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Spectral Graph Neural Networks with Hermite Polynomials: A Comprehensive Study

Shuang Wu

arXiv:2609.28979v1cs.LG

TL;DR

Graph-filter learning leaves open when a simple fixed polynomial family is useful. The paper proposes HermNet, combining a nodewise predictor with normalized Hermite propagation, and analyzes its coordinates and optional functional penalties. Controlled short-budget comparisons find a Hermite advantage in a bounded regime, while longer training and real-data results delimit it.

  • Problem

    Polynomial spectral graph neural networks learn graph-frequency responses through sparse propagation, but when a simple fixed polynomial family is useful for learning a graph filter remains an open design question.

    Source: §1 · p. 1

  • Method

    HermNet combines a nodewise predictor with normalized probabilists’ Hermite propagation, while the paper analyzes its scope and separates optional calibration and functional regularization.

    Source: §1 · pp. 1–2

  • Results

    At five updates, plain Hermite reduces MSE 9.4% versus Bernstein, and the controlled advantage also appears with a jointly learned nonlinear predictor.

    Source: §1 · p. 2, §5.2 · p. 9

  • Takeaways & Limitations

    Hermite coordinates can help under limited training budgets despite spanning the same degree-bounded filter space as other complete polynomial bases.

    Source: §1 · p. 2

    The plain Hermite lead disappears with longer training, and matched real-data comparisons report accuracy deficits.

    Source: §5.2 · p. 10, §5.5 · p. 12

Abstract

from arXiv · show

We study spectral graph neural networks built from Hermite polynomials and propose HermNet, a simple model that combines a nodewise predictor with normalized Hermite propagation. Its sparse recurrence requires neither eigendecomposition nor a learned basis. We distinguish the basic model from optional coordinate calibration, response normalization and Gaussian derivative regularization. Hermite and other complete polynomial bases span the same degree-bounded filter space, but their coordinates can produce different optimization behavior under limited training budgets. We analyze this behavior through spectral signal energy, label sampling, changes in learned features and the bias--variance trade-off of regularization. Controlled synthetic experiments identify a regime in which plain HermNet outperforms matched polynomial-basis alternatives, including with a jointly trained nonlinear predictor. Curvature regularization further improves HermNet when the same functional penalty is available to every comparator. Fixed-predictor controls support the advantage under short training budgets, but longer training removes the plain-model lead. Matched real-data comparisons show accuracy deficits, and architectural and numerical studies identify further limits. Together, the analysis and experiments clarify when Hermite propagation is useful and how calibration and regularization affect its performance.

1 Introduction

The paper asks when a fixed polynomial family is useful for learning graph filters and proposes HermNet, separating its basic propagation from optional calibration and regularization. Its analysis and experiments identify a limited-budget regime where Hermite coordinates help, while longer training and real-data comparisons bound that advantage.

Motivation

  • The motivating question is when a simple fixed polynomial family is useful for learning a graph filter.Earlier Hermite-filtering applications are reviewed separately; the favorable experiment assesses update-budget advantage, native-model performance and fitting cost separately.Source: §1 · pp. 1–2

Contributions

  • HermNet combines a nodewise predictor with normalized Hermite propagation; calibration, response normalization and filter penalties are optional extensions.The plain model uses prescribed coordinates and unrestricted coefficients, without reference calibration or a filter penalty.Source: §1 · pp. 1–2

Properties and scope

  • Hermite coordinates can change finite-step optimization although complete degree-bounded polynomial bases span the same filter space.The analysis relates their usefulness to spectral signal energy, label coverage and changes in learned features.Source: §1 · pp. 1–2

Contributions

  • After five updates, HermNet has lower prediction error than five matched complete polynomial bases, including with a jointly learned nonlinear predictor.A common curvature penalty also improves HermNet; longer-budget, native-model and numerical controls delimit the advantage.Source: §1 · p. 2

2 Related Work

Related work places HermNet among polynomial filters, adaptive bases and spectral extensions. The paper distinguishes its focus on fixed Hermite coordinates and controlled comparisons from those neighboring approaches.

Polynomial filters and adaptive bases

  • Polynomial-filter models use varied parameterizations, from localized Chebyshev filters and signed propagation weights to adaptive bases and graph-dependent propagation.Examples include ChebNet (Defferrard et al., 2016), GPR-GNN (Chien et al., 2021), FavardGNN and OptBasisGNN (Guo and Wei, 2023a), and AdaptKry and UniFilter (Huang et al., 2024a,b).Source: §2 · p. 2

Hermite filtering and evaluation

  • HermNet uses a fixed Hermite recurrence within the same complete degree-bounded polynomial space, focusing on learning procedure and when its coordinates are useful.Hermite filtering also appears in cortical learning, LSAP diffusion and PolyCF recommendation (Huang et al., 2020; Sim et al., 2024; Qin et al., 2024).Source: §2 · pp. 2–3

Regularization and spectral extensions

  • Related regularization work includes graph-dependent shape priors and filter smoothness, while this paper uses Gaussian derivative penalties as function-space preferences transformable across bases.The transformation permits controlled comparisons without changing the underlying preference.Source: §2 · p. 2

Evaluation context

  • The cited literature also motivates evaluating coordinate choice, regularization, architecture and fitting cost separately; the reviewed literature is broader than the experimental baseline roster.Related benchmark work raises questions about spectral-GNN explanations, efficiency and heterophily evaluation (Liao et al., 2025; Platonov et al., 2023).Source: §2 · p. 3

3 Simple HermNet

Simple HermNet applies polynomial graph filters to nodewise predictor outputs and uses normalized probabilists’ Hermite polynomials for propagation. Its recurrence spans the complete degree-bounded polynomial filter space with sparse graph operations.

3.1 Learning problem and spectral signal energy

  • The learning setup applies polynomial filters to nodewise predictor outputs, with graph-frequency signal energy defined from node signals such as predictor outputs, inputs or frozen references.The spectral measure describes energy over graph frequencies, not the marginal distribution of node features.Source: §3.1 · p. 3

3.2 Hermite propagation

  • HermNet uses normalized probabilists’ Hermite polynomials on the scaled Laplacian and unrestricted channelwise coefficients, evaluated by a sparse three-term recurrence.The plain model uses prescribed coordinates and no filter penalty; its response can be low-pass, high-pass or mixed.Source: §3.2 · p. 4

3.3 Function class, locality and cost

  • At degree K, Hermite responses form a complete polynomial basis and the propagation is K-hop local, with cost O(KmC) and O(nC) additional inference memory.The propagation cost excludes the nodewise predictor and optional calibration, and its asymptotic order does not establish a runtime advantage.Source: §3.3 · p. 4

4 Properties, Regularization and Calibration

The analysis separates polynomial coordinates from filter function class and penalty, then studies how optimization, label sampling, representation limits and calibration affect HermNet. It gives conditions and boundaries rather than a distribution-free performance guarantee.

4.1 Gaussian alignment and finite-step optimization

  • A complete basis transformation preserves predictions but induces a positive-definite preconditioner for gradient descent, so equivalent exact solutions can yield different finite-step predictions.The identity does not cover general adaptive optimizers or guarantee neural-model performance.Source: §4.1 · p. 5

4.2 Labeled nodes and changes in the predictor

  • Label-sampling conditioning depends on row concentration and feature changes, while the probability bound requires a label-independent reference or a new fitting mask.A predictor fitted with the same labels is not independent of the mask; the result does not guarantee unrestricted end-to-end performance.Source: §4.2 · pp. 5–6

4.3 Limits imposed by degree, scale and representation

  • A graph spectrum bounded by [−r, r] constrains how closely its Hermite Gram can match Gaussian orthogonality at degree K; small scale can increase polynomial magnitudes and sensitivity.When r ≤1 and ϵ ≤0.2, the stated condition excludes K ≥2.Source: §4.3 · p. 6
  • For a fixed input, an invertible complete-basis change preserves the response space and representation floor, while increasing degree cannot increase that floor.Conditioning alone cannot remove fixed-input representation error; changing the learned predictor may change the space.Source: §4.3 · p. 6

4.4 Gaussian derivative and curvature penalties

  • Gaussian derivative and curvature penalties express function preferences, with energy, derivative and curvature corresponding to q = 0, 1 and 2.Enhanced comparisons transform the same functional penalty into each comparator basis, including the shared curvature prior in the nonlinear experiment.Source: §4.4 · p. 7

4.5 Regularization risk and sensitivity

  • Under the stated fixed-design assumptions, small high-order coefficients and noisy labels can favor derivative shrinkage, though a sharp response can incur too much bias.The fixed-filter perturbation bound requires the same coefficients and coordinates; retraining or recalibration requires an additional term.Source: §4.5 · pp. 7–8

4.6 Optional coordinate calibration

  • Optional coordinate calibration selects coordinates by reference-Gram discrepancy, while response normalization rescales columns and empirical QR orthogonalizes independent responses.These procedures can benefit any basis, add computation and are absent from the primary plain-HermNet comparison.Source: §4.6 · p. 8

5 Experiments

The experiments test when HermNet’s short-budget prediction advantage holds, how it relates to spectral signal energy and optimization, and where it is limited. The advantage appears in controlled synthetic settings, but longer training and real-data comparisons constrain its scope.

5.1 Research questions and evaluation design

  • The proposed comparisons hold predictor, degree, coordinate menu, optimizer rule, function-space initialization and validation criterion common across five complete polynomial-basis controls.Synthetic conditions are paired across methods; positive one-sided paired-t lower bounds resolve mean improvements under the stated familywise adjustment.Source: §5.1 · pp. 8–9

5.2 Downstream prediction with a jointly learned predictor

  • Plain Hermite reaches MSE 0.34671 versus Bernstein’s 0.38266 (9.4% lower); enhanced Hermite reaches 0.33048 versus the best enhanced comparator’s 0.37405 (11.6% lower).These results come from an independent five-update evaluation with 80 paired draws and a jointly learned nonlinear predictor; all eleven adjusted contrasts have positive lower bounds.Source: §5.2 · p. 9, Figure 2 · p. 10

5.3 Fixed-predictor controls and the update budget

  • With the broad scale menu, fixed-predictor Hermite reaches MSE 0.00511 versus Bernstein’s 0.02165 (76.4% lower), but its resolved lead disappears with more updates.Response normalization leaves the selected Hermite mean unchanged; with normalization available, Bernstein has a lower mean from 20 updates onward. In the initial two-scale study, empirical QR also slightly outperforms Hermite: 0.00657 versus 0.00670.Source: §5.3 · p. 10, Figure 3 · p. 11

5.4 Gaussian connection and diagnostic controls

  • Gaussian-weighted inner products can explain favorable Hermite conditioning, but sampling, predictor drift, degree and input representation restrict the mechanism.A dominant-coordinate intervention changes spectral scale and task difficulty as well as shape, so it does not isolate a causal effect of Gaussianity.Source: §5.4 · p. 11

5.5 Real-data performance and native architectures

  • On citation benchmarks, plain Hermite trails the best means by 3.37, 2.07 and 2.63 percentage points; three seeds on one split do not establish noninferiority.The table reports test accuracy over three final seeds on one fixed split; the separate calibrated native-model comparison uses a different protocol and an enhanced model.Source: Table 1 · p. 12, §5.5 · p. 12
  • Transfer to other architectures is mixed: enhanced Hermite leads one common-basis ring-classification comparison, while native JacobiConv reaches 80.59% versus Hermite’s 73.90%.A subsequent shared-dropout comparison removes the Hermite lead, and expanded real-data comparisons include larger Hermite deficits.Source: §5.5 · p. 12

5.6 Scope of the empirical findings

  • The independent nonlinear experiment resolves both the five-update plain-Hermite advantage and its incremental curvature benefit, while fixed-predictor derivative and curvature studies do not resolve that incremental effect.A bounded Gaussian prior meets none of eight prespecified practical-improvement criteria.Source: §5.6 · p. 12
  • The favorable learned-predictor study lacks a full native-model panel and wall-clock comparison; measured CPU QR/Hermite total-time ratios are 0.9311–0.9964.Fewer updates therefore do not establish lower total fitting cost.Source: §5.6 · p. 12

A Implementation Details

The appendix separates optional calibration, response transforms, functional priors and architectural controls from the plain model. It also specifies training safeguards and computational costs.

A.1 Coordinate calibration and response transforms

  • Coordinate calibration selects a pair from declared finite grids using reference-Gram discrepancy, while response normalization rescales response columns and QR orthogonalizes them when independent.These options can benefit any basis, add computation, and leave the complete polynomial space unchanged.Source: §A.1 · p. 15

A.2 Functional priors and architectural controls

  • Functional priors are compared across complete bases, while the odd-hop control changes the function class and channel-filter restrictions couple orders and input channels.A bounded Gaussian measure changes the functional prior and generally makes its derivative matrix nondiagonal; unrestricted channel matrices enlarge the architecture for every complete basis.Source: §A.2 · p. 15

A.3 Training and computational cost

  • Training restores the validation-selected configuration and checkpoint before test evaluation, with no global convergence assumed.For m = nnz(L) + n, propagation costs O(KmC); calibration adds candidate- and reference-dependent Gram costs, and reported timings include applicable costs.Source: §A.3 · p. 16

B Proofs

The proofs establish the polynomial and recurrence properties behind HermNet and derive Gaussian derivative penalties. They also delimit the coordinate-update identity’s scope.

B Proofs

  • The appendix proves the paper’s statements from Sections 3–4 and the product-graph identity used in Section 5.4.Source: B Proofs · p. 16

B.1 Polynomial completeness and coordinate updates

  • The Hermite responses span the degree-K polynomial space, and the sparse three-term recurrence gives K-hop locality with one sparse multiplication per step.The coordinate-gradient identity follows by the chain rule for a fixed invertible transform; its proof does not cover adaptive optimizers or changing transforms.Source: §B.1 · p. 16

B.2 Proof of the Gaussian derivative identity

  • Gaussian orthonormality eliminates cross terms in the derivative identity, yielding the stated energy, derivative and curvature penalties.The same proof establishes invariance under the consistently transformed complete objective.Source: §B.2 · pp. 16–17

B.3 Proof of the label-sampling bound

These appendices prove bounds and decompositions used in the analysis, then detail the experimental construction and equivalent-coordinate optimization checks.

B.3 Proof of the label-sampling bound

  • The label-sampling proof centers sampled Grams, bounds their summands and variance, applies matrix concentration uniformly over candidates, and controls changed designs.The resulting quadratic-loss gradient descent contracts when the stated lower eigenvalue bound is positive.Source: §B.3 · p. 17

B.4 Proof of the risk decomposition; B.5 Proof of the representation decomposition

  • The risk proof derives prediction risk from the normal equations and noise covariance; the representation proof shows complete coordinate changes preserve the design’s column space and that increasing degree cannot increase best-fit error.Without positive definiteness, equal objective values alone do not specify a common fitted signal away from labeled nodes.Source: §B.4 · p. 18, §B.5 · p. 18

B.6 Proof of the perturbation bound

  • The perturbation proof bounds signal differences by expressing operators in mixed eigenbases, summing entrywise bounds, and adding feature perturbation.Source: §B.6 · pp. 18–19

B.7 Proof of the degree–scale constraint

  • The condition is necessary, but it does not guarantee the existence of an aligned measure on a given graph.Source: §B.7 · p. 19

B.8 Proof of the product-graph moments

  • The product-graph identity concerns the unweighted spectral measure; realized signals have different energy weights, so their moments are measured separately.Source: §B.8 · p. 19

C Experimental Details; C.1 Graph, signal, and target construction

  • Experimental details distinguish fixed-design diagnostics, learned predictors and native architectures; real-data variability comes from three optimizer seeds on one fixed split per dataset.Those standard deviations describe optimization variability on the selected graphs, not uncertainty across graph populations.Source: C Experimental Details · p. 20

C.1 Graph, signal, and target construction

  • The signal construction prescribes energy on graph eigenvalues, while fixed-feature studies reserve validation and test nodes and use noisy labels for training and selection.The prescribed energy densities do not describe the distribution of graph eigenvalues.Source: §C.1 · p. 20

C.2 Calibration and equivalent-coordinate optimization

  • The coordinate study checks numerical rank and exact cross-coordinate prediction agreement across 3,072 trajectories, then evaluates five prespecified criteria in both graph families.No Gaussian width/degree pair meets all criteria in both families.Source: §C.2 · p. 21

C.3 Neural classification and native architectures

The section details experimental protocols for learned predictors, derivative-prior selection, expanded model comparisons, and numerical diagnostics. It also identifies development-data reuse and other boundaries on interpreting the evaluations.

C.3 Neural classification and native architectures

  • Classification experiments use paired graph conditions and shared predictor initialization for frozen and jointly trained modes; citation benchmarks reuse development graphs and CPU timings are not GPU-comparable.Native architectures and safeguards are retained, so model capacities and computation can differ.Source: §C.3 · pp. 21–22

C.4 Learned-feature regression

  • Learned-feature regression compares four matched bases on shared predictors and objectives, then evaluates on new graphs after validation-based configuration selection.The stopping rule bounds training duration without guaranteeing convergence.Source: §C.4 · p. 22

C.5 Derivative-prior selection and error decomposition

  • Derivative-prior selection tunes strength and degree on noisy validation data before new-graph evaluation, with practical gains defined against four comparators in each family.The practical target requires at least 0.005 absolute MSE and 5% relative MSE gain; these are descriptive criteria, not significance tests.Source: §C.5 · p. 23
  • An oracle diagnostic projects known noiseless targets onto fixed-degree designs after evaluation, separating representation error from within-space effects without changing fitted models.The within-space term includes bias, noise sensitivity, sampling, and selection.Source: §C.5 · p. 23

C.6 Expanded common-model and native comparisons

  • Expanded comparisons preserve native architectures and method-specific capacities, while descriptive standard deviations and development evaluations do not provide independent confirmatory tests.The common-architecture roster includes six polynomial bases; the native roster includes ten graph-learning methods plus MLP.Source: §C.6 · p. 23
  • CiteSeer uses a previously developed public partition, and early pruning can discard configurations that improve with longer training.Equal update allocations also leave computation cost and search coverage unequal across native architectures.Source: §C.6 · p. 24

C.7 Channel-filter, rounding and sparse-projection protocols

  • Channel-filter, coefficient-rounding, sparse-dictionary, and analytic sign-agreement diagnostics use fixed targets or filters; timing results do not establish generic-graph or GPU preprocessing costs.Orthogonality alone is insufficient to predict rounding error, which also depends on coefficient range and error direction.Source: §C.7 · pp. 24–25

D Additional Results and Ablations

  • Bold values mark best displayed means, while † identifies principal HermNet evidence; neither marking implies statistical significance.Lower MSE, distortion, and finite optimization counts are better; higher accuracy is better.Source: D Additional Results and Ablations · p. 25

D.1.1 Learned predictors: independent evaluation and earlier study

An independent five-update evaluation reports positive adjusted bounds across planned comparisons for a jointly trained nonlinear predictor. Earlier learned-predictor and fixed-predictor studies examine selected tasks and update budgets, with the plain Hermite advantage dependent on the setting.

D.1.1 Learned predictors: independent evaluation and earlier study

  • All 11 adjusted paired-t lower bounds were positive in the independent five-update nonlinear-predictor follow-up across 80 paired draws.Nonzero curvature was selected in 38 of 80 Hermite fits, and every selected Hermite predictor changed from initialization.Source: §D.1.1 · p. 25, Table 3 · p. 26, Table 2 · p. 26
  • At 20 updates, the earlier selected task showed gains of 12.0% over the best plain rival, 16.7% over the best enhanced rival, and 11.5% over plain Hermite.Neither arm resolved superiority over every rival at 20 updates, although the incremental Hermite benefit was resolved.Source: §D.1.1 · p. 25, Figure 4 · p. 26

D.1.2 Plain propagation and update budgets

  • The initial fixed-predictor screen found no clear Hermite advantage at 100 updates, then selected a degree-four sine task for evaluation on 40 new draws per weight condition.The primary follow-up budgets were five and 20 updates.Source: §D.1.2 · p. 27
  • The broader fixed-predictor follow-up compares bases under shared scales and step fractions, with a second menu adding analytical unit-RMS columns.Table 4 reports five-update means; the supplied passages do not provide the cell values.Source: §D.1.2 · p. 27, Table 4 · p. 27, Figure 5 · p. 28

D.1.3 Derivative, curvature and odd-hop controls

Derivative and curvature penalties improve comparisons against enhanced rivals in selected short-budget settings, but their incremental gains over plain Hermite remain unresolved. Odd-hop restrictions did not meet the prespecified selection thresholds.

D.1.3 Derivative, curvature and odd-hop controls

  • At five updates, derivative-regularized Hermite led enhanced rivals, but its gain over plain Hermite was unresolved; Bernstein became better with longer training.The adjusted lower bound for the incremental gain was −0.00026, and nonzero strength was selected on 14 of 40 draws.Source: §D.1.3 · p. 28, Figure 6 · p. 29
  • At five updates, curvature resolved comparisons against five rivals but not improvement over plain Hermite in the selected 16-label, noise-SD-1.2 setting.The adjusted lower bound against plain Hermite was −0.00319.Source: §D.1.3 · p. 28, Table 6 · p. 29
  • No odd-hop candidate met both prespecified 10% gain thresholds, so independent confirmation and the planned cosine-mismatch evaluation were not conducted.The restriction uses the same odd subspace across bases, with normalization recomputed and no derivative penalty.Source: §D.1.3 · p. 28

D.2.1 Plain citation benchmarks

Across matched and native controls, Hermite’s advantage depends on the task, coordinates, and training budget; the conditioning and prior studies also identify limits to that advantage.

D.2.1 Plain citation benchmarks

  • The plain citation comparison matches predictor, dropout, degree, and optimizer settings across six bases, with one fixed split per graph and separate CE-first sensitivity results.The reference-moment arm differs from degree-specific calibration, and its CiteSeer failure is retained.Source: §D.2.1 · p. 29, Table 7 · p. 30, Table 8 · p. 30

D.2.2 Ring classification and architectural controls

  • Enhanced Hermite leads the ring/uniform/wave common-basis comparison, but native JacobiConv scores higher; common dropout removes the earlier Hermite lead.Plain stochastic-block/uniform/heat ties Jacobi at 89.72%.Source: §D.2.2 · p. 30, Figure 7 · p. 31, Figure 8 · p. 32

D.2.3 Calibrated citation and synthetic neural comparisons

  • JacobiConv has the largest mean on all three calibrated citation datasets; development ablations show no consistent Hermite gain from reference changes or longer training.A conditionally selected positive derivative strength increased MSE in all eight learned-regression conditions.Source: Table 10 · p. 32, §D.2.3 · p. 32

D.2.4 Expanded product, citation and WebKB comparisons

  • Across expanded comparisons, results include unfavorable Hermite outcomes, while checkpoint-selection means are descriptive rather than population rankings.The reported product-graph procedures use accuracy-first or cross-entropy-first checkpoint selection on the same selected configuration.Source: §D.2.4 · p. 33, Table 13 · p. 34, Table 14 · p. 34

D.2.5 Channel-specific filters

  • Replacing factorized channel filters with unrestricted full filters improves some bases, but shared configurations and selection criteria complicate ranking comparisons.Some references share a configuration and trajectory and are not independent fits.Source: Table 17 · p. 36, Table 18 · p. 36, §D.2.5 · p. 36

D.3.1 Coordinate conditioning

  • Narrow energy and degree two favor Hermite conditioning, but no width/degree pair meets every stability, optimization, and adequacy criterion in both graph families.Counts above 2,000 are censored; whitening remains fixed under mismatch.Source: §D.3.1 · p. 37, Figure 9 · p. 37, Table 19 · p. 38

D.3.2 Prior support and representation floors

  • The oracle decomposition attributes most primary bounded-prior loss to representation floors, which rise under input mismatch.The representation floor accounts for 80.86%/85.32% of primary bounded-prior loss on stochastic-block/geometric graphs, rising to 84.89%/88.55% under mismatch.Source: §D.3.2 · p. 39, Figure 10 · p. 39
  • Hermite’s primary bounded-prior comparisons never meet both gain thresholds, and only three of 160 remaining contrasts exceed the harm threshold.All three harmful contrasts are the stochastic-block/G.4/cutoff/30-label/ρ = 0 condition.Source: Table 21 · p. 39, §D.3.2 · p. 39

D.4 Precision, sparsity and computational cost

Numerical and computational controls show conditional differences among constrained polynomial representations, without establishing a Hermite total-time advantage.

D.4 Precision, sparsity and computational cost

  • The CPU measurements do not establish a Hermite total-time advantage because they use prepared inputs and a fixed update count, not a common accuracy target.Repetition variability and the small setup fraction further limit inferences about calibration cost.Source: §D.4 · p. 40, Figure 11 · p. 41
  • Degree-four coarse-rounding gains depend on the target and coordinate choice, while higher degree and dominant weights reverse aggregate orderings.Fixed Legendre slightly improves wave distortion (0.2905% versus Hermite 0.2930%); moment Legendre improves smooth distortion (0.3740% versus 0.3782%).Source: Table 22 · p. 40, §D.4 · p. 40, Figure 11 · p. 41
  • Sparse Hermite often improves on sparse QR, but standard-coordinate controls perform better throughout tested settings except numerical ties.These constrained dictionaries differ despite spanning the same unrestricted polynomial space.Source: §D.4 · p. 40, Figure 11 · p. 41
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