PaperlayerView exact revision on arXiv

Source-linked AI summary

When Unpaired Sets Support Shared-Corruption Calibration: Moment Geometry and Two-Sample Precision

Shuheng Cao, Zhenhao Zhang, Ruiqi Chen, Renjie Cao, Siyu Zhang, Zhaoxiang Feng, Lingwei Dang, Haoyang Wu

arXiv:2609.26209v1stat.MEcs.AImath.ST

TL;DR

When observations share a corruption but only an unpaired clean reference set is available, it is unclear which correction directions fixed moments can support. The paper develops a two-sample moment calibration and rank-aware information report. Controlled validation distinguishes moment-visible directions from unresolved ones, while global planar-similarity identification is conditional on stated assumptions.

  • Problem

    For shared-corruption calibration with an unpaired clean reference population, the paper asks when fixed moments support a prescribed low-dimensional correction and which directions remain unresolved.

    Source: §1 · p. 1

  • Method

    The paper matches corrected-observation and clean-reference summaries, then reports rank, scaled moment sensitivity, source-separated two-sample uncertainty, and compatibility.

    Source: §1 · p. 1, §3 · p. 3

  • Results

    Controlled validation shows error decreases along sensed directions while an injected kernel component remains unresolved, and applications report color, channel-response, and paired geometric results.

    Source: §5 · p. 5, §7 · p. 7

  • Takeaways & Limitations

    The framework distinguishes locally identifiable corrections from weak or unresolved information instead of treating every fitted correction as identified.

    Source: §1 · p. 1, §7 · p. 7

    Global planar-similarity identification requires matched populations, no clipping, and a nonzero ensemble moment; when that moment vanishes, the prescribed pair does not identify orientation and scale.

    Source: §1 · p. 1, §4 · p. 4

Abstract

from arXiv · show

Collections of diverse observations often share one acquisition, processing, geometric, or channel corruption, while only an unpaired clean reference set is available. For a prescribed low-dimensional correction shared across observations, the observed and clean reference sets support inference only through the response of fixed moments. We formulate this problem as two-sample moment calibration and report a rank-aware information state combining local rank, scaled moment sensitivity, source-separated covariance, and a moment compatibility residual. Full rank gives local moment identifiability, whereas kernel directions remain unresolved to first order. A unified linearization separates observed-set and reference-set uncertainty. Under covariance weighting, the weakest scaled singular value determines worst-direction asymptotic amplification. For an orientation-preserving planar-similarity correction shared across observations, ensemble centroids and a nonzero third-order complex moment yield closed-form global population identification of translation, rotation, and isotropic scale under matched-population and no-clipping assumptions. Controlled validation tests the predicted $N^{-1}$ and $σ_{\min}^{-2}$ laws, Gaussian efficiency, and interval coverage. Bounded applications report color corrected-output quality, channel magnitude-response calibration, and a separate paired geometric de-beautification result. The framework therefore reports missing or weak information instead of treating every fitted correction as identified.

1 Introduction

The paper frames shared-correction calibration with an unpaired clean reference set as matching fixed moments, then audits whether those moments support inference. Its estimator and information state distinguish locally visible directions, sensitivity, and uncertainty from both finite sets.

1 Introduction

  • Shared corruption can be calibrated from an observed set and an independent clean reference sample, but only correction directions exposed by moment responses gain information from more samples.The section identifies the missing tool as an operational report that combines compatibility, local rank, scaled sensitivity, and two-sample precision.Source: §1 · p. 1
  • The contributions combine a rank-aware information state and controlled validation with a conditional global planar-similarity result and bounded color, channel, and paired geometric applications.The paired geometric application is separate from validation of the unpaired planar-similarity result.Source: §1 · p. 1

2 Related Work

  • Unlike prior blind-inverse, distribution-matching, and moment-identification work, this setting audits selected moments for a prescribed correction against an independent reference population.The paper presents the integration as an operational reporting role, not as a new general GMM, rank, or covariance result.Source: §2 · pp. 1–2

3 Problem Setup and Estimator

  • The population restriction matches summaries of corrected observations and clean references, without requiring paired samples or equality of their full corrected and clean distributions.The estimator conditions its theory on a consistent selected interior local minimizer.Source: §3 · pp. 2–3
  • For geometric correction, coordinate-sensitive spatial moments retain geometric information removed by pooled appearance moments, and equivariant moments represent translation, rotation, and isotropic scale.The correction acts spatially as Ta,b(z) = az + b; real and imaginary parts of (µ, κ) provide four real moment coordinates.Source: §3 · p. 3
  • Two-sample linearization retains fluctuations from both sets, while ordinary local parameter inference requires full column rank and rests on regularity and consistency assumptions.The stated assumptions include independent i.i.d. sets, finite second moments, local convergence, and a correctly specified restriction.Source: §3 · p. 3
  • The information state reports rank, the scaled weighted Jacobian spectrum, residual, and covariance; rank-deficient fits report unresolved directions rather than ordinary parameter intervals.The dimension condition q ≥ p is necessary but insufficient; pilot checks test independent moment responses, and data-selected moments are frozen using a pilot split or valid cross-fitting.Source: §3 · p. 3

4 Moment Geometry for Identifiability, Stability, and Two-Sample Precision

The corrected-moment Jacobian determines which parameter directions are locally identifiable, while the two-sample covariance describes how observed-set and reference-set noise affect precision. Under stated assumptions, a nonzero ensemble moment also yields global population identification for a shared planar similarity.

Local identifiability

  • Full-column-rank Jacobians give local moment identifiability, whereas Jacobian-kernel directions remain unresolved to first order.Increasing either sample size reduces sampling noise but does not change the population derivative.Source: §4 · pp. 3–4

Two-sample precision

  • The leading covariance separates observed-set and reference-set contributions, both amplified through the same Jacobian.The respective terms are BVYB⊤/N and BVXB⊤/M.Source: §4 · p. 4

Stability

  • Worst-direction asymptotic amplification scales as σ_min(KW)^-2 under covariance weighting.Smaller scaled singular values amplify parameter error according to σ_min^-2.Source: §4 · pp. 4–5

Planar similarity

  • A nonzero third-order ensemble moment globally identifies shared planar translation, rotation, and isotropic scale under matched-population and no-clipping assumptions.If the moment vanishes, the prescribed moment pair cannot identify orientation and scale; more samples cannot create the missing response.Source: §4 · pp. 4–5
  • With a nonzero ensemble moment and finite second moments, the planar-similarity plug-in estimator is consistent and has separate N^-1 and M^-1 first-order covariance contributions.The four-real-parameter moment map has full column rank.Source: §4 · p. 4

Gaussian mean shift

  • For a Gaussian mean shift with known mean, the estimator Y−µ attains the Cramér–Rao lower bound V Id/N.The independent-set mean difference has covariance V(1/N + 1/M)Id.Source: §4 · pp. 4–5

5 Experiments

Experiments test the predicted limits of moment visibility and two-sample precision, then report bounded color, channel, and paired geometric applications. Sampling improves estimation in sensed directions, while kernel directions and application-specific scope boundaries remain.

Controlled moment visibility and precision

  • Squared mean Euclidean error decreased from 1.9011 to 0.0280 across observed-set sizes; fitted slopes for set size, signal strength, and variance were −1.017, −2.068, and 1.000.These exponents were tested in the exact linear/Gaussian construction, not asserted for nonlinear operator instances.Source: §5 · p. 5

Color corrected-output calibration

  • Across shared-filter sets, LPIPS fell from 0.1788 at N = 1 to 0.0578 at N = 30, while mixed filters raised the N = 30 value to 0.1150.The color protocol used identity-disjoint training, test, and reference sets; test filters were from the same pilgram family as training filters.Source: §5 · p. 5
  • Moment calibration achieved LPIPS 0.0547 versus 0.0954 for filtered input, 0.1023 for ARColorOpV2, and 0.0761 for Reinhard transfer.Harm frequency was 1% for moment calibration, versus 70% for ARColorOpV2 and 17% for Reinhard.Source: §5 · p. 5

Controlled moment visibility and precision

  • Row-space error fell from 0.5314 to 0.0631 while the injected unit kernel component stayed 1.000; a rank-four draw reduced total error from 0.6878 to 0.0901.The k = 2 moment map retained only two directions for four parameters; Figure 2 summarizes the sensed-versus-kernel behavior.Source: §5 · p. 5, Figure 2 · p. 6
  • For 20,000 exact Gaussian draws, mean-shift MSE followed dV (1/N + 1/M) within 0.58%, and empirical interval coverage ranged from 94.84% to 95.28%.The mean-shift MSE approached the observed-set floor dV/N as reference size M increased.Source: §5 · p. 5, Figure 3 · p. 6
  • With N = 256, MSE followed (14) as σmin(KW) varied from 1 to 0.05, with slope −1.999 and R2 = 1.000.The separate known-mean study reported CRB ratios from 0.986 to 1.008.Source: §5 · p. 5, Figure 3 · p. 6

Channel magnitude-response calibration

  • The channel instance estimates sampled magnitude response, but finite-record boundary effects prevent exact realization of the stationary identity; phase and reconstructed-signal quality were not evaluated.The correction is defined by the reciprocal of the sampled magnitude response.Source: §5 · p. 6

Paired geometric de-beautification

  • On 72 held-out paired faces processed by one commercial operator, Ours reached PSNR 39.83 and LPIPS 0.0049 versus 38.34 and 0.0061 for Direct U-Net; SSIM tied at 0.972.Figure 4 shows representative outputs; this paired per-image protocol is separate from the unpaired estimator and Proposition 2 assumptions.Source: §5 · p. 6, Figure 4 · p. 7

6 From Moment Geometry to Calibration Decisions

The calibration decision procedure freezes design choices before fitting, then uses rank, spectrum, residual, and covariance to select what can be reported. It withholds ordinary intervals for rank-deficient fits and treats residual size as a compatibility diagnostic, not proof of correct specification.

Decision procedure

  • Freeze the correction family, moments, parameter scales, and weight before fitting; report estimates and projected intervals only for adequately conditioned full-rank fits.For weak full-rank fits, flag the precision cost; for rank deficiency, report the fitted correction, residual, scaled spectrum, and unresolved right-singular subspace instead of ordinary intervals.Source: §6 · p. 7
  • A large residual flags tension among the correction family, moments, and reference population, while a small residual does not establish correct specification.Separate observed-set and reference-set uncertainty terms can identify which set offers the larger precision gain for a target contrast and interval width.Source: §6 · p. 7

7 Scope and Conclusion

The paper concludes that fixed-moment response governs support for shared corrections, and that the framework should report weak or missing information rather than presume identification. Its application evidence is bounded and does not extend to clipped, arbitrary nonrigid, or phase-recovery settings.

Scope and conclusion

  • Full rank supports local moment identifiability, while Jacobian-kernel directions remain unresolved to first order.Increasing observed or reference sample size reduces sampling noise but does not change the population derivative.Source: §7 · p. 7
  • Global population identification for a shared orientation-preserving planar similarity requires matched populations, no clipping, and a nonzero third-order moment.The paired geometric result does not extend the unpaired theory to nonrigid deformation.Source: §7 · p. 7
  • Applications report bounded color, channel magnitude-response, and paired in-distribution geometric outcomes; cross-operator generalization and arbitrary geometric inversion are not claimed.Clipped, arbitrary nonrigid, and phase-recovery settings remain outside the framework's general claims.Source: §7 · p. 7

Supplementary Material for “When Unpaired Sets Support Shared-Corruption

The supplementary-material front matter lists the paper's contributors and their contribution markers.

Supplementary material

  • Shuheng Cao, Zhenhao Zhang, and Ruiqi Chen are marked co-first authors, while five contributors are marked equal contributors.The listed equal contributors are Renjie Cao, Siyu Zhang, Zhaoxiang Feng, Lingwei Dang, and Haoyang Wu.Source: Supplementary Material for “When… · p. 9

1 Claim–Evidence Closure

The closure table links formal results to their supporting arguments, checks, and operative boundaries, then summarizes three bounded applications.

1 Claim–Evidence Closure

  • Moment calibration reports LPIPS 0.0547 with 1% harm.The table also reports that channel magnitude-response error decreases by 99.5% from N = 1 to N = 256.Source: §1 · p. 9
  • Paired Ours improves Direct U-Net by 1.49 dB PSNR while reducing LPIPS by 19.7%.This is a separate paired geometric de-beautification result.Source: §1 · p. 9
  • Table S1 maps formal results to complete arguments, controlled checks when claimed, and operative boundaries.It also summarizes the three bounded applications.Source: §1 · p. 9

2 Estimator Construction

The estimator construction projects summary influences and covariance calculations into the parameter space, reducing covariance estimation and inversion to p × p matrices.

2 Estimator Construction

  • The projected sandwich estimator uses the projected summary influences and sample covariances from the observed and reference sets.The passages introduce the projected sandwich estimator through bA = bJ⊤W bJ.Source: §2 · p. 9
  • After projection, covariance estimation and inversion involve only p × p matrices.Source: §2 · p. 9
  • For mean and standard-deviation matching, the pooled denominator-L −1 correction preserves the first-order law.It differs from the population-form standard deviation by a known 1 + O(L−1) factor.Source: §2 · p. 9

3 Complete Proofs and Derivations

The derivations establish local identification through moment-Jacobian rank, separate uncertainty from the two samples, and give a closed-form global result for a shared planar similarity under stated conditions.

3 Complete Proofs and Derivations

  • Full column rank of the corrected-moment Jacobian gives local moment identifiability through a nonsingular selected submap.The inverse-function theorem makes that selected map locally one-to-one.Source: §3 · p. 9
  • The unified linearization yields separate first-order covariance contributions from the observed and reference sets.Independence of the sets gives the leading covariance.Source: §3 · p. 9
  • Table S1 maps formal claims to their arguments, controlled checks when claimed, and operative boundaries.It also summarizes the bounded applications.Source: Table S1 · p. 10
  • The weakest scaled singular value controls worst-direction asymptotic amplification.The displayed variance in direction u is 1/(Nσ_min(KW)^2).Source: §3 · p. 10
  • A nonzero third-order complex moment supports global identification of a shared orientation-preserving planar similarity.The map a ↦ |a|²a is bijective away from zero with positive real Jacobian determinant, and the centroid equation supplies an identity block for translation.Source: §3 · p. 10
  • Replacing population moments with sample averages gives consistency and separate N −1 and M −1 first-order covariance contributions.These conclusions are stated for the nonzero-moment case.Source: §3 · p. 10
  • With a known mean in the stated Gaussian model, the estimator for s attains the Cramér–Rao lower bound.The Fisher information is NId/V.Source: §3 · p. 11

4 Additional Experimental Details

The controlled studies test visibility, scaling, two-sample precision, and conditioning, while the application studies report bounded color and channel results and a separate paired geometric result.

4 Additional Experimental Details

  • Mean row-space error decreases from 0.5314 to 0.0631 while the injected unit kernel component remains 1.000.A separate rank-four draw reduces total error from 0.6878 to 0.0901.Source: §4 · p. 11
  • Empirical parameter MSE differs from dV (1/N + 1/M) by at most 0.58% and approaches dV/N as M increases.Across tested N and M, empirical coverage ranges from 94.84% to 95.28%; weak-coordinate MSE has slope −1.999 as σ_min(KW) varies from 1 to 0.05.Source: §4 · p. 11
  • Squared mean Euclidean error falls from 1.9011 to 0.0280 across tested N values, with log-log slope −1.017.The controlled scaling studies also report Neff slopes of −2.068 against S and 1.000 against V.Source: §4 · p. 11
  • The color correction applies the same stages to every image, uses 70 parameters, and matches pooled RGB and VGG-16 summaries against 200 clean reference faces.The correction clamps outputs to [0, 1].Source: §4 · p. 11
  • The color test filters were held out within the pilgram implementation family, not drawn from an independent app or engine.Calibration used filtered observed images without their clean counterparts; queries were evaluated against clean ground truth.Source: §4 · p. 11
  • At N = 30, moment calibration attains the lowest LPIPS among the reported information regimes; shared-filter LPIPS is 0.0578 versus 0.1150 with mixed filters.The mixed-filter result marks the condition that the calibration set share the corruption being calibrated.Source: §4 · p. 12, Figure S3 · p. 13
  • On the in-distribution paired test set, Ours achieves PSNR 39.83 and LPIPS 0.0049, versus Direct U-Net at 38.34 and 0.0061.This paired, per-image nonrigid result supports the de-beautification application, not validation of the unpaired moment estimator or Proposition 2.Source: §4 · p. 13
  • Channel relative magnitude-response MSE decreases from 0.2340 at N = 1 to 0.0012 at N = 256, with slope −0.971 for N ≥ 2.The instance estimates magnitude while leaving phase unresolved and has finite-length boundary effects.Source: §4 · p. 13

5 End-to-End Interpretation and Reproducibility

The interpretation links predicted information and precision laws to bounded color, channel, and paired-geometry findings, while distinguishing unpaired calibration from paired nonrigid restoration.

5 End-to-End Interpretation and Reproducibility

  • Channel calibration reports stable magnitude recovery across a 256-fold set-size range.The shared-set color ablation directly tests the framework’s central premise.Source: §5 · p. 13
  • The separate paired geometry model improves every aggregate metric except tied SSIM.This result is not conflated with unpaired calibration or paired nonrigid restoration.Source: §5 · p. 13
  • The interpretation connects Jacobian rank, projected influence, and the weakest scaled singular value to the controlled studies’ predicted laws.The evidence chain distinguishes local visibility, uncertainty from the two sampling sources, and weakest-direction amplification.Source: §5 · p. 13
  • Protocol records specify repetitions, seeds, disjoint color data identities, and evaluation-filter separation.Observed clean counterparts do not enter fitting; query clean images serve only as ground truth.Source: §5 · p. 13
Loading 2609.26209v1…