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Is your uncertainty map wrong, or is its target? Exact diagnostics for the Tweedie diagonal, and a gradient-free alternative

Vicent Ribas, Anna Oliveras Tous

arXiv:2609.33786v1cs.AI

TL;DR

Clinicians need spatial uncertainty maps that rank where predicted follow-up scans may be wrong, but it is unclear whether their estimator or the Tweedie-diagonal target limits usefulness. The paper diagnoses the exact diagonal and introduces a gradient-free residual probe; the diagonal reverses against error in four conditions, while clinical comparisons vary by task and region.

  • Problem

    Common Tweedie-diagonal uncertainty maps are often evaluated against another approximation, leaving unclear whether estimator error or the target limits their usefulness for ranking prediction error.

    Source: §1 · p. 2

  • Method

    The paper computes exact Tweedie diagonals across checkpoint–corpus conditions and introduces T-PT, which repeatedly corrupts one model-supported prediction and measures voxel-wise variance in the denoiser’s residual response.

    Source: §1 · p. 2

  • Results

    In four of fourteen conditions the exact diagonal is anti-correlated with denoising error, while on brain MRI T-PT leads the twenty-chain ensemble on five of eight tissue endpoints and trails on none at 16× fewer network evaluations.

    Source: §1 · p. 2, §5.2 · p. 8

  • Takeaways & Limitations

    The exact diagonal can faithfully encode an unsuitable error-ranking target, while T-PT is offered as a feasible instrument rather than a better approximation.

    Source: §1 · p. 2, §7 · p. 9

    Both T-PT and the reference lose most discrimination in the changed region, and the reason for the diagonal’s anti-correlation with error remains open.

    Source: §6 · p. 9

Abstract

from arXiv · show

A diffusion model can predict a follow-up medical scan from a baseline, but a clinician needs a per-voxel map of where that prediction can be trusted. Many such maps approximate the diagonal of the Tweedie posterior covariance, and are evaluated against another approximation of it, so whether the estimator or the target limits them is unclear. We compute the exact diagonal on six checkpoints across fourteen model-corpus conditions. Hutchinson at M=200 tracks it at rank agreement of at least 0.92 everywhere, yet in four of the fourteen the exact diagonal is anti-correlated with the denoising error, reaching -0.13, so a faithful estimator reproduces that reversal. All four are real-image conditions; on the models' own samples the reversal does not appear, so evaluating on generated samples flatters this family. What limits these maps is the target, not the estimator. We then introduce Tweedie Probe-Tangent (T-PT), a gradient-free residual probe that corrupts one model-supported prediction repeatedly and measures the voxel-wise variance of the denoiser's response. T-PT reads a different functional of the same Jacobian, and its exact second-order form ranks with the diagonal wherever the diagonal reverses; at thirty probes it returns a map too unstable to reproduce that ranking, while Hutchinson at M=5 already reproduces it, so T-PT there is not evidence against the reversal. We offer it as an instrument, not a better approximation. On brain MRI at full resolution, where every Jacobian-based estimator we test runs out of memory, T-PT leads a twenty-chain Monte-Carlo ensemble on five of eight endpoints inside tissue and trails it on none, at 16x fewer network evaluations; over the whole volume the ensemble leads, and fifty chains close the tissue gap. On lung CT the ensemble is ahead throughout. Both lose most of their discrimination where the change is, which remains open.

1 INTRODUCTION

A useful clinical uncertainty map must rank where a predicted scan is likely to be wrong, but common Tweedie-diagonal maps are usually evaluated against another approximation. The paper tests whether the estimator or the diagonal target limits these maps, then introduces T-PT as a gradient-free instrument.

1 INTRODUCTION

  • A clinician needs a per-voxel ranking of locations where a follow-up prediction is most likely to be wrong, rather than merely a plausible predicted scan.The study focuses on spatial rankings, not calibrated predictive standard deviations.Source: §1 · p. 1
  • Existing Tweedie-diagonal methods are commonly compared with another approximation, leaving uncertainty about whether estimator error or an unsuitable target limits their maps.The exact diagonal requires one Jacobian–vector product per input coordinate.Source: §1 · p. 2
  • Four of fourteen conditions show the exact diagonal anti-correlated with denoising error, even though Hutchinson tracks it at rank agreement of at least 0.92 in every condition.On model-generated samples the reversal does not appear; the paper concludes that the target, not estimator fidelity, limits these maps.Source: §1 · p. 2
  • Tweedie Probe-Tangent (T-PT) repeatedly corrupts one model-supported prediction and measures voxel-wise variance in the denoiser’s residual response.On full-resolution brain MRI, it matches or beats a twenty-chain ensemble inside tissue at 16× fewer network evaluations, while the ensemble leads over the whole volume.Source: §1 · p. 2

2 RELATED WORK AND EVALUATION GAP

Prior spatial uncertainty approaches use chain-to-chain variation, propagated noise-prediction uncertainty, or approximations to Tweedie posterior covariance. The paper distinguishes its contribution from these targets and introduces a residual-variance probe that reads a different Jacobian functional.

Sampling-based uncertainty

  • Monte-Carlo diffusion methods estimate spatial uncertainty from independent reverse-chain spread, with cost linear in the number of chains; the 20-chain reference costs 1,220 network evaluations per volume.Other families target network uncertainty or coverage and generally require extra training or multiple models.Source: §2 · p. 2

Analytical variance propagation

  • Analytical methods propagate uncertainty through reverse diffusion; U2Diff accumulates noise-prediction uncertainty along DDIM, U2Diffine adds first-order denoiser sensitivity, and MUMINS extends U2Diff to 3-D medical synthesis.The cited work establishes that the MUMINS recursion is expansive, motivating its delayed start.Source: §2 · p. 2

Tweedie covariance estimators

  • Tweedie covariance methods approximate denoiser-Jacobian posterior covariance, while T-PT adds no new estimand or improved approximation and instead supplies the exact object for diagnosis.T-PT repeatedly corrupts one input and takes voxel-wise variance of the denoiser’s residual response, reading a different functional of the same Jacobian.Source: §2 · pp. 2–3

3 DIAGNOSIS: IS THE EXACT DIAGONAL A USEFUL TARGET?

The diagnosis computes the exact Tweedie-covariance diagonal across six checkpoints and fourteen model–corpus conditions, then tests its validity, estimator fidelity, and relationship to denoising error.

3 DIAGNOSIS: IS THE EXACT DIAGONAL A USEFUL TARGET?

  • Fourteen checkpoint–corpus conditions are evaluated using the exact diagonal and denoising error, with corpora including held-out photographs, model samples, and held-out real images.The experiment uses 100 images per condition and matches the probe timestep to the clinical runs’ forward marginal noise scale.Source: §3 · p. 3

3.1 VALIDITY OF THE IMPLIED COVARIANCE

  • Up to 5.7% of coordinates have negative implied covariance on the class-conditional checkpoint; these exact negative entries indicate the learned denoiser implies a non-covariance, so the study clamps them to zero.The probe’s sample-standard-deviation map has no negative entries to clamp.Source: §3.1 · p. 3

3.2 FIDELITY IS NOT THE BINDING CONSTRAINT

  • At M=200, Hutchinson matches the exact diagonal at rank correlation 0.92–0.996 across all fourteen conditions, but its error correlation converges only to the diagonal’s own value.The row-sum surrogate does not converge to the diagonal and has negative or null error correlation in ten conditions, reaching −0.232.Source: §3.2 · p. 4

3.3 THE FAILURE MODE, AND WHAT IT DEPENDS ON

  • Four conditions have negative exact-diagonal/error correlation; real-image corpora show the reversals, while model samples do not, and the reversal is specific to the diagonal estimand.For inet, the correlations are −0.132 on COCO, −0.131 on Imagenette, and +0.058 on self samples; estimators of the diagonal inherit the reversal, unlike Monte-Carlo sampling or variance recursion.Source: §3.3 · p. 4

4 RESPONSE: TWEEDIE PROBE-TANGENT, A ONE-CHAIN RESIDUAL PROBE

Tweedie Probe-Tangent (T-PT) uses repeated residual probes at a model-supported prediction to estimate a different Jacobian functional from the Tweedie diagonal. Its second-order ranking follows the diagonal where that target reverses, but at K=30 the probe is too unstable to resolve the ranking; probe-point adequacy also matters.

4.1 ESTIMATOR AND COST

  • T-PT uses K residual evaluations at one timestep without differentiation; on the clinical tasks it costs 91 forwards on lung CT and 76 on brain MRI, versus 1,220 for MC-DDIM.The fixed clean prediction is repeatedly corrupted, and the per-voxel map is the sample variance of the denoiser’s residual responses.Source: §4.1 · pp. 4–5

4.2 GEOMETRIC INTERPRETATION

  • T-PT estimates the covariance of the injected noise left unidentified by the noisy observation; on a flat manifold, that covariance is proportional to the tangent-space projector.Its sample covariance converges almost surely to Ct(x0), under the hypotheses in Appendix B.Source: §4.2 · pp. 5–6

4.3 CONDITIONAL PREDICTION AND THE PROBE POINT

  • The probe requires a model-supported clean state; in conditional prediction, using v0=0 probes an unseen point and yields lung-CT error correlation ¯ρ≈0.004, versus 0.65 with an additional DDIM reconstruction.Probe adequacy is validated against a reference rather than derived from theory; the implemented residual is claimed only to be predominantly fibre-aligned.Source: §4.3 · p. 6

4.4 TWO ESTIMANDS THAT SHARE A REVERSAL

  • On four reversed conditions, T-PT’s second-order map ranks identically to the exact diagonal, but at K=30 it does not resolve that ranking.At that budget, Hutchinson at M=200 shows four harmful conditions while T-PT shows none; the probe’s flatness is budget-specific.Source: Figure 1 · p. 5, §4.4 · pp. 6–7
  • At the probe point, the second-order expansion adds Jacobian row energy to a term involving the Jacobian diagonal, so T-PT and the posterior-mean-Jacobian diagonal are distinct functionals.At a=0.32, the quadratic term cannot be dropped; in the ideal flat case the two diagonals rank alike, while on reversed conditions the row energy ranks with the diagonal.Source: §4.4 · p. 7
  • At K=30, the residual-variance probe has 26% relative error and near-zero reproducibility where the diagonal reverses, while Hutchinson at M=5 reproduces all four reversals.As T-PT’s budget grows, the anti-correlation emerges on the two inet conditions; on cifar neither reproducibility nor anti-correlation emerges at tested budgets.Source: §4.4 · p. 7

5 CLINICAL EVALUATION: BRAIN MRI AND LUNG CT

The clinical comparison favors different estimators by task and region: T-PT leads on several brain-tissue endpoints, while the 20-chain ensemble leads throughout lung CT and across the whole brain volume. At full brain resolution, tested Jacobian-based estimators exhaust memory while T-PT completes.

5.1 DESIGN

  • The study evaluates two retrained 3-D U-Nets on patient-disjoint lung-CT and brain-MRI test splits, comparing T-PT and analytical variance propagation with a 20-chain MC-DDIM reference.The tasks comprise 130 lung-CT test pairs from 47 participants and 211 OASIS-3 MRI pairs from 63 participants; maps are ranked against a common ensemble-mean error map.Source: §5.1 · pp. 7–8, Table 1 · p. 8

5.2 RESULTS

  • On brain MRI, T-PT leads on five of eight endpoints inside tissue and trails on none, while MC-DDIM leads over the whole volume by 0.038 and wins 210 of 211 pairs.On lung CT, MC-DDIM leads on every endpoint; its tissue difference is −0.022 [−0.037, −0.007] in its favour.Source: §5.2 · pp. 8–9, Table 1 · p. 8

5.3 COST IS A FEASIBILITY BOUNDARY, NOT A RATIO

  • At 128^3 on a 40 GB accelerator, T-PT peaks at 33.8 GiB and completes in 66.7 s, while the row-sum surrogate and Hutchinson at M=50 and M=500 exhaust memory.The reported boundary is for released implementations with activation checkpointing disabled for the tangent pass.Source: §5.3 · p. 9

6 DISCUSSION AND LIMITATIONS

The maps lose discrimination specifically in the changed tissue, and that weakness also appears in the 20-chain reference. The clinical scope is limited to two tasks and specified preprocessing, while the cause of the diagonal’s reversal remains open.

6 DISCUSSION AND LIMITATIONS

  • In the top decile of changed OASIS tissue, T-PT’s Spearman falls from 0.349 to 0.128, and the 20-chain reference falls to 0.133.Random tissue subsets of the same size score 0.378, indicating the drop is specific to where the change is.Source: §6 · p. 9
  • The clinical numbers are scoped to OASIS resampled at 1.5 mm isotropic, and the clinical evidence covers two tasks from two MUMINS checkpoints.Whether the reversal occurs in clinical checkpoints remains unanswered; the only medical checkpoint with a computable diagonal is two-dimensional and unconditional.Source: §6 · p. 9
  • The ranking depends on a probe point on the data manifold and one timestep, while T-PT at K=30 is barely reproducible where the diagonal reverses.The reason diag(J) is anti-correlated with error rather than merely uninformative remains open.Source: §6 · p. 9

7 CONCLUSION

The paper concludes that the failure in four of fourteen conditions belongs to the diagonal target, not its approximation, and presents T-PT as an instrument rather than a better approximation.

7 CONCLUSION

  • In four of fourteen conditions, the exact diagonal is anti-correlated with error, and exact evaluation identifies the failure as belonging to the target.The second-order T-PT ranking follows the diagonal where it reverses, but at thirty probes the map is too unstable to reproduce that ranking.Source: §7 · p. 9
  • At 128^3, T-PT is feasible where every tested Jacobian-based estimator runs out of memory.Where its own map resolves, T-PT contains more information about the error than a Hutchinson estimate of the diagonal.Source: §7 · p. 9
  • The changed region remains an open problem because the maps lose discrimination there.The conclusion does not claim T-PT resolves that limitation.Source: §7 · p. 9

REPRODUCIBILITY STATEMENT

The release provides the code, configurations, protocols, and preprocessing needed to reproduce the estimator, analyses, and experiments, while withholding data and brain-model weights under stated access constraints.

REPRODUCIBILITY STATEMENT

  • The supplementary release includes source code for the estimator, exact-denoiser and synthetic-manifold checks, scoring pipeline, and statistical analysis.Training configurations and preprocessing are also included.Source: REPRODUCIBILITY STATEMENT · p. 10
  • Appendix M records checkpoints, samplers, schedules, probe settings, and test splits for the medical experiments.It also operationally defines every mask and metric; Appendix R states the resampling unit behind each interval and validates it on simulated data.Source: REPRODUCIBILITY STATEMENT · p. 10
  • Brain checkpoints and image data are not redistributed; the release provides a retraining recipe and documents how to request data access and reproduce preprocessing.The OASIS-3 data-use agreement does not cover derivatives of that cohort.Source: REPRODUCIBILITY STATEMENT · p. 10

ETHICS STATEMENT

The study uses de-identified, publicly released medical imaging datasets under their data-use agreements, but its uncertainty ranking is not calibrated or validated for clinical decisions.

ETHICS STATEMENT

  • The study uses two de-identified, publicly released medical imaging datasets under their respective data-use agreements, with no new data collection or human-subject experimentation.Source: ETHICS STATEMENT · p. 10
  • The estimator provides an uncertainty ranking, not a calibrated predictive standard deviation, and is not validated for clinical decision-making.It indicates where a model’s prediction is least constrained.Source: ETHICS STATEMENT · p. 10
  • An off-manifold probe produced a smooth, plausible map with correlation 0.004 to true error, so deployment requires validation against a reference.Source: ETHICS STATEMENT · p. 10

USE OF AI TOOLS

The authors used language models as assistants for writing, coding, proof drafting, and adversarial review. The appendices organize theory, computable-truth tests, experimental design, and detailed results and sensitivity analyses.

USE OF AI TOOLS

  • Language models assisted with prose, analysis-code implementation and debugging, proof arguments checked by the authors, and adversarial review of theoretical claims and experimental protocol.The exact-denoiser validation was implemented with LLM assistance under author direction.Source: USE OF AI TOOLS · p. 10

APPENDIX ROADMAP

  • Appendices A–G contain theory, H–I test claims against computable truth, J–M describe experimental design, and N–U report detailed results and sensitivity analyses.Source: APPENDIX ROADMAP · p. 13

A THE DDIM RECURSION, THE DELAYED START, AND THE TOY OVERSHOOT

In affine additive covariance accumulation, the recursion is expansive wherever its transition coefficient exceeds one, and a schedule-calibrated delayed start bounds—but does not eliminate—inflation. A toy-manifold experiment shows substantial overshoot even after delayed start.

A THE DDIM RECURSION, THE DELAYED START, AND THE TOY OVERSHOOT

  • In the affine additive class, the recursion’s linear part has spectral radius a^2_s independently of the additive variance terms, and is non-contractive wherever |a_s| > 1.The claim covers the affine additive class used by the cited implementations, not state-dependent or Jacobian-linearised recursions.Source: A THE DDIM RECURSION, THE DELAYED… · pp. 13–14
  • The DDIM setup uses T steps, cumulative products ¯α_t, a cosine schedule, and skip interval ζ; its reverse update combines a clean prediction, noise prediction, and stochastic injection.The text distinguishes forward marginal scale σ_t from reverse-transition injection scale ˜σ_t.Source: A THE DDIM RECURSION, THE DELAYED… · p. 14
  • For the cosine schedule, a^2_s = ¯α_(t−ζ)/¯α_t > 1 at every transition, so no non-expansive accumulation window exists; delayed start bounds total inflation.Source: A THE DDIM RECURSION, THE DELAYED… · p. 14
  • A user-chosen amplification budget κ > 1 sets the latest admissible start from the schedule alone in O(T/ζ) scalar operations, without a denoiser call or data.The tolerance is interpretable but not data-optimal, and delayed start is not the only possible response.Source: A THE DDIM RECURSION, THE DELAYED… · p. 14
  • At T = 300 and ζ = 5, an illustrative one-per-cent budget gives sκ = 16; unmasked accumulation ends at V ≈ 1.5 × 10^3 versus 2.5 × 10^-3 at sκ.A late mask at t = 64 still inflates variance by a further 40×; the budget was not fixed independently of the validation-likelihood choice, so Eq. (8) is not claimed to predict sκ = 16.Source: A THE DDIM RECURSION, THE DELAYED… · p. 14, Figure 2 · p. 15
  • 13.7× overshoot remains after applying delayed start at 60% of the schedule on a one-dimensional analytic manifold in R1024.The true per-pixel variance is measurable by Monte Carlo, and propagation uses the same chains under the same independence assumption.Source: A THE DDIM RECURSION, THE DELAYED… · p. 14

B PROOF OF PROP. 2

For the affine covariance recursion, additive terms cancel when comparing two accumulated covariances, leaving an exact multiplicative gain. The proof also distinguishes this result from flat-manifold residual covariance claims and states what convergence of the probe covariance does—and does not—establish.

B PROOF OF PROP. 2

  • For the affine recursion, the difference between two covariance states is scaled exactly by a_s^2, regardless of the additive forcing term.The forcing term may collect the variance head, injected sampler noise, and additive corrections.Source: B PROOF OF PROP. 2 · p. 15
  • Across a window with |a_s| > 1, the accumulated covariance is multiplied by the cumulative gain and inflates monotonically.The gain over the window telescopes to the ratio of the corresponding cumulative noise schedules.Source: B PROOF OF PROP. 2 · p. 15
  • A state-dependent correction can restore contractivity while preserving positivity, but it falls outside the additive forcing rule.The example changes the recursion’s linear factor to c, with 0 < c < 1, and requires a propagation rule that rescales using a_s.Source: B PROOF OF PROP. 2 · p. 15
  • The residual sample covariance converges almost surely to the population covariance, but that convergence alone does not make the covariance a projector.The projector result is exact in the stated flat-manifold cases; departures from flatness are associated with curvature.Source: B PROOF OF PROP. 2 · p. 16
  • Under a proper Gaussian prior, the flat-manifold projection shrinks, with its eigenspace remaining the tangent space.The common eigenvalue reaches one only as τ approaches infinity or σ_t approaches zero.Source: B PROOF OF PROP. 2 · p. 16
  • With a translation-invariant tangent prior, the residual covariance equals the tangent projector, but this idealization has no proper probability measure on a non-compact subspace.The result is interpreted as the improper-prior limit of the proper Gaussian case.Source: B PROOF OF PROP. 2 · p. 16

C PROOF OF THEOREM 5

This section develops analytical results on covariance and variance behavior under stated assumptions, clarifying exact identities, approximations, and their limits. It also examines conditional interpretations and delayed-start masking, including tolerance-driven steps and ways to avoid a propagation issue.

C PROOF OF THEOREM 5

  • Under (H1)–(H4), the leading residual covariance is (I + hWν)^−2 PT + O(σ^2), with tangent eigenvalues (1 + hκ_i)^−2.The expansion applies to probes offset from the manifold; on-manifold probes give PT to this order, regardless of curvature.Source: C PROOF OF THEOREM 5 · pp. 17–19, REMARKS ON THE LEADING TERM AND THE… · p. 20
  • The order-σ bias mixes curvature with density drift, and the unit-circle check gives ∥E[r]∥ = 1.00 σ with fitted exponent 1.03.Unlike covariance, which drops the bias, this check can detect an incorrect curvature power.Source: C PROOF OF THEOREM 5 · p. 19

REMARKS ON THE LEADING TERM AND THE REMAINDER

  • The second-order form tracks the predicted eigenvalue to 1% over a factor of 2.3, with disagreement scaling as σ^2; at σ_t*=0.32, the finite-tangent-prior term is 0.19.The theorem reports curvature as seen from the offset, while the remainder also includes tangent-density effects; the first-order form leaves about a third of the leading term unaccounted for.Source: REMARKS ON THE LEADING TERM AND THE… · pp. 19–20

D THE FIBRED RESIDUAL, AND WHAT MEASUREMENT INJECTION DOES

  • Under a compatible local-product structure, the leading covariance is block diagonal and the residual-component cross-covariance is O(σ_t^2), but measurement injection only approximately cancels horizontal residuals.The fibre-uncertainty interpretation requires clean-baseline conditioning and a typical baseline; the implemented estimator may include horizontal contamination and has no exact-rank guarantee.Source: D THE FIBRED RESIDUAL, AND WHAT… · pp. 20–21

E WHEN THE DENOISER IS NOT OPTIMAL

  • A validation-distribution mean residual estimates global model-error bias, but the ideal-residual/model-error cross term is unknown and is not assumed zero.On toy manifolds at K = 2000 and d = 2 in codimension 1022, sampling alone gives eigenvalues 1.03 ± 0.03 and 0.97 ± 0.03; a measured 0.839 is roughly five standard deviations low.Source: E WHEN THE DENOISER IS NOT OPTIMAL · p. 21

F THE DELAYED-START MASK IN CLOSED FORM

  • The delayed-start mask cannot make the reverse-chain covariance recursion contractive: under the cosine schedule at T = 300 and ζ = 5, its homogeneous part is expansive at 100% of steps.Instead, the uniquely determined start step for a chosen amplification budget bounds cumulative inflation, whose gain telescopes; the tolerance remains the user's choice.Source: F THE DELAYED-START MASK IN CLOSED FORM · pp. 21–22
  • At T = 300 and ζ = 5, κ = 1.005, 1.01, and 1.02 yield sκ = 11, 16, and 24, respectively.The reported ˆs = 15 is close to the one-percent point, but that budget was not fixed before seeing it; the map uses O(T/ζ) scalar operations, with no autograd, denoiser call, or data. Proposed escapes include changing the schedule and retraining, leaving the affine class, tracking higher moments, or stopping recursion; the paper takes the last option.Source: F THE DELAYED-START MASK IN CLOSED FORM · p. 22

G MAGNITUDE AND RANKING IN ANALYTICAL VARIANCE PROPAGATION

  • Uniform sampler-noise injection preserves voxel ordering, including ties, and therefore leaves rank-based correlations and selective prediction unchanged under consistent tie handling.This applies to the fixed trajectory and active transitions.Source: G MAGNITUDE AND RANKING IN ANALYTICAL… · p. 22
  • The rank guarantee does not ensure calibrated intervals or good error ranking; changing the sampler, active window, or trajectory can change the ordering.A spatially constant injection floor alone cannot explain lost ranking performance.Source: G MAGNITUDE AND RANKING IN ANALYTICAL… · p. 23

H TESTING THE CURVATURE TERM AGAINST AN EXACT DENOISER

Exact-denoiser and synthetic-manifold tests probe the curvature correction and the limits of residual-covariance tangent recovery. Across the appendix experiments, the probe’s geometry depends on using a model-supported input, while finite-noise and finite-sample effects remain measurable.

H TESTING THE CURVATURE TERM AGAINST AN EXACT DENOISER

  • At h = 0.3, the exact covariance eigenvalue is 0.592 versus 0.490 from squaring the first-order projection, and measurement favors the exact form.The relative disagreement divided by σ^2 stays between −0.87 and −0.96 as σ falls from 0.20 to 0.05, consistent with an O(σ^2) remainder.Source: H TESTING THE CURVATURE TERM AGAINST… · p. 23

I.1 Cov(r) RECOVERS THE TANGENT EIGENSPACE AND EFFECTIVE RANK

  • On S1 and S2, residual covariance recovers the analytic tangent eigenspace to four and three decimal places, respectively, with the correct effective rank.This validates tangent support, not equality to the exact projector at finite noise; Proposition 1(iii) rules out that equality.Source: §I.1 · p. 24, I SYNTHETIC VALIDATION ON MANIFOLDS… · p. 24

I.2 THE CORRECTION TERMS ARE MEASURABLE, AND NOT ZERO

  • The measured residual covariance approximates the tangent projector only to within tens of per cent at tested noise scales, with a roughly 17% discrepancy persisting as noise shrinks.The experiment cannot distinguish denoiser error from schedule discretization, higher-order geometry, and finite-sample covariance error.Source: §I.2 · p. 25

I.3 NAIVE VARIANCE PROPAGATION, AGAINST GROUND TRUTH

  • With affine variance propagation delayed-started at t = 60 of 100 on S1, propagated per-pixel variance averaged over 20 unconditional DDPM runs overshoots the empirical per-pixel variance across those samples by 13.7×.The empirical reference is calculated across 20 samples, and the ratio compares means over the full 32 × 32 frame. The estimate uses seeds 100–119 and one manifold/seed set.Source: §I.3 · p. 25

I.4 ON- AND OFF-MANIFOLD BEHAVIOUR

  • Moving the probe query off-manifold separates residual drift by 4.54× on S1 and 2.79× on S2; probing a model reconstruction restores error information absent off-manifold.The degradation is monotone with distance to the manifold, and the medical claim is only that approximate DDIM reconstructions land near enough for the projector reading to survive.Source: §I.4 · pp. 25–26

J COST ACCOUNTING

  • At 128^3 resolution on a 40 GB accelerator, three Jacobian-based estimators exhaust memory before one volume, while the gradient-free probe completes.The released implementations disable activation checkpointing for the tangent pass; this is an implementation boundary, not a mathematical one.Source: Table 5 · p. 26, K WHY FORWARD-MODE DIFFERENTIATION… · p. 26

L PROTOCOL FOR THE CHECKPOINT WITHOUT A VARIANCE HEAD

  • The medical checkpoint without a variance head uses the same architecture and data as the hybrid PNG checkpoint, but emits C rather than 2C channels and omits the L_vlb loss term.It is trained with an ℓ1 noise-prediction objective and evaluated with the shared ζ = 5, η = 1 sampler.Source: L PROTOCOL FOR THE CHECKPOINT WITHOUT… · p. 27

N THE FOURTEEN CONDITIONS

  • Across fourteen model–corpus conditions, Hutchinson at M = 200 tracks the exact diagonal with at least 0.92 rank agreement in every condition.The supplied results describe the cross-condition approximation agreement, not a guarantee that the target diagonal ranks denoising error well.Source: Table 8 · p. 29, Table 7 · p. 29

O THE DIAGONAL REFERENCE AGAINST THE PROBE, IN FULL

The clinical comparison finds that T-PT can outperform a 20-chain reference inside brain tissue on selected endpoints, though the ensemble leads over the whole volume and lung CT. Probe-budget tests show that low-budget T-PT maps can be too unstable to interpret where the diagonal reverses.

O THE DIAGONAL REFERENCE AGAINST THE PROBE, IN FULL

  • The exact second-order probe map ranks identically to the exact diagonal on all four reversed conditions, so their shared reversal reflects distinct functionals rather than a shared estimand.On those conditions the row energy mirrors the diagonal, while the finite-budget probe map is barely reproducible at K = 30.Source: Table 11 · p. 30
  • At K = 30, probe maps on reversed conditions are too unstable to resolve their ranking; Hutchinson at M = 5 is already reproducible enough to reproduce the harmful reversal.On the two inet conditions, increasing probe budget improves reliability and the error correlation moves toward the exact second-order map’s −0.131.Source: O THE DIAGONAL REFERENCE AGAINST THE… · p. 31

P THE OPEN QUESTION, AND TWO THINGS THAT ARE NOT CRITERIA

  • The change-region loss is not explained by the top-10% threshold: on lung CT, probe correlations remain below MC-DDIM at top-5%, 10%, and 20% change thresholds.On brain MRI both maps are far below their tissue-wide correlations in changed regions; proposed explanations remain untested.Source: P THE OPEN QUESTION, AND TWO THINGS… · p. 32

O THE DIAGONAL REFERENCE AGAINST THE PROBE, IN FULL

  • On brain MRI, T-PT tracks ventricular and cortical boundaries consistently with MC-DDIM (ρ = 0.70 on this slice; 0.709 cohort-wide) and leads the ensemble on five of eight tissue endpoints, with three unresolved.On lung CT, T-PT reproduces MC-DDIM’s spatial pattern (ρ = 0.79 on this volume) at 13.4× fewer network evaluations.Source: Figure 3 · p. 33

Q QUALITATIVE COMPARISON

  • On lung CT, the three uncertainty maps concentrate on the nodule region, and T-PT’s spatial pattern agrees with MC-DDIM at ρ = 0.79 on the example volume.The overlays are individually quantile-normalized, so they show spatial ordering rather than comparable magnitudes.Source: Q QUALITATIVE COMPARISON · p. 32, Figure 3 · p. 33
  • On brain MRI, T-PT and MC-DDIM both follow ventricular and cortical boundaries, with cross-agreement ρ = 0.70 on the example and 0.709 across the cohort.The representative volumes were selected for anatomical clarity, not favorable numbers.Source: Q QUALITATIVE COMPARISON · pp. 32–33, Figure 3 · p. 33

R THE DEPLOYMENT COMPARISON, AND PAIRED TESTS

Deployment comparisons favor MC-DDIM over the whole volume, but T-PT leads inside brain tissue; larger ensembles narrow that gap, while probe settings affect the cost and regional performance of T-PT.

R THE DEPLOYMENT COMPARISON, AND PAIRED TESTS

  • Scoring each map against its own prediction raises whole-volume Spearman from 0.518 to 0.523 but lowers tissue Spearman from 0.349 to 0.343.The whole-volume increase reflects shared background agreement, while the tissue score better exposes the primary contrast.Source: R THE DEPLOYMENT COMPARISON, AND… · pp. 33–34
  • In simulation, participant-level bootstrap coverage was 95%, versus 57% for volume-level resampling of the true effect.The reported paired comparisons resample participants, preserving the pairing between estimators.Source: R THE DEPLOYMENT COMPARISON, AND… · pp. 34, 36
  • On brain MRI, T-PT leads inside tissue but trails MC-DDIM over the whole volume; on lung CT, MC-DDIM is significantly better despite T-PT’s 13.4× lower evaluation cost.Which estimator wins depends on dataset and region; in the deployment setting, MC-DDIM leads brain MRI over the whole volume by 3.2 Spearman points.Source: Table 14 · p. 34, R THE DEPLOYMENT COMPARISON, AND… · p. 36

S WHAT A LARGER MONTE-CARLO ENSEMBLE BUYS

T SENSITIVITY TO THE PROBE TIMESTEP

  • On lung CT, tissue ρ changes by less than 0.01 from t*=15 to 120, while edge and change-region scores improve but remain below MC-DDIM.The t*=60 setting was fixed before these runs.Source: Table 22 · p. 39

U HOW MANY PROBES?

  • On lung CT, K=10 probes recover 96.9% and K=15 recover 98.4% of the K=30 result.The sweep reuses the same reverse chain and covers the complete test set.Source: Table 23 · p. 40
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