observation-noise variance can conceal that loss. The paper analyzes this absorption and tests scale-free residual-spectrum shape across repeated problems. The test detects structured out-of-basis variation, but not Gaussian variation that is isotropic in the complement.

  • Problem

    Basis-restricted priors can lose credible-set coverage when the true field has components outside the basis.

    Source: §1 · p. 1

  • Method

    The paper projects repeated observations onto the model-range complement and tests residual-spectrum shape with John’s statistic, using scale-free direction after fitting noise.

    Source: §1 · p. 2, §4 · p. 5

  • Results

    Observation coverage moved by 0.02 while field coverage fell by more than 0.5, and the scale-free test detects structured out-of-basis variation.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Residual-spectrum shape can reveal structured misspecification that noise-scale absorption hides from observation-space checks.

    Source: §1 · p. 2

    The real-data evidence covers one field, one basis construction, and three operators; the test also cannot detect Gaussian isotropic complement energy.

    Source: §8 · p. 9

Abstract

from arXiv · show

Basis-restricted priors in Bayesian inverse problems can lose coverage when the truth has components outside the basis. We show that estimating the observation-noise variance can hide this loss. Under a linear forward model, when the in-span prior variance dominates the noise, the maximum-likelihood noise estimate absorbs the out-of-basis energy in the complement of the model range. Residual-magnitude and observation-coverage checks then stay near nominal while field coverage falls. We study repeated problems sharing one forward operator and one basis, fixed independently of the tested data. After projection onto the complement, and conditionally on the fitted noise scale, every exact test is a test of the scale-free direction of the residuals. We test the shape of their sample spectrum with John's sphericity statistic. Under Gaussian noise its null model is exact at finite sample size, and we derive its null mean and its power at proportional dimension. On synthetic problems and in a preregistered GEBCO topography study, the test detects structured out-of-basis variation that cross-validation and observation-coverage checks largely miss. It cannot detect Gaussian out-of-basis variation that is isotropic in the complement, since that is indistinguishable from a change of noise scale.

1 Introduction

Basis restriction can oversmooth fields with out-of-basis components, while fitting noise variance can conceal the resulting field-coverage loss. The paper explains this absorption and tests residual shape across repeated problems to detect structured misspecification.

1 Introduction

  • When truth has components outside the fixed basis, the restricted prior oversmooths and its credible sets can have degenerating frequentist coverage (Knapik et al., 2011).Source: §1 · p. 1
  • With in-span prior scale dominating noise, the fitted noise scale absorbs out-of-basis energy in the model-range complement, hiding misspecification from observation-space checks.Source: §1 · pp. 1–2
  • Observation coverage moved by 0.02 while field coverage fell by more than 0.5, and held-out observation checks therefore pass despite field-coverage loss.The paper attributes this dissociation to fitted noise absorbing out-of-basis energy.Source: §1 · pp. 1–2
  • The method requires repeated observations through one operator and basis, with the basis fixed independently of the tested data; it does not apply to one observation vector.Source: §1 · p. 2
  • The paper identifies noise-scale absorption as the channel hiding basis misspecification, while pinning the true noise scale removes concealment but not field-coverage collapse.Source: §1 · p. 2
  • Conditionally on fitted noise, every exact test on the complement tests residual direction; John’s statistic tests sample-spectrum shape, with finite-sample null results and proportional-dimension power derived.Source: §1 · p. 2

2 Setup

The setup studies repeated observations of independent fields through a shared linear forward operator, with a basis fixed independently of the tested realizations. It separates modeled-range variation from unmodeled energy in the complement.

2 Setup

  • The model uses one forward operator, noise scale, and basis for N observation vectors, each arising from an independent field realization.Source: §2 · p. 3
  • The basis must be fixed independently of tested realizations; learning it from those same data makes the complement data-dependent and invalidates the finite-sample null exactness.Source: §2 · p. 3
  • The field decomposes into an in-basis component and an orthogonal residual; the observable law identifies coefficient and basis-restricted residual covariances only through their sum.The paper therefore reports no separate covariance recovery; its contribution concerns basis restriction, fitted noise, and the complement test.Source: §2 · p. 3
  • Projecting observation space onto the complement of range(HΦ) removes modeled structure, leaving the out-of-basis observation energy as the unmodeled component.Source: §2 · p. 3

3 Absorption of the Noise Scale

Because the model has only noise variance in the complement of its range, its likelihood separates into modeled-range and complement terms. When in-span covariance dominates, the complement residual magnitude determines the fitted noise variance, which absorbs out-of-basis energy.

3 Absorption of the Noise Scale

  • Under the stated linear, basis-restricted covariance model, the likelihood separates into range and complement terms because the complement covariance is σ²P⊥.Source: §3 · pp. 3–4
  • Consequently, in the dominance regime ˆσ²/σ² equals ρ in expectation and ˆσ/σ is approximately √ρ, absorbing excess energy into the fitted noise scale.Source: §3 · p. 3
  • When in-span covariance dominates and the complement residual magnitude meets the stated condition, the exact maximum-likelihood estimate is ˆσ² = ∥P⊥y∥²/(n − r).Source: §3 · p. 4
  • Under the true law, the complement residual contains projected out-of-basis field energy plus projected noise, so its squared magnitude per dimension has expectation σ²⊥.Source: §3 · p. 4
  • Across 31 sweep points, the implemented noise update’s ˆσ/σ matched √ρ within 3.4%, or within 0.5% after a finite-n degrees-of-freedom correction.The implemented update differs from the exact maximizer by a factor involving (n − r)/n.Source: §3 · p. 4

4 A Scale-Free Test on the Complement

The complement residuals retain scale-free directional information after noise-scale fitting, so the paper tests their sample-spectrum shape with John’s sphericity statistic. Its null model is exact for Gaussian noise at finite sample size, but isotropic Gaussian out-of-basis energy is undetectable.

4 A Scale-Free Test on the Complement

  • Conditionally on the fitted noise scale, every exact test on complement residuals is a test of their scale-free direction; John’s statistic tests the shape of their sample spectrum.Source: §4 · p. 4
  • Information about σ that escapes absorption—such as a known instrument noise level, repeat measurements, or noise-only runs—can constrain the noise scale; the study setting has none.With unconstrained coefficient covariance, range(G) supplies only a one-sided bound on σ².Source: §4 · p. 5
  • Under Gaussian noise and correct specification, the complement null model is exact for every finite (N, p), without asymptotic or p/N conditions.The implemented test uses a parametric Monte Carlo bootstrap at the matching (N, p), which carries Monte Carlo error.Source: §4 · p. 5
  • For k equal covariance spikes, detectability depends on (ρ − 1)p/k; the observable detection threshold eR needs neither σ nor k.The null median of eR is 1 at every finite (N, p).Source: §4 · p. 5
  • If out-of-basis energy is Gaussian and isotropic in the complement, the covariance remains spherical and the scale-invariant statistics have exactly their null distributions at any ρ.Source: §4 · p. 6
  • The test compares John’s statistic with its bootstrap null at matching (N, p), and its null mean formula matched bootstrapped means across 59 pairs to 0.041% median relative error.Source: §4 · p. 6

5 Synthetic Experiments

The synthetic experiments show that observation-space checks can miss severe field-coverage loss, while the spectrum-shape test detects many structured alternatives but not isotropic Gaussian excess. Its power is calibrated under the exact null and remains high across a broad range of dimensions.

Dissociation and benchmarks

  • The study spans nine benchmarks across four operator classes and p from 16 to 135, plus a 31-point sweep varying ρ by a factor of 306.The sweep uses one operator family.Source: §5 · p. 6
  • Field coverage fell from 0.851 to 0.331 while observation coverage stayed near nominal; pinning σ removed concealment but not the collapse.At ρ = 1.32, detection was 1.00 when field coverage was already 0.547.Source: §5 · p. 6

Power

  • 39 of 40 misspecified cells were detected at 1.000 across p/N from 0.01 to 10, including all six cells with p > N.The null-cell false-positive rate was 0.0509 against a nominal 0.05.Source: §5 · p. 6

Blind spot

  • Across 98 isotropic cells, median detection was 0.045, near the nominal rate; at (N, p) = (800, 96), power stayed at 0.8 until excess spread over more than 83% of directions.At N = 100, that width fell to 50%.Source: §5 · p. 6

Power

  • Figure 2 maps detection and relative excess-estimation error for a single spike at p = 96; its two correctly specified benchmarks lie left of the 0.8 detection contour.Simulated cells use population excess, while benchmark positions use measured excess above the null as a proxy.Source: Figure 2 · p. 7

Power law

  • Theorem 13 predicts detection from the law of John’s statistic at proportional dimension, with mean absolute error 0.011 for ν ≥ 10, or 0.008 with a third-order term.The result applies to Gaussian data with any spectrum of bounded norm.Source: §5 · p. 7

6 Real Topography

In the preregistered GEBCO study, structured complement variation was common, and the spectrum-shape test detected harmed cells that observation-space checks often missed. Observation coverage remained near nominal despite substantial field-coverage loss.

Data and design

  • The study used GEBCO 2024 topography in 32 × 32 tiles, with principal bases built from separate blocks and design and admissible outcomes preregistered before data were read.The 36 cells combine three linearized operators, six basis dimensions, and two noise scales.Source: §6 · p. 7

Real misspecification

  • Across 18 operator-and-d cells, PR/p ranged from 0.036 to 0.709, indicating concentrated rather than isotropic misspecification; it rose with d and was smallest for gravity, largest for pointwise subsampling.The operator ordering followed how strongly each operator suppresses high wavenumbers.Source: §6 · p. 7
  • The test detected at 0.95 or more in 35 of 36 cells even after tile complement directions were randomized to make population covariance isotropic.Per-tile complement energy had median coefficient of variation 2.85 versus 0.108 for Gaussian noise; a Gaussian with the same covariance was detected at 0.95 or more in 34 of 36.Source: §6 · p. 7

Detection, harm and competitors

  • Observation coverage stayed in 0.924–0.978 while field coverage ranged from 0.583 to 0.954, and the fitted noise scale reached 185 times σ.The complement estimate had a median ratio of 1.009 to ρ across 36 cells.Source: §6 · p. 8
  • 31 of 32 harmed cells were detected at N = 100 and all 32 at N ≥800; the only miss was gravity at d = 96, with population excess of 4%.All four unharmed cells also triggered, so specificity was not estimable on these data.Source: §6 · p. 8
  • Cross-validated predictive coverage and held-out observation coverage detected only 8 and 5 of 32 harmed cells, respectively; the full-covariance test rejected all 36 cells, including unharmed ones.Inflating field variance by 15% with shape held correct made the full-covariance test reject on 99% of draws.Source: §6 · p. 8

7 Related Work

The paper places basis-restricted prior calibration alongside work on credible sets, covariance misspecification, residual diagnostics, and invariant spectrum tests. Its stated contribution is to connect noise-scale absorption in observation space to hidden field-space failure and develop a general-spectrum power law.

Credible sets in inverse problems

  • Prior and truth regularity jointly affect coverage in linear inverse problems with Gaussian priors, while adaptive constructions and reduced-rank or constrained Gaussian-process priors address related settings.Observation-space predictive checks are the diagnostics the paper’s mechanism defeats.Source: §7 · p. 8

Variance absorption under misspecification

  • Earlier Gaussian-process work studied variance absorption under covariance misspecification; here the absorbing parameter is in observation space while the concealed failure is in field space.The paper states that pinning σ identifies absorption as the channel between them, and that ˆσ/σ ≈ √ρ requires the in-span prior scale to dominate noise.Source: §7 · p. 8

Lack-of-fit and residual diagnostics

  • Classical lack-of-fit tests obtain σ from repeated measurements or near-replicate clusters, whereas these repeated inverse problems use different fields through one design and supply no such external scale.Residual-pattern checks need external σ or read magnitude, which is constant when referred to a same-sample fitted scale.Source: §7 · p. 8

Sphericity and invariant detection

  • John’s statistic belongs to a literature on sphericity and spiked-alternative theory, while invariant detectors test scale-invariantly after projecting out known interference subspaces.With multiple snapshots, the cited work identifies spectrum shape as the maximal invariant.Source: §7 · pp. 8–9

Error contrasts and prior–data conflict

  • The complement residual is an error contrast, and checking a sampling model through data conditional on a minimal sufficient statistic is an established diagnostic principle.Source: §7 · p. 9

Sphericity and invariant detection

  • The paper derives the conditional direction result through completeness, reduces testing to spectrum shape through invariance, and gives a proportional-dimension power law for general Gaussian spectra.It distinguishes its range(HΦ) interference subspace from conventional known interference subspaces and reports that the cited power-law results do not cover general spectra at proportional dimension.Source: §7 · p. 9

8 Discussion and Limitations

Across the misspecification sweep, observation coverage stays near nominal while field coverage falls sharply; the discussion also identifies boundaries on what the test and theory establish. The appendices detail the absorption law, coverage decomposition, sweep results, and exact null mean.

8 Discussion and Limitations

  • On nine benchmarks, the rank-d restriction beats a full-rank prior on two with ρ ≈ 1, by 8.2× and 3.9× in field RMSE, and loses on seven out-of-basis problems.The authors propose no threshold on ρ.Source: §8 · p. 9
  • The real-data evidence covers one field, one basis construction, and three operators; the test also depends on linear, known H and has a Gaussian isotropic-complement blind spot.The power law is proved for Gaussian data, while non-Gaussian behavior is measured rather than derived.Source: §8 · p. 9

AI Use Statement

  • The paper reports that generative AI assisted with discussion, proof checking, literature searches, drafting, editing, and organization; authors reviewed and verified the assisted material.AI was not used to generate experimental data or determine reported results.Source: AI Use Statement · p. 9

Reproducibility Statement

  • Training uses convergence criteria, metrics use held-out splits, and released scripts generate figures and tables from stored JSON artifacts checked against quantitative claims.The data are regenerated by two release scripts rather than shipped.Source: Reproducibility Statement · pp. 9–10

A The Absorption Law in Detail

  • Across all 31 sweep points, measured ˆσ/σ matched √ρ within 3.4%, with no residual trend; tightening the marginal-likelihood convergence criterion tenfold shifted residual only from 3.0% to 2.6%.The residual is a finite-n effect: for n ∈ {25, 50, 100, 200}, shortfalls were −7.7%, −3.7%, −2.4%, and −1.3%, tracking −r/(2n).Source: A The Absorption Law in Detail · p. 11

B Field Coverage in Two Terms

  • The two-term coverage prediction requires ˆσ to remain small relative to the prior scale; absorption makes its error grow, reaching −28% at ρ = 306.The coverage account separates basis restriction, which survives pinning σ, from absorption; the two shortfalls largely cancel in the coverage ratio.Source: B Field Coverage in Two Terms · pp. 12–13

C The Misspecification Sweep

  • Field coverage falls from 0.851 to 0.331 with σ free while observation coverage stays within 0.878–0.897; pinning σ removes concealment, not the collapse.At ρ = 306, field coverage is 0.024 with σ pinned; the authors caution that the free-versus-pinned comparison is an attribution, not a causal decomposition.Source: C The Misspecification Sweep · pp. 13–14

D Proof of Lemma 4

  • The exact null-mean proof uses independence of the Gaussian matrix norm and direction, making the scale-invariant statistic pivotal at every (ν, p), including p > ν.At (p, ν) = (5, 7), simulation gives 0.7575 against the exact value 0.7568; the χ2 approximation gives 0.5714.Source: D Proof of Lemma 4 · p. 15

E The Test on the Benchmarks

On the benchmarks, the test is calibrated and detects at the sweep’s first misspecified point; power-study results show strong detection across many excess shapes and dimensions, with isotropic excess as the blind spot.

E The Test on the Benchmarks

  • At ρ = 1.32, detection is 1.00 and remains 1.00 at every larger ρ; the test finds the sweep’s first misspecified point in 2 of 10 seeds.Against an exact null, the false-positive rate was 0.025 at nominal 0.05.Source: E The Test on the Benchmarks · pp. 15–16
  • The leading-eigenvalue prediction had 3.0% median and 9.6% worst error across seven misspecified benchmarks; the second eigenvalue also confirmed k = 1.At ρ = 1.32, the predicted value was 31.7 against 32.4 measured.Source: E The Test on the Benchmarks · p. 16
  • Sensors retained 1.00 detection with p = 16 and N ≈ 800; its spectrum had one direction at 215 times the bulk.The remaining fifteen directions spanned 0.76–1.53.Source: E The Test on the Benchmarks · p. 16

F The Power Study

  • At (N, p) = (800, 96), detection was 1.000 across tested anisotropic shapes and ρ values, but isotropic excess had median detection 0.045 across 98 cells.For ρ = 1.32, the 0.8-power spread boundary was about 90% of complement directions at N = 800, 70% at N = 200, and 50% at N = 100; these are lower bounds.Source: F The Power Study · pp. 17–18
  • Across p/N from 0.01 to 10, 39 of 40 misspecified cells detected at 1.000, including all six singular cases with p > N.Across 98 exact-null cells, the false-positive rate was 0.0509 against nominal 0.05; calibration uses a bootstrap matched to each (N, p).Source: F The Power Study · p. 17
  • For a five per cent excess at ρ = 1.05, detection was 1.000 at every N ≥100 for k = 1, and 0.63 at N = 25.At ρ = 1.10, detection was 1.000 for N ≥100 and 0.965 at N = 25.Source: F The Power Study · p. 17

G The Auxiliary Statistics ˆρ, R and eR

The auxiliary statistics estimate excess size and characterize spectral departures, but their interpretation depends on how the excess is distributed and on the sample dimensions.

G The Auxiliary Statistics ˆρ, R and eR

  • On benchmarks over ρ ∈[1, 5], ˆρ had 2.08% median relative error and 4.02% worst error because the excess occupied one complement direction.At k = 48 = p/2, the median lands in the inflated half and ˆρ recovers almost none of the excess, although the test still detects with certainty.Source: The magnitude ˆρ · p. 18
  • Across all 527 cells with detection ≥0.8, ˆρ −1 understated ρ −1, with median error −19.1%.The estimate was tight for concentrated misspecification and became vacuous as the excess spread.Source: The magnitude ˆρ · p. 18
  • At ρ = 306, ˆρ drifted to −9.6%; the bulk estimate rose from 0.962 to 1.093, accounting for almost all of the drift.The fraction of complement eigenvalues above 1.05× the bulk remained flat at 0.462 across the sweep.Source: The magnitude ˆρ · p. 18
  • R is not comparable across (N, p): under sphericity its median ranged from 1.56 at (1600, 96) to 5.70 at (100, 96).Below N = p/2, more than half the sample spectrum is zero, so eR should not be read.Source: The magnitude ˆρ · pp. 18–19
  • At the 0.8-detection crossing, eR was the only candidate variable that collapsed across 21 series, with CV 0.068 across shapes and 0.113 across dimensions.An eR above 1.3 indicated power of at least 0.8 anywhere in N > p/2; below that, the threshold depends on N/p.Source: The magnitude ˆρ · p. 19, Table 5 · p. 19

H Asymptotics

The section examines how John’s test detects nonwhite complement covariance and how detection relates to field error, the forward operator, and coefficient fit. It also describes a case at d = 96 where the test misses a harmed cell despite 24.4 m field error, while pointwise sampling detects it in every run; the contrast is attributed mainly to instrument gain.

H Asymptotics

  • At fixed p, John’s test is consistent as N grows exactly when Σ is not proportional to the identity, so isotropic unmodelled energy in the complement is its blind spot.This is the shared condition for finite-sample power, consistency, and the blind spot: Upop>0, or equivalently nonwhite complement energy.Source: H Asymptotics · pp. 19–20
  • Wang & Yao (2013) and Li & Yao (2016) express power through Upop in different regimes; Theorem 13 covers general bounded spectral alternatives at proportional p/ν for Gaussian data.The cited laws agree to leading order but differ in the (1 + Upop) correction, which matters for large misspecification; non-Gaussian behavior is measured rather than derived.Source: H Asymptotics · p. 20
  • Under bounded spectra and proportional p/ν, Theorem 13 gives Gaussian-data power through Upop and κ; fixed alternatives have power tending to one, while local power has a Gaussian limit.At local alternatives the κ correction vanishes; at fixed alternatives it sets the fluctuation scale, and the corollary’s bootstrap-estimated power is asymptotically accurate.Source: A power theorem at proportional p/ν,… · p. 22
  • For ν≥10, the third-order power law’s error in the transition band was 0.021, close to 0.019 expected from binomial noise; all laws failed at ν≤4.At ν≤4 they were optimistic; at N=3, p=96, one spike, and ρ=1.32, predictions were 0.85, 0.73, and 0.67 against measured power 0.50.Source: A power theorem at proportional p/ν,… · p. 23
  • The third-order law’s eR crossing had median ratio 1.000 to the measured crossing and fell below it in 48% of series, versus 95% for (12).The law reproduces the measured threshold spread, with CV 0.115; deriving the eR band in closed form remains open because it depends on the largest eigenvalue.Source: A power theorem at proportional p/ν,… · pp. 23–24

From field error to detection

  • On GEBCO, the exact identities connecting field-error covariance to Upop held to rounding in all 36 cells; basis-split predictions had mean absolute error 0.017 and the right detection call in 35 cells.The prediction used the nominal noise scale and was conservative because the basis-split estimate of ρ−1 was a median 0.85 of its evaluation value.Source: From field error to detection · p. 24
  • Detection requires sufficient complement anisotropy and field-error magnitude, with the forward operator carrying the error into observations and the coefficient fit not absorbing it.The condition is expressed through gain g and leakage ℓ; no field-error-only bound guarantees detection.Source: From field error to detection · p. 24
  • As the basis grew from d=4 to 96, leakage into range(G) rose and Upop fell under all three operators, so large field error could remain undetected.Errors in null(H) leave observations unchanged, while errors mapped into range(G) can be absorbed by coefficients; the gravity-sweep decline in ρ−1 was a factor of 598.Source: From field error to detection · p. 25
  • At d = 96, the harmed cell has 24.4 m field error and is detected in only 0.20 of runs at N = 100; pointwise sampling detects it in 1.00.At the same d and field error, pointwise sampling has 2.1×104 times the gain.Source: From field error to detection · p. 25
  • Under the gradient operator, leak spans 0.100 to 0.241 over the same range, falling to 0.051 at d = 8 before rising.The passage describes the leak dose-response as real but secondary.Source: From field error to detection · p. 25
  • The passage attributes the missed detection mainly to instrument gain rather than model leak, and says no test on the data could recover that gain.It characterizes gain as a property of the instrument.Source: From field error to detection · p. 25

I Real Topography: Design and Further Results

In the GEBCO study, the sphericity test detected structured complement variation even where field coverage rose as reconstruction error worsened, while coverage checks missed the gradient arm below d = 96. The study also found that the rank-d prior’s relative performance tracked whether problems were in-basis, whereas the constrained prior over-covered without restoring accuracy.

I Real Topography: Design and Further Results

  • The preregistered prediction of operator-dependent isotropy was refuted: PR/p was below 1 under every operator and at every d, indicating concentrated complement excess.The evaluation set was enlarged from 720 to 8,108 tiles because PR/p had not converged; three of 18 cells remained unconverged at the 1% tolerance.Source: I Real Topography: Design and Further… · p. 25
  • The real-topography evaluation used direct-measurement GEBCO tiles, separated basis and evaluation blocks, and matched signal-to-noise ratios across three operators.Whole-block assignment and a 32-cell buffer prevented basis and evaluation tiles from being adjacent or sharing the same continental margin at the seam.Source: I Real Topography: Design and Further… · pp. 25–26
  • The GEBCO noise estimate was the closed-form complement estimator, and agreement in ρ degraded with geographic separation between the tiles defining it.The out-of-span part of regional mean shifts carried at most 2.2 × 10^-4 of the complement excess in any cell and arm.Source: I Real Topography: Design and Further… · p. 26, Table 7 · p. 26
  • At N = 100, measured detection differed from the power-law prediction by a mean absolute 0.012 across 36 cells; complement log-energies remained correlated at 0.625–0.894 across tile separations.Thinning to 120 km retained detection at 1.000 in four of five cells, though it discarded 98.5% of tiles.Source: I Real Topography: Design and Further… · pp. 26–27
  • Field coverage was 0.821–0.954 across gradient cells despite RMSE reaching 27.0 times the in-span floor, while the sphericity test detected all twelve cells at every N ≥25.Cross-validated predictive coverage and held-out observation coverage missed the entire gradient arm below d = 96.Source: I Real Topography: Design and Further… · pp. 26–27

J Experimental Setup

  • The synthetic experiments used a closed-form E-step; a Metropolis-adjusted Langevin E-step was 40–55× slower for results identical to three decimals.Six of seven benchmarks other than Trimodal regenerated bit-exactly.Source: J Experimental Setup · pp. 28–29

K Scoping and the Constrained Prior

  • Across nine benchmarks, the rank-d restriction won on the two with ρ ≈1, by 8.2× on syn_inbasis and 3.9× on Darcy (2D), but lost on the seven out-of-basis problems.The full-rank prior covered at 0.92–1.00, while the rank-d prior covered at 0.18–0.87; the suite does not establish a general threshold on ρ.Source: K Scoping and the Constrained Prior · p. 29

The constrained prior

  • The constrained prior over-covered rather than under-covered and did not restore accuracy: field RMSE was 1.9–6.5× the full-rank baseline on six of eight benchmarks.It enforces an operator-induced linear constraint rather than collapsing to rank d.Source: The constrained prior · p. 30

L Negative Results

The rank-d restriction, rather than the coefficient prior, accounts for reconstruction results: a GP with the same kernel performs similarly. The restriction also imposes an out-of-span error floor that neither restricted estimator can overcome.

L Negative Results

  • Across nine benchmarks, the latent apparatus measurably improves results only on Darcy (2D), while the same-kernel GP matches the model on five benchmarks and syn_inbasis to three significant figures.On Darcy (2D), RMSE is 0.196 versus 0.220; on syn_inbasis it is 0.0190 versus 0.0189.Source: L Negative Results · p. 30
  • The coefficient prior affects reported latent structure, but not reconstruction, coverage, or either diagnostic.The appendix attributes the reconstruction wins to the rank-d basis restriction, which plain GP regression with the same kernel also achieves.Source: L Negative Results · pp. 30–31
  • On Bimodal, the in-span excess over the floor is 0.411 for the model and 0.410 for same-kernel GP regression; complement statistics cannot reach this in-span error.The floor itself is 0.162; unrestricted gpr-fullrank reaches 0.158 because it is not confined to the basis span.Source: L Negative Results · p. 31

M Status of Every Result

The paper distinguishes exact, approximate, asymptotic, and empirical claims, with each tied to its stated conditions or measurement grid. Its reported empirical regularities are measurements on the tested suite, not universal laws.

M Status of Every Result

  • Results are labeled exact, approximate, asymptotic, or empirical; empirical claims describe stated measurements and are not extrapolated beyond them.Exact claims hold at every finite (N, p); asymptotic claims apply only in a stated limit.Source: Table 16 · p. 32
  • The test’s asymptotic power tends to 1 at fixed p as N grows, while the finite-sample null law depends on the entry fourth moment.This asymptotic claim does not establish power at finite (N, p).Source: M Status of Every Result · p. 33
  • For the 949 Gaussian cells, mean absolute errors are 0.016 and 0.011, compared with 0.019 for approximation (12).The null law of U varies with (N, p), so bootstrap calibration is recalibrated at each pair.Source: M Status of Every Result · p. 33
  • The reported detection band at power 0.8 is 1.17–1.26, with median 1.19; the source labels these as empirical regularities, not universal cutoffs.The detection threshold varies with sample size and dimension, and the stated empirical grid is limited.Source: M Status of Every Result · p. 33
  • On the measured suite, observation coverage stayed at 0.878–0.897 while field coverage fell from 0.851 to 0.331.These controlled measurements cover 31 sweep points and 10 seeds in one model family.Source: M Status of Every Result · p. 33
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