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REVIEW 3 major objections 6 minor 79 references

Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A two-step probabilistic DeepONet models uncertainty in a low-dimensional coefficient space and recovers cross-location output covariance in a single forward pass, with an error bound tied to four identifiable factors.

desk verdict A genuinely useful extension of Prob-DeepONet: the paper moves Gaussian NLL training into a rotated low-dimensional coefficient space and thereby induces non-diagonal conditional covariance in the output field, but the fixed-basis diagonal-coefficient assumption is a structural prior that the experiments never directly test. read the letter →

arxiv 2608.09071 v1 pith:B2SHNH2G submitted 2026-08-10 math.NA cs.AIcs.NA

Yupei Nie , Lei Wang , Jiasen Liu This is my paper · ORCID
classification math.NAcs.AIcs.NA MSC 68T0765C20
keywords uncertaintypropagationoperatorlearningDeepONetprobabilisticsurrogatecovariancerecoverytwo-steptrainingrandominputfields
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to establish that a probabilistic operator surrogate can represent structured, cross-location predictive covariance without parameterizing a full high-dimensional covariance matrix. The proposed two-step MV-DeepONet performs Gaussian modeling in a low-dimensional, rotated coefficient space and maps the resulting diagonal coefficient covariance back through a shared orthonormal basis, producing a generally non-diagonal conditional predictive covariance in the physical output space while retaining single-pass inference. If the claim is right, uncertainty propagation for PDE-driven engineering systems gains a lightweight surrogate whose uncertainty bands and correlation patterns reflect the physical coupling of the output field rather than isolated pointwise variance. The paper also derives a Frobenius-norm error decomposition and tests the method on reaction-diffusion, Burgers, Darcy, and hypersonic aerothermal problems.

What carries the argument

The load-bearing object is the covariance propagation identity $\widehat{\Sigma}_s(u) = \widetilde{Q}^* \widehat{\Lambda}_{c,\theta}(u)(\widetilde{Q}^*)^\top$, which converts a diagonal Gaussian assumption on the rotated modal coefficients into structured output covariance through the shared orthonormal trunk basis $\widetilde{Q}^* \in \mathbb{R}^{M \times p}$ with $p \ll M$. The machinery also includes the two-step training schedule (trunk basis learning first, branch coefficient regression second, with QR orthogonalization and empirical coefficient-covariance diagonalization), and the error analysis built on the Eckart-Young-Mirsky truncation, a spectral-gap bound relating trunk reconstruction risk to subspace distance, and a Davis-Kahan-type finite-sample bound for the empirical eigenspace. Together these components support the claim that covariance recovery is governed by low-rank compressibility, subspace quality, sample size, and branch covariance estimation.

What would settle it

For a fixed test input $u$, draw many solver evaluations (or use a reference surrogate), center the outputs, project them onto the learned rotated basis $\widetilde{Q}^*$, and estimate the conditional coefficient covariance $C = \mathrm{Cov}(c \mid u)$; if the off-diagonal entries of $C$ are not negligible compared with its diagonal entries, then Eq. (30) will miss output correlations and the claimed recovery fails even with a perfect trunk subspace.

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Extended reading notes

Core claim

The central claim is that probabilistic modeling in a rotated low-dimensional coefficient space, followed by linear mapping through a shared trunk basis, induces a generally non-diagonal conditional predictive covariance in the physical output space: $\widehat{\Sigma}_s(u) = \widetilde{Q}^* \widehat{\Lambda}_{c,\theta}(u)(\widetilde{Q}^*)^\top$, where $\widehat{\Lambda}_{c,\theta}(u)$ is diagonal. The shared basis is the orthonormal trunk basis after QR orthogonalization and an empirical-covariance eigendecomposition, so each modal coefficient fluctuation reaches multiple output locations and creates cross-location dependence without explicit covariance parameterization. This relaxes the pointwise conditional-independence assumption of Prob-DeepONet while keeping single-pass inference, and the paper argues that the total predictive covariance decomposes by the law of total covariance into a mean-induced part and this structured conditional part. The theoretical analysis further claims that the Frobenius-norm recovery error is bounded by four terms: low-rank truncation of the true output covariance, trunk subspace projection error, finite-sample statistical error, and branch coefficient-covariance estimation error.

Load-bearing premise

The load-bearing premise is that, after the rotation, the modal coefficients are conditionally independent given the input, so the coefficient covariance $\widehat{\Lambda}_{c,\theta}(u)$ can be taken as diagonal; the rotation diagonalizes only the empirical covariance averaged over training samples, not the conditional covariance at each fixed input.

Editorial extensions

If this is right

  • A diagonal Gaussian in coefficient space becomes a generally non-diagonal conditional predictive covariance in output space via the shared basis, so cross-location dependence is represented without a full $\mathbb{R}^{M \times M}$ covariance matrix.
  • Single-pass inference is retained: the branch network predicts $p$ coefficient means and variances, and the output covariance is obtained by the low-rank product $\widetilde{Q}^* \widehat{\Lambda}_{c,\theta}(u)(\widetilde{Q}^*)^\top$.
  • The Frobenius error bound implies that fast spectral decay of the true output covariance, a well-learned trunk subspace separated by a spectral gap, and accurate branch covariance regression are the practical conditions for reliable covariance recovery.
  • On the tested reaction-diffusion, Burgers, Darcy, and hypersonic aerothermal problems, the method reports lower out-of-distribution mean errors, more structured uncertainty bands, and accurate multi-anchor correlation maps relative to Prob-DeepONet.
  • Conformal calibration brings interval coverage to nominal levels while the two-step model maintains noticeably narrower intervals than the baseline in the reported experiments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The rotation in the paper diagonalizes the unconditional empirical coefficient covariance over training data; whether the conditional coefficient covariance for a fixed input is also diagonal is a modeling assumption the paper does not prove, and non-diagonal conditional coefficients would change the recovered output correlations.
  • A natural extension is to relax the diagonal Gaussian coefficient model to non-Gaussian or correlated latent distributions, since the covariance propagation identity would still map latent dependence into output covariance; the paper lists this direction as future work.
  • The framework suggests a general design pattern for operator surrogates: choose any low-dimensional latent distribution, propagate it through a learned linear basis, and obtain an output covariance with structure inherited from the basis, so the method's expressiveness is tied to the quality of the trunk subspace.
  • The rank-one aerothermal case illustrates that low-rank covariance is not always easier to recover: with a single dominant mode, Frobenius error is highly sensitive to bias in that mode's variance even when the correlation structure is accurate, so metrics should be read together.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes two-step MV-DeepONet, a probabilistic operator-learning model that combines the two-step DeepONet training strategy of Lee and Shin with Gaussian modeling in a low-dimensional rotated coefficient space. The key algebraic device is Eq. (30), where a diagonal input-dependent coefficient covariance is mapped through a learned orthonormal output basis to produce a generally non-diagonal conditional predictive covariance in the physical output space. The authors also provide a Frobenius-norm error decomposition for total predictive covariance recovery, and they validate the method on reaction-diffusion, Burgers, Darcy, and hypersonic aerothermal problems, comparing against Prob-DeepONet in terms of generalization, uncertainty bands, covariance recovery, and calibrated prediction intervals.

Significance. If the central claim holds, the paper offers a lightweight alternative to full covariance parameterization in probabilistic operator learning: it obtains structured conditional covariance while retaining single-pass inference. The construction in Eqs. (22)-(30) is algebraically correct, and the paper is unusually explicit about the decomposition of covariance recovery error into truncation, subspace, statistical, and coefficient-regression terms via Eckart-Young and Davis-Kahan arguments. The empirical study is broad, including three PDE benchmarks and a CFD-based aerothermal problem with a validated solver. The paper also makes a fair comparison setup: covariance recovery is evaluated against held-out empirical covariances, and the branch network minimizes a Gaussian NLL rather than the Frobenius covariance error directly, so the central mechanism is not circular. The main weakness is that the claimed advantage over Prob-DeepONet concerns the conditional predictive covariance, while the numerical evaluation targets the total predictive covariance, for which no ground-truth conditional object exists in the deterministic benchmarks.

major comments (3)
  1. [Section 2.4.2, Eq. (27)] The diagonal form of the conditional coefficient covariance is an input assumption, not a consequence of the rotation in Eq. (22). The rotation diagonalizes the unconditional empirical coefficient covariance over the training set; for a fixed input u it does not diagonalize the conditional covariance of the coefficient residuals, and the eigenvectors of that conditional covariance may vary with u. The model class in Eq. (30) is therefore restricted to matrices of the form eQ* diag(sigma^2) eQ*^T. If the true conditional coefficient covariance has non-negligible off-diagonal entries, the predicted off-diagonal output correlations are misspecified. Since the misspecification is absorbed into the unquantified term eta_B_app in Eq. (65), the analysis does not control the error caused by the central modeling assumption. A concrete diagnostic would be to estimate the conditional covariance of the coefficient residuals in fixed input bins, or to test on a synthetic problem with known non-diagonal conditional coefficient covariance.
  2. [Section 3, Eq. (70) and Sections 3.1.2-3.4.2] The numerical evaluation compares the total predictive covariance bSigma_2step of Eq. (37) with the empirical total covariance of the test outputs, not the conditional covariance bSigma_s(u) of Eq. (30). Because the PDE and CFD maps in these benchmarks are deterministic, there is no ground-truth conditional covariance of s(u)|u against which to validate Eq. (30); the non-diagonality of the reference covariance arises from variation of the conditional means across inputs, i.e., the first term in the law of total covariance, Eq. (14). The reported Frobenius errors and correlation maps therefore cannot substantiate the paper's claim that the method recovers off-diagonal conditional dependence. I recommend adding either a benchmark with repeated output realizations for the same input (for example noisy observations or stochastic PDE outputs), or a held-out residual analysis that decomposes the total covariance error into the contributions of bB_mu and bB_sigma, together with a correlation-map comparison against Prob-DeepONet's total covariance.
  3. [Section 2.4.4, Eqs. (65)-(69)] The presentation calls Eq. (69) an upper bound, but eta_B_app and eta_B_opt are defined as the deviations of the trained model from a best-in-class model and are never estimated or dominated by computable quantities. The recovered error hierarchy T1 << T2, T3 << T2 reported in Sections 3.1.2, 3.2.2, and 3.3.2 is inferred from the gap between the oracle curve and the final model curve; that gap is precisely the unquantified model-dependent error. The theory would be substantially stronger if the approximation term were bounded or estimated, or if the misspecification in Eq. (27) were isolated in a separate term that the experiments explicitly control.
minor comments (6)
  1. [Eqs. (32) and (41)] The prediction equations describe the centered output only; the paper should state explicitly that the reported mean predictions in Figures 4, 9, 14, and 21 are obtained by adding the training mean from Eq. (19) to eQ* times the predicted coefficient mean.
  2. [Section 2.4.3, Eq. (31)] The argument that the covariance is nonzero excludes only the case where all basis functions vanish simultaneously at one of the two locations; a complementary pattern in which some modes vanish at y_i and other modes vanish at y_j is still logically possible. A generic-density or non-vanishing-product assumption would make the argument complete.
  3. [Section 3, ensemble usage] The paper does not report the network widths, depths, learning rates, optimizer settings, or the ensemble aggregation rule for the eight-member ensembles shown in the training-loss figures; because the numerical results appear to depend on these ensembles, the experiments are difficult to reproduce without this information.
  4. [Figures 2, 3, 7, 8, 12, 13, 18, 19] The axis labels use a pound sign where a multiplication sign is intended; the labels should read 'Epochs (x 10^3)' or similar.
  5. [Section 3.1.1, Table 2] The 'mean error' over out-of-distribution correlation lengths is an unweighted average over an irregular grid of ell values; the averaging convention should be stated.
  6. [Section 3.4 and Appendix B] The aerothermal example varies only the freestream Mach number, and Appendix B shows that the effective stochastic dimension is one; the paper should acknowledge that this case is not a test of high-dimensional input-field uncertainty propagation in the same sense as the PDE benchmarks.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the non-diagonal conditional covariance in Eq. (30) follows by exact covariance propagation from a diagonal coefficient-space model; the rotation is a standard training-set representation and is not used as a substitute for the prediction target.

full rationale

The paper's central construction is algebraic rather than circular. Eq. (30), Sigma_s(u) = eQ* Lambda_c,theta(u) (eQ*)^T, is the definition of the output covariance of the linear map x_hat(u) = eQ* c_hat(u); a diagonal Lambda produces a generally non-diagonal output covariance, so the stated result is a consequence of the model specification, not a hidden reuse of the target. The rotation in Eq. (22) diagonalizes the unconditional empirical coefficient covariance over training outputs; this is representation learning (a PCA-type coordinate change), and the paper never claims it makes the conditional coefficient covariance diagonal. The diagonal conditional coefficient covariance in Eq. (27) is an explicitly stated modeling assumption, and Section 4 identifies non-Gaussian conditional structure as future work; an unverified assumption is a limitation of model fidelity, not circularity. The error bound in Eqs. (50)-(69) is a standard triangle-inequality decomposition into truncation, subspace, statistical, and branch terms; it does not assume the quantity being bounded. The two-step training and Theorem 3.5 are cited to Lee and Shin [45], an external reference, and no self-citation is load-bearing. Numerical evaluation compares the model covariance to the held-out test empirical covariance, so the covariance claim is independently checked rather than fitted by construction.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The method introduces no new physical entities. The free parameters are hyperparameters selected for the experiments. The main modeling axiom is the diagonal conditional coefficient covariance, which is not guaranteed by the unconditional rotation. The concentration-rate assumption in the theory is stated without the effective-rank caveat, which weakens the theoretical bound as a general statement.

free parameters (4)
  • Number of retained modes p = varies per case (e.g., up to 120 for reaction-diffusion, 200 for Burgers)
    Rank of the low-dimensional coefficient space, chosen by the user to balance subspace expressiveness and regression complexity; the paper sweeps p in the error curves and tables.
  • VAE latent dimension dz = 64
    Selected by sensitivity study in Appendix A.2 among {56, 64}; affects the aerothermal branch input.
  • KL regularization weight lambda_KL = 1e-4
    Selected in Appendix A.2 by sensitivity study among {0, 1e-6, 1e-4}.
  • Physics-informed weighting strength epsilon = 1.0
    In Eq. (92), controls the W_delT weighting in the aerothermal trunk reconstruction loss; set to 1.0, giving a bounded weight between 1 and 2.
assumptions (3)
  • domain assumption The true covariance spectral gap gamma_p = lambda_p - lambda_{p+1} is non-vanishing.
    Invoked in Sections 2.4.4.3 and 2.4.4.5 to apply Davis-Kahan type bounds; the paper states this is assumed and only the empirical gap is verified numerically.
  • domain assumption Standard covariance concentration conditions give ||Sigma_hat - Sigma||_2 = OP(K^{-1/2}).
    Section 2.4.4.5, Eq. (69). This is not generally true for high-dimensional M without an effective-rank condition; the resulting Frobenius bound OP(sqrt(M/K)) is therefore not fully justified as a general statement.
  • ad hoc to paper The conditional covariance of modal coefficients in the rotated space is diagonal (independence of coefficients given input).
    Eq. (27) imposes a diagonal Gaussian model on the rotated coefficients. The rotation in Eq. (22) diagonalizes the unconditional empirical coefficient covariance, not the conditional covariance, so this is a modeling restriction rather than a consequence of the rotation.

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Cite this review

Pith. "Pith review of Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields." pith.science (2026). https://pith.science/paper/B2SHNH2G

@misc{pith2026260809071,
  author       = {Pith},
  title        = {Pith review of: Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2SHNH2G}},
  note         = {Machine review of arXiv:2608.09071}
}
read the original abstract

Forward uncertainty propagation in complex physical systems can induce structured covariance across field-valued outputs. For a probabilistic surrogate, the total predictive covariance comprises the covariance of conditional means across input realizations and the average conditional predictive covariance. Probabilistic DeepONet (Prob-DeepONet) provides lightweight uncertainty quantification by predicting pointwise Gaussian means and variances in a single forward pass, but its conditional predictive covariance is restricted to a diagonal form. To represent cross-location conditional dependence without explicitly parameterizing a full high-dimensional covariance matrix, we develop a two-step mean-variance DeepONet (two-step MV-DeepONet) through two principal modifications. First, two-step training is used to decouple output-basis learning from the input-to-coefficient mapping, together with basis orthogonalization and subspace rotation. Second, Gaussian probabilistic modeling is transferred from the high-dimensional physical output space to the low-dimensional rotated coefficient space. Mapping these probabilistic coefficients through the shared basis induces a generally non-diagonal conditional predictive covariance in the physical output space while retaining single-pass inference. A Frobenius-norm error decomposition and corresponding upper bound identify low-rank covariance compressibility, trunk-subspace approximation, finite-sample statistical error, and coefficient-space covariance estimation as the principal factors governing covariance recovery. Numerical experiments on three representative problems governed by partial differential equations (PDEs) and a hypersonic blunt-body aerothermal problem show improved generalization, more structured uncertainty bands, and accurate recovery of off-diagonal correlation patterns compared with Prob-DeepONet.

Figures

Figures reproduced from arXiv: 2608.09071 by the authors.

Figure 1
Figure 1. Schematic of the two-step MV-DeepONet framework. Step 1 learns the output-space basis and applies QR orthogonal￾ization and the subspace rotation Qe∗ = Q∗VbQ. Step 2 predicts the coefficient mean µˆc,θ(u) and diagonal coefficient covariance Λbc,θ(u). Finally, the resulting coefficient distribution is mapped back to the physical output space through the shared rotated basis, producing the output mean and structured c… view at source ↗
Figure 2
Figure 2. Training loss curves of Prob-DeepONet: individual ensemble members (left) and the ensemble mean with a ±1 standard deviation band (right). The quantitative errors under the same input distribution are summarized in [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Training loss curves of two-step MV-DeepONet for the trunk (a) and branch (b) networks: individual ensemble members (left) and the ensemble mean with a ±1 standard deviation band (right) [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Predictions and predictive uncertainty bands of Prob-DeepONet (left column) and two-step MV-DeepONet (right column) for representative out-of-distribution samples at different input correlation lengths. facilitates training-data fitting but may produce a less stable re…
Figure 5
Figure 5. Figure 5: Normalized Frobenius covariance error as a function of the number of retained modes. (1) Oracle SVD curve. The Oracle SVD error decreases rapidly and reaches approximately 10−13 for p ≥ 15, indicating that the low-rank compressibility assumption introduced in Section 2…
Figure 6
Figure 6. Figure 6: Multi-anchor correlation maps showing the reference correlations, predicted correlations, and corresponding absolute errors for three anchor points, where erow denotes the relative L2 error of the predicted correlation map for each anchor point. location x = xa, respon…
Figure 7
Figure 7. Figure 7: Training loss curves of Prob-DeepONet: individual ensemble members (left) and the ensemble mean with a ±1 standard deviation band (right). 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Training loss curves of two-step MV-DeepONet for the trunk (a) and branch (b) networks: individual ensemble members (left) and the ensemble mean with a ±1 standard deviation band (right) [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Predictions and predictive uncertainty bands of Prob-DeepONet (left column) and two-step MV-DeepONet (right column) for representative out-of-distribution samples at different values of the random-field parameter τ. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 9
Figure 9. Figure 9: Predictions and predictive uncertainty bands of Prob-DeepONet (left column) and two-step MV-DeepONet (right column) for representative out-of-distribution samples at different values of the random-field parameter τ (continued) [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: Normalized Frobenius covariance error as a function of the number of retained modes. 2 × 10−1 to 8 × 10−3 . The enlarged inset shows that the Final model prediction remains approximately 1–2 × 10−3 above the Learned-subspace oracle. Two observations follow: 1. The con…
Figure 11
Figure 11. Figure 11: Multi-anchor correlation maps showing the reference correlations, predicted correlations, and corresponding absolute errors for three anchor points, where erow denotes the relative L2 error of the predicted correlation map for each anchor point [PITH_FULL_IMAGE:figur…
Figure 12
Figure 12. Figure 12: Training loss curves of Prob-DeepONet: individual ensemble members (left) and the ensemble mean with a ±1 standard deviation band (right). 0.0 0.5 1.0 1.5 2.0 2.5 Epochs (£10 3 ) 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Training loss Individual ensemble members Member 1 Mem…
Figure 13
Figure 13. Figure 13: Training loss curves of two-step MV-DeepONet for the trunk (a) and branch (b) networks: individual ensemble members (left) and the ensemble mean with a ±1 standard deviation band (right) [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Predictions and predictive uncertainty bands of Prob-DeepONet (left column) and two-step MV-DeepONet (right column) for representative out-of-distribution samples at different input correlation lengths. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 14
Figure 14. Figure 14: Predictions and predictive uncertainty bands of Prob-DeepONet (left column) and two-step MV-DeepONet (right column) for representative out-of-distribution samples at different input correlation lengths (continued). 3.3.2 Analysis of Output Covariance Recovery Accuracy…
Figure 15
Figure 15. Figure 15: shows the three normalized Frobenius covariance-error curves as a function of the number of retained modes p, and [PITH_FULL_IMAGE:figures/full_fig_p026_15.png]
Figure 16
Figure 16. Figure 16: Multi-anchor correlation maps showing the reference correlations, predicted correlations, and corresponding absolute errors for five anchor points, where erow denotes the relative L2 error of the predicted correlation map for each anchor point. 3.3.3 Prediction Interv…
Figure 17
Figure 17. Figure 17: Comparison of wall heat flux distributions obtained from the present CFD simulation, the LENS-XX experiment, and NASA DPLR. The angle θ is measured along the hemispherical surface from the stagnation point, with θ = 180sarc/(πR), where sarc is the surface arc length a…
Figure 18
Figure 18. Figure 18: Training loss curves of Prob-DeepONet: individual ensemble members (left) and the ensemble mean with a ±1 standard deviation band (right). 0 40 80 120 160 200 Epochs (£10 3 ) 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Training loss Individual ensemble members Member 1 M…
Figure 19
Figure 19. Figure 19: Training loss curves of two-step MV-DeepONet for the trunk (a) and branch (b) networks: individual ensemble members (left) and the ensemble mean with a ±1 standard deviation band (right) [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]
Figure 20
Figure 20. Figure 20: presents the reference fields, predicted fields, and relative error distributions for these samples [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: Predicted mean and uncertainty bands for the training and in-distribution test samples with the largest relative L2 errors. The uncertainty bands correspond to ±2σ. The first and second rows correspond to Prob-DeepONet and two-step MV-DeepONet, respectively. Regarding…
Figure 22
Figure 22. Figure 22: Normalized Frobenius covariance error as a function of the number of retained modes [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]
Figure 23
Figure 23. Figure 23: Multi-anchor correlation maps showing the reference correlations, predicted correlations, and corresponding absolute errors for three anchor points, where erow denotes the relative L2 error of the predicted correlation map for each anchor point. 3.4.3 Prediction Inter…

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