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A Product-Grassmann Tucker Surrogate for Parametric Field Prediction

Melika Baghi, Xiao Liu, Kamran Paynabar

arXiv:2610.34559v1cs.CE

TL;DR

Many-query applications require repeated full-field predictions from expensive PDE simulations. The paper proposes a Tucker surrogate that interpolates mode-wise subspaces and models aligned core variation. Across five PDE benchmarks, it reports substantial reductions in interpolation size and stored model size with close accuracy, while full-field reconstruction limits end-to-end speedup.

  • Problem

    Many-query applications need repeated full-field predictions from expensive PDE simulations at low cost.

    Source: §1 · p. 2

  • Method

    The surrogate keeps spatial directions and time as separate Tucker modes, interpolates their subspaces on a product Grassmann manifold, and models Procrustes-aligned core variation with a low-rank model.

    Source: §5 · p. 27

  • Results

    Across five PDE benchmarks, interpolation size and stored model size fall by more than an order of magnitude and cached query time by at least threefold, while accuracy remains close to the single-subspace surrogate.

    Source: §5 · p. 27

  • Takeaways & Limitations

    Full-field reconstruction limits the realized end-to-end speedup of the mode-wise surrogate.

    Source: §3.7 · p. 12, §5 · p. 28

    The implementation uses fixed Tucker ranks, grid-axis modes, and one reference tangent space, and the main numerical study covers scalar fields on regular grids with two or three parameters.

    Source: §5 · p. 28

Abstract

from arXiv · show

Many-query applications such as design optimization, uncertainty quantification and digital twins call for repeated full-field predictions from expensive partial differential equation (PDE) simulations. Grassmannian reduced-order models interpolate parameter-dependent subspaces but commonly vectorize the spatial field, and the ambient dimension then grows with the product of the grid sizes. Tensor-based models preserve the multiway structure but often use shared mode-wise bases or a fixed core across parameters. We propose a Tucker-based surrogate that keeps the spatial directions and time as separate modes. The factor subspaces are interpolated on a product Grassmann manifold, Procrustes alignment places the Tucker cores in one coordinate system, and a truncated singular value decomposition models their variation. Across five PDE benchmarks in two and three spatial dimensions, including public computational fluid dynamics data, the interpolation size is reduced by factors of 13 to 482 and the stored model by factors of 13 to 464 relative to a single-subspace Grassmannian surrogate with the same local interpolation. Cached queries run 3.3 to 8.5 times faster, the mean errors over ten resampled splits differ by at most 0.22 percentage points (0.36 when every model choice is re-selected inside each split), and the prediction-error terms of the two methods differ by less than 0.1 percentage points. The remaining accuracy differences are consistent with the larger representation error of the Tucker format. In a three-dimensional grid study the interpolation-size ratio rises to 2,260 and the query speedup to 11.9 at 64^3.

1. Introduction

Many-query applications need repeated, low-cost predictions of high-dimensional fields, while vectorized Grassmannian models interpolate in an ambient space that grows with the product of spatial grid sizes. This work keeps spatial directions and time as separate Tucker modes, interpolates their subspaces on a product Grassmann manifold, and models aligned core variation.

1. Introduction

  • Vectorized Grassmannian models operate in an ambient space of dimension XY, so the interpolated object grows with the product of the spatial grid dimensions.The study compares this cost and accuracy with keeping tensor modes separate.Source: §1 · p. 2
  • The surrogate preserves the field's multiway structure by decomposing each training field into Tucker factors for the X, Y, and T modes.Their column spaces define a point on a product Grassmann manifold.Source: §1 · p. 2
  • Orthogonal Procrustes rotations align training factors to a common reference before a truncated SVD models variation in their rotated Tucker cores.The factor subspaces and core coordinates are predicted with the same local parameter-space weights; the implementation uses one reference tangent space.Source: §1 · p. 3
  • The main contributions include a product-Grassmann Tucker surrogate, a reference-aligned low-rank model of parameter-dependent cores, and cost and error analyses on five PDE benchmarks.The benchmarks include public CFD data and a three-dimensional scaling study.Source: §1 · pp. 3–4

2. Related Work

Prior work includes Grassmannian reduced-order models for parameter-dependent spaces and tensor methods that preserve multiway structure, but their treatment of spatial modes and parameter variation differs. The proposed construction assigns each Tucker mode its own parameter-dependent subspace and models aligned core variation.

2.1. Parametric reduced-order models and Grassmann interpolation

  • Grassmannian reduced-order models interpolate parameter-dependent subspaces, while vectorized spatial fields produce an ambient dimension that grows with the product of grid dimensions.Related approaches also include learned parameter-to-subspace maps, adaptive sampling, and interpolation of dynamic mode decomposition modes.Source: §2.1 · p. 4

2.2. Tensor-structured reduced-order modeling

  • Tensor-structured methods preserve multiway organization, but many share physical-mode bases or carry parameter dependence in coefficients, a core, or a parameter-mode factor.Closest space-time methods vary spatial and temporal bases, but a single basis still spans all spatial degrees of freedom; the present construction extends to each spatial axis and time.Source: §2.2 · p. 5

2.3. Basis alignment

  • Tucker factors can rotate without changing the reconstructed tensor, so equivalent bases must be aligned before cores are compared across parameters.Orthogonal Procrustes alignment places equivalent bases in a common orientation.Source: §2.3 · p. 6

2.4. Limits of existing approaches

  • The proposed surrogate gives every Tucker mode its own parameter-dependent subspace on a product Grassmann manifold and varies the core after aligning it to a common reference.This addresses the reviewed approaches' use of a product-sized vectorized space, shared physical-mode bases, or one basis spanning all spatial degrees of freedom.Source: §2.4 · p. 6

3. Methodology

The surrogate preserves spatial and temporal modes by representing each training field with Tucker factors and a core, then predicts a new field by interpolating the factor subspaces and core. Reference alignment makes the cores comparable, while mode-wise interpolation avoids the product-sized ambient space used by a single-subspace representation.

3.1. Problem setup and Tucker representation

  • Each training field is Tucker-decomposed into mode-wise factors and a core, separating changes in dominant subspaces from interactions among coordinates in those subspaces.The ranks determine both compression and the size of the online problem.Source: §3.1 · p. 7

3.2. Product-Grassmann representation of the factor subspaces

  • The surrogate represents the X-, Y-, and T-mode factor subspaces as a product-Grassmann point, while the comparison method reshapes the spatial dimensions into one axis and interpolates in dimension XY.A third spatial axis adds one Grassmann factor and one index to the core without otherwise changing the construction.Source: §3.2 · p. 7

3.3. Reference alignment of Tucker factors and cores

  • Orthogonal Procrustes rotations align each training factor to a fixed reference and transfer the same rotations to its core, so aligned-core differences reflect parameter dependence rather than factor orientation.Any reference satisfying the stated nonsingularity condition supports the construction; the mean is used to keep training angles small.Source: §3.3 · pp. 8–9

3.4. Tangent-space prediction of the factor subspaces

  • Each mode’s factor subspace is mapped to tangent coordinates at a reference, predicted there by a weighted average of training coordinates, and returned to the manifold by the exponential map.One set of parameter-space weights serves all three modes and also drives core prediction.Source: §3.4 · pp. 9–10

3.5. Low-rank prediction of the aligned core

  • A truncated SVD of vectorized residuals about the mean aligned core retains shared core-variation patterns, with each training run represented by coordinates in those patterns.At a query, the coordinates are predicted while the pattern basis remains fixed.Source: §3.5 · pp. 10–11

3.6. Compatible reconstruction

  • The predicted factors and aligned core share the reference orientation, so they can be combined directly to reconstruct the field without further query-time alignment.Remaining errors include factor-coordinate interpolation, core-coordinate interpolation, and residual-core truncation; field Tucker truncation is a separate representation error.Source: §3.6 · p. 11

3.7. Computational complexity

  • The query-cost and storage comparison separates Grassmann interpolation from field reconstruction; reconstruction has the same order in both methods and bounds the end-to-end speedup.Table 1’s interpolation-size row excludes core and coefficient terms, which favors the single-subspace method.Source: §3.7 · p. 12, Table 1 · p. 12
  • At comparable ranks, the product-Grassmann method interpolates XrX + YrY + TrT values versus XY rs for the single-subspace method, with the ratio about XY/(X + Y + T).A third spatial axis increases the ratio because the single-subspace size grows with the product of spatial sizes, while the mode-wise size grows with their sum.Source: §3.7 · p. 12

4. Numerical Experiments

Across five PDE benchmarks, the numerical experiments assess how M1 balances accuracy against interpolation size and cached-query time relative to M2. They also examine factors affecting predictions, including parameter-varying cores, alignment, and the methods’ representation and prediction error terms.

4.1. Experimental protocol

  • The experiments compare M1 with M2 using the same training data and local interpolation, and assess full-tensor test error, interpolation size, cached query time, and representation versus prediction error.Each method was tuned on its own training data, and accuracy comparisons refit both methods on ten resampled splits using the selected ranks and hyperparameters.Source: §4.1 · pp. 13–15

4.2. Benchmark problems

  • The study uses five parameterized PDE benchmarks spanning differences in mode-wise compressibility, subspace variation, and spatial dimensionality; four were constructed for the study and one uses public CFDBench data.Table 3 summarizes the governing problems, grids, parameter dimensions, data splits, and benchmark purposes.Source: §4.2 · p. 16, Table 3 · p. 16

4.3. Case Study I: anisotropic diffusion-reaction

  • On the easily compressed diffusion-reaction field, M1 reached M2’s best mean error of 1.18 percent to within 0.01 points at r = 6 with 107 times fewer interpolated values.The methods had the same mean test error at their selected ranks, while fixing the core at its training mean more than tripled error.Source: §4.3 · p. 17, Figure 3 · p. 18

4.4. Case Study II: double-slit wave interference

  • On the wave field, M1 remained 0.09 points above M2’s best mean error of 22.28 percent at r = 43 while interpolating 19 times fewer values.A fixed core gave three times the methods’ error; both methods left narrow residual bands consistent with small predicted-phase errors.Source: §4.4 · p. 18, Figure 5 · p. 19

4.5. Case Study III: multimodal forced heat equation

  • At r = 13, M1 reached M2’s best mean error of 7.82 percent to within 0.01 points using 100 times fewer interpolated values.The heat field has fixed spatial subspaces but parameter-varying temporal subspaces; freezing the core increased M1’s error more than fivefold.Source: §4.5 · p. 20, Figure 7 · p. 21

4.6. Case Study IV: tube flow (CFDBench)

  • At r = 67, M1 matched M2 within 0.01 points using 17 times fewer interpolated values; increasing its core rank later lowered error to 1.72 percent, below M2’s 1.76 percent.M2’s rank ceiling makes its representation exact in this case, and the prediction-error terms nearly coincide; the remaining gap is consistent with M1’s Tucker representation error.Source: §4.6 · pp. 21–22, Figure 9 · p. 22

4.7. Case Study V: three-dimensional advection-diffusion

  • At r = 15, M1 remained 0.21 points above M2’s best mean error while using 447 times fewer interpolated values; its +0.22-point resampled gap stayed within tolerance.The 3D study found the largest interpolation reduction and query speedup among the five benchmarks; the gap was unchanged on finer grids, and query speedup grew more slowly than interpolation savings. The fixed-core and unaligned variants raised error by factors of 2.8 and 2.5, respectively.Source: §4.7 · pp. 23–24, Figure 11 · p. 24, Table 4 · p. 24

4.8. Accuracy and cost across the benchmarks

  • Both M1 ablations were worse on every case: the fixed-core ablation had errors 2.8 to 5.4 times higher and the unaligned-core ablation 2.3 to 10.6 times higher.The largest interpolation-size reductions occurred on the three-dimensional benchmark; the smallest occurred on the wave problem, where 500 time steps enlarge the temporal mode.Source: §4.8 · p. 25
  • Across five benchmarks, M1 stayed within the prespecified 1-percentage-point accuracy tolerance while reducing interpolation size by factors of 13 to 482; M2 was 0.07 to 0.22 points more accurate on Cases II to V.Every upper limit was below the relative margin of 10% of the M2 error.Source: Table 5 · p. 25, §4.8 · p. 25
  • Cached-query speedups were 3.3 to 8.5, smaller than Grassmann-stage ratios of 3.9 to 28.1 because full-field reconstruction remains necessary and limits end-to-end gains.M1’s Grassmann stage was faster on every benchmark, while reconstruction was slower except on Case III.Source: §4.8 · p. 25, Table 6 · p. 26
  • M1’s cached-query savings recovered its longer offline training after about 1,500 queries on Case III and fewer on Cases II and V; M1 was ahead from the first query on Cases I and IV.The Grassmann stage drives the speedup on Case V, whereas reconstruction accounts for most of M1’s query cost on Case I.Source: §4.8 · pp. 25–26
  • Under nested model re-selection, every upper limit remained below tolerance, although the Case II mean gap rose to 0.36 points; widening Case V’s design box to 0.40 raised its gap to 5.30 points.Across rank budgets, M1 matched M2’s best mean error within 0.01 points on Cases I, III, and IV, and remained 0.09 and 0.21 points higher on Cases II and V at the largest budgets; at matching points it used 17 to 107 times fewer interpolation values and answered 2.9 to 6.5 times faster on Cases I, III, and IV.Source: §4.8 · p. 26

4.9. Error attribution

  • The methods’ prediction terms differed by less than 0.1 percentage points on every benchmark, while M1’s remaining error differences were consistent with its larger representation term.The representation term was at most about one eighth of total error on Cases I to III, where methods were very close, and about one quarter on Cases IV and V, which had the largest M1-minus-M2 differences relative to error.Source: §4.9 · pp. 26–27, Table 7 · p. 27

5. Conclusion

The surrogate keeps spatial directions and time as separate Tucker modes, predicts their subspaces on a product Grassmann manifold, and models aligned-core variation. Across five PDE benchmarks it retains close accuracy while reducing computational costs, with implementation and study boundaries that motivate further validation.

5. Conclusion

  • The method keeps spatial directions and time as separate Tucker modes, predicts factor subspaces on a product Grassmann manifold, and models Procrustes-aligned core variation with a low-rank model.The predicted factors and core combine directly in reconstruction.Source: §5 · p. 27
  • Across five PDE benchmarks, the product-Grassmann surrogate is close in accuracy to a single-subspace Grassmannian surrogate while reducing interpolation size, stored model size, and cached query time by more than an order of magnitude or at least threefold, respectively.The full-field reconstruction limits the realized end-to-end speedup.Source: §5 · pp. 27–28
  • The current implementation uses fixed Tucker ranks, grid-axis-tied modes, and one reference tangent space, requiring training and query subspaces to remain close to that reference.The main numerical study covers scalar fields on regular grids with two or three parameters; six-parameter and widened-design-box studies report accuracy challenges.Source: §5 · p. 28
  • Future work includes using multiple local reference spaces and validating on larger three-dimensional finite-volume or finite-element simulations and measured spatiotemporal fields.Source: §5 · p. 28

CRediT authorship contribution statement

The authors divide conceptualization, methodology, software, validation, investigation, supervision, and writing responsibilities.

CRediT authorship contribution statement

Funding

The research reports no specific grant funding.

Funding

  • The research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.Source: Funding · p. 29

Data and code availability

The authors describe access to implementation and experiment materials during review and planned public release upon acceptance.

Data and code availability

  • During review, the implementation, experiment scripts, stored run outputs, and build file are available from the corresponding author on request.Source: Data and code availability · p. 29
  • Upon acceptance, the repository is to be made public with generator scripts, stored fields, preprocessing and case files, the software environment, and files behind reported tables and figures.Source: Data and code availability · p. 29
  • The appendix provides bounds, proofs, and algorithms for the methodology, including a proof of compatibility between predicted factors and the predicted aligned core.That compatibility lets them combine without online realignment; the appendix is written for the third-order case.Source: Data and code availability · p. 30

Appendix A.1. Representation error

The appendix characterizes Tucker representation error, defines the product-Grassmann geometry and local maps, and shows how Procrustes alignment gives factors and cores compatible coordinates.

Appendix A.1. Representation error

  • The truncated HOSVD error is bounded using discarded singular-value tails across modes, and the experiments report realized truncation error for the implemented decompositions.The randomized mode-wise SVDs used in Cases I–IV distinguish these realized errors from the theoretical bound.Source: Appendix A.1 · p. 30

Appendix A.2. Product Grassmann geometry

  • The product manifold represents separate X, Y, and T subspaces, with tangent dimension rX(X − rX) + rY(Y − rY) + rT(T − rT), rather than rs(XY − rs) for one spatial subspace.Its reference is the Riemannian center of mass of the training product points.Source: Appendix A.2 · p. 31

Appendix A.3. Grassmann logarithm and exponential maps

  • Grassmann logarithm and exponential maps are applied independently to each mode, assuming training and query points remain in a common neighborhood where the maps are unique.The implementation uses 0.9(π/2) as a diagnostic and clipping threshold; training angles may exceed it while remaining below π/2.Source: Appendix A.3 · pp. 31–32

Appendix A.4. Procrustes alignment and coordinate compatibility

  • Procrustes rotations align training factors to the reference frame and counter-rotate their cores, making alignment an exact coordinate change with no reconstruction error.The predicted factors and aligned training factors share the same reference frame, so the predicted core combines with them without another rotation.Source: Appendix A.4 · pp. 32–33

Appendix A.5. Aligned-core bound

The appendix bounds aligned-core truncation error, summarizes the higher-dimensional reconstruction, and documents the numerical settings and data used across five case studies.

Appendix A.5. Aligned-core bound

  • At the selected core ranks, the aligned-core truncation-error bound is 0.34, 4.51, 0.83, 1.62, and 1.34 percent of the mean training field norm for Cases I–V.This is a worst-case bound over an arbitrary weight vector.Source: Appendix A.5 · p. 34

Appendix A.6. Algorithmic summary

  • Case V adds a third spatial factor and core index, so its reconstruction applies mode-wise steps over X, Y, Z, and T.Cases I–IV use randomized mode-wise SVDs, while Case V uses full mode-wise SVDs.Source: Appendix A.6 · p. 34

Appendix B. Supplemental Numerical Model Details

  • The supplemental section records experimental constants, designs, data provenance, and implementation details behind the numerical experiments.The governing equations are in the main text.Source: Appendix B · p. 34

Appendix B.1. Case Study I: anisotropic diffusion-reaction

  • Case I uses 40 Latin-hypercube parameter samples, with 30 for training and 10 interior points for testing, on a 256 × 256 grid sampled at 100 times.The fields use a Fourier spectral method with an integrating-factor step.Source: Appendix B.1 · pp. 34–35

Appendix B.2. Case Study II: double-slit wave interference

  • Case II uses a 6 × 6 amplitude-frequency grid, training on 30 runs and testing on six interior grid points.The archived data omit the integration scheme and slit and boundary specifications.Source: Appendix B.2 · p. 35

Appendix B.3. Case Study III: multimodal forced heat equation

  • Case III uses 40 uniform parameter draws split into 30 training and 10 test cases, with fields on a 128 × 128 grid at 101 times.The initial field has eight sine modes, and the forcing comprises nine Gaussian-in-time pulses.Source: Appendix B.3 · p. 36

Appendix B.4. Case Study IV: tube flow (CFDBench)

  • Case IV uses 98 CFDBench runs of common length 15, training on 78 and testing on 20, including eight boundary cases.The study uses fluid-property variation and reduces the two velocity components to magnitude.Source: Appendix B.4 · pp. 36–37

Appendix B.5. Case Study V: three-dimensional advection-diffusion

  • Case V tests on a 32 × 32 × 32 × 24 tensor of 786,432 entries, generated on demand from a periodic 32^3 Fourier grid.Its 50 Latin-hypercube samples are split into 40 training and 10 interior test cases; no prediction extrapolates.Source: Appendix B.5 · p. 37

Appendix B.6. Implementation and hyperparameter selection

The appendix documents interpolation and rank-selection procedures, including nested resampling and an alternative truncation-floor rule, and reports how choices affect compression and error.

Appendix B.6. Implementation and hyperparameter selection

  • Interpolation uses Gaussian weights over the k nearest parameters in standardized space, while Tucker factors use randomized mode-wise SVDs in Cases I–IV and full mode-wise SVDs in Case V.The selected ranks have mean training truncation errors of 0.10, 2.16, 0.88, 0.47, and 1.88 percent for Cases I–V.Source: Appendix B.6 · p. 38
  • Across ten resampled splits, every model choice is re-selected within each training split; the two selection bands choose the same ranks on Cases I and IV and nine Case V splits.On the wave and heat cases, the narrower band raises ranks on every split and shifts the mean gap by at most 0.05 points.Source: Appendix B.6 · p. 39, Table B.2 · p. 39
  • Under the reported band, Case II has a median interpolation-size ratio of 29.0× across splits, with range [12.9, 29.0].Its selected single-subspace rank is 20 or 45 across those splits.Source: Appendix B.6 · p. 39
  • The truncation-floor rule selects the same ranks as leave-one-out on Cases I, III, and V, while leave-one-out gives up compression for accuracy on Cases II and IV.Case IV's truncation-floor gap is +1.00 points, at the prespecified tolerance.Source: Appendix B.6 · pp. 39–40, Table B.3 · p. 40

Appendix B.7. Timing and rank sensitivity

The appendix describes cached timing measurements and rank sensitivity, showing how measured interpolation ratios relate to grid shape and how rank choices alter compression.

Appendix B.7. Timing and rank sensitivity

  • Realized interpolation ratios fall within a small factor of grid-based estimates, with rank selection correcting the ratios; M1 training is faster in Cases I and IV and slower in II, III, and V.The largest break-even count is 1,500 queries on Case III; M1 is ahead from the first query in two cases.Source: Appendix B.7 · p. 41, Table B.4 · p. 41
  • On Cases I, III, and V, increasing M2 rank above its selected value leaves error flat while inflating the interpolation ratio; on the wave case, rank 20 to 60 lowers error by 0.05 points as the ratio triples.Case IV candidates above rank 15 fit the same model because 15 is its matrix-rank ceiling.Source: Appendix B.7 · p. 41

Appendix B.8. Resampling

Across ten paired resampled splits, every benchmark met both stated tolerance criteria for the M1–M2 error gap; targeted core and temporal variants clarify the comparison.

Resampling design

  • The study refit each method on ten seed-redrawn splits, used identical test samples for M1 and M2, and reported mean gaps, paired 95% t intervals, and a margin of 10% of M2 error.Ranks and hyperparameters were fixed; the redraw pools differed by case.Source: Appendix B.8 · pp. 41–42, Table B.5 · p. 42

Resampling results

  • Every benchmark met both paired-interval tolerance criteria; on four benchmarks the interval lay above zero, with M2 more accurate and cluster-bootstrap intervals at least threefold below tolerance.The widest cluster interval was [+0.08, +0.29] points on Case V.Source: Appendix B.8 · p. 42, Table B.5 · p. 42

Core ablations

  • Across the resampled splits, the fixed core was 2.9–5.0 times worse than M1 and the unaligned core 2.2–6.5 times worse; Procrustes alignment made the cores comparable enough to model.The original-split fixed-core errors were 3.66, 67.67, 35.72, 6.84, and 19.12% on Cases I–V; unaligned-core errors were 2.37, 112.72, 53.42, 14.81, and 17.37%.Source: Appendix B.8 · p. 42

Temporal-basis check

  • On the wave case, M2's gap was +0.07 ± 0.03 percentage points over ten splits; giving M2 a global rank-rT temporal basis brought it level with M1 without changing interpolation-size ratios.On heat, the same variant cost M2 0.217 points, exceeding the +0.13-point gap; the original-split errors were 6.63% versus 6.43%, and resampled errors were 8.03 ± 1.64% versus 7.82 ± 1.71%.Source: Appendix B.8 · pp. 42–43

Appendix B.9. Generalization and robustness checks

The checks vary parameter dimension, test-set location and training perturbations, and the width of the reference neighborhood. Accuracy gaps generally remain small, but wider neighborhoods expose a regime where the mode-wise method loses faster.

Parameter dimension

  • As parameter dimension rose from 2 to 6, both methods lost accuracy together, from 6.6 and 6.4% to 19.7 and 18.0%; the gap reached +0.98 points at d = 6, exceeding tolerance by its upper confidence bound.Training truncation error stayed below 0.9%, so the observed loss was prediction error; the gap was +0.23 points or below through d = 5.Source: Appendix B.9 · p. 43

Boundary and noise checks

  • Boundary splits raised errors for every method by factors of 1.5–3.8 for M1, while M1 and M2 stayed within 0.1 points on 2D cases and 0.21 on the 3D case.With 3% relative Gaussian training noise, M1 changed by less than 0.01 points on Cases I–III and V and 0.13 on the external case; M2 degraded on every case, with excess degradation up to +1.52 points. Under spatially correlated perturbation, the separation persisted on four cases, while wave degraded alike.Source: Appendix B.9 · pp. 43–44

Reference-neighborhood width

  • When the transverse design-box width reached 0.30–0.60, M1's gap over M2 widened to 0.62, 5.30, 3.11 and 2.28 points, as training truncation error stayed at 1.9–2.0%.At width 0.60, errors were 16.41% for M1 and 14.13% for M2; the mode-wise construction lost faster when neighboring low-rank transverse subspaces turned through most of a right angle, the regime targeted by several local references.Source: Appendix B.9 · p. 44, Table B.6 · p. 44

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