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Adaptive Multiresolution Diffusion Operators: A Variational Theory on Evolving Multiresolution Spaces

Christian Tantardini, Stig Rune Jensen, Roberto Di Remigio Eikås, Joakim Henrik Beck

arXiv:2610.01809v1math.NAcs.NA

TL;DR

Existing state-dependent diffusion approaches leave the representation on which the operator acts fixed, while adaptive methods generally represent an independently specified operator. This paper derives diffusion from the evolving adaptive multiresolution state and formulates coupled updates variationally. In denoising, it reports retaining structural reconstruction quality close to the best full-space baseline with approximately one tenth of the active representation.

  • Problem

    Prior state-dependent diffusion allows interaction weights to depend on the signal, but the representation remains fixed; adaptive methods likewise typically use an independently specified operator.

    Source: §1 · p. 2

  • Method

    The paper generates an intrinsic diffusion operator from the adaptive state and couples tree and function updates through a time-discrete variational principle.

    Source: §1 · p. 3, §10 · p. 33

  • Results

    AMDI retains structural reconstruction quality close to the best full-space baseline with approximately one tenth of the active representation, but does not outperform full-resolution total-variation denoising in unrestricted accuracy.

    Source: §9.4 · p. 27, §10 · p. 33

  • Takeaways & Limitations

    The denoising results support retaining high structural fidelity with a small fraction of the full active representation.

    Source: §9.4 · p. 27, §10 · p. 33

    AMDI does not universally outperform total-variation denoising in raw accuracy, and its accuracy–compression balance varies with initialization complexity budget.

    Source: §9.4 · p. 27, §9.6 · p. 30

Abstract

from arXiv · show

We develop a variational framework for state-dependent diffusion on adaptive multiresolution representations in which the diffusion operator is generated by the adaptive state itself. The state consists of an admissible multiresolution tree, its active approximation space and basis, and the corresponding coefficient representation. It determines a symmetric nonnegative interaction form and an associated positive semidefinite intrinsic diffusion operator. In contrast to classical adaptive wavelet methods, where a prescribed operator is represented on an evolving approximation space, refinement and coarsening here modify simultaneously the representation, interaction graph, and operator. Because the adaptive hierarchy evolves through discrete topological changes, the coupled dynamics are formulated through a time-discrete variational principle rather than a differential evolution on a fixed space. We establish existence of the discrete updates, a discrete energy inequality, the coefficient-space null mode, and contractivity for frozen adaptive states. For regularized inverse problems, the construction yields Adaptive Multiresolution Diffusion Imaging (AMDI), combining data fidelity, intrinsic diffusion, coefficient sparsity, and tree complexity in a state-dependent energy. Numerical experiments verify the assembled operator identities, examine refinement-commutator decay, and confirm discrete energy dissipation. Adaptive Haar and higher-order multiwavelet calculations demonstrate localization of resolution on heterogeneous data. In denoising, AMDI retains high structural reconstruction quality with less than 10\% of the full active representation, with stable behavior across held-out noise realizations.

1. Introduction.

The paper makes the adaptive multiresolution representation part of the operator state, so refinement and coarsening change both the representation and the diffusion process. It develops this coupled construction through a time-discrete variational evolution and extends it to inverse problems and adaptive multiresolution diffusion imaging.

1. Introduction.

  • Earlier adaptive wavelet methods represent operators specified independently of the adaptive hierarchy, while the proposed construction makes the representation itself part of the operator state.The paper also distinguishes its approach from operator compression, operator-induced bases, learned surrogates, and state-dependent diffusion with a fixed representation.Source: §1 · pp. 1–2
  • The framework generates the diffusion operator from the adaptive state, coupling the active multiresolution set, approximation space, interaction topology, weights, and coefficients.This treats the operator as part of the evolving mathematical object rather than a prescribed operator approximated on an adaptive space.Source: §1 · pp. 2–3
  • The adaptive state determines a symmetric positive semidefinite interaction form and intrinsic diffusion operator through its tree, basis, geometry, and coefficients.As the tree and coefficients evolve, the operator is a state-dependent dynamical quantity.Source: §1 · p. 3
  • Because adaptive hierarchies change through discrete topological events, the coupled evolution updates the tree and represented function by minimizing a common incremental functional.The paper does not assume or establish a continuous-time limit as the time step tends to zero.Source: §1 · p. 3
  • The framework establishes structural properties and variational dynamics, then extends the construction to inverse problems and adaptive multiresolution diffusion imaging.The paper also develops the ambient setting and transfer tools for comparing states associated with different adaptive trees.Source: §1 · pp. 3–4, §2 · p. 4

2. Adaptive Multiresolution Function Spaces.

The paper places finite adaptive approximation spaces inside a common ambient Hilbert space and builds admissible states from finite hierarchical trees and their coefficients. Stable transfer operators then allow represented functions from different adaptive spaces to be compared.

2.1. Ambient Hilbert space and multiresolution hierarchy.

  • The multiresolution dictionary is assumed to be a Riesz basis of the ambient Hilbert space, giving unique coefficient representations; its fixed normalization makes coefficient differences unambiguous.All adaptive approximation spaces are finite-dimensional subspaces of the same ambient space.Source: §2.1 · pp. 4–5

2.2. Admissible adaptive trees.

  • An admissible finite tree contains the coarsest-level indices and preserves the parent–child hierarchy, so every active detail function has its ancestral path.The resulting structure is a finite rooted graph, or a rooted forest when there are multiple coarsest-level functions.Source: §2.2 · p. 5

2.3. Adaptive multiresolution state.

  • An adaptive state is determined by its admissible tree and coefficient vector, with its space, basis, and hierarchical geometry fixed by that tree.The state space is a union of finite-dimensional sets indexed by discrete tree configurations, while represented functions remain in the common ambient Hilbert space.Source: §2.3 · p. 6

2.4. Transfer between adaptive states.

  • Uniformly stable projectors define transfers between adaptive spaces; nested refinement introduces no transfer error, and nonnested transfers remain uniformly stable in the ambient norm.The stability constant is independent of the admissible trees.Source: §2.4 · pp. 6–7

3. Intrinsic Adaptive Multiresolution Diffusion Operators.

The intrinsic operator is built from state-evaluated interactions among active basis functions and acts linearly when the adaptive state is fixed. The resulting operator is symmetric and positive semidefinite, with a uniform coefficient-vector null mode, under the stated structural assumptions.

3.1. Adaptive interaction geometry.

  • The interaction geometry can connect active basis functions through hierarchy, spatial support, neighboring supports, cross-scale relations, or shared parents; refinement decay can suppress newly introduced cross-level couplings.The weights are fixed during the operator's action at a fixed state, though they may depend nonlinearly on coefficients for a fixed tree.Source: §3.1 · pp. 7–8

3.2. Intrinsic Dirichlet form.

  • For a fixed state, the adaptive interaction form is a symmetric bilinear form, and nonnegative weights make its interaction energy nonnegative.Evaluating the interaction energy at the state coefficient vector gives the energy of the adaptive state itself.Source: §3.2 · p. 8

3.3. Intrinsic diffusion operator.

  • The intrinsic diffusion operator is an operator family indexed by adaptive state; at each fixed state it is self-adjoint and positive semidefinite on the active space.Its dependence on coefficients may be nonlinear through the state-dependent interaction weights, despite its linear action at a fixed state.Source: §3.3 · pp. 8–9

3.4. Structural assumptions on the adaptive operator family.

  • Uniform interaction bounds and a uniform interaction-degree bound restrict the operator family to uniformly local, sparse graph constructions.The assumptions exclude unrestricted all-to-all coupling unless it satisfies the same degree bound; refinement consistency is also specified along compatible nested states.Source: §3.4 · pp. 9–10

3.5. Elementary operator properties.

  • For each fixed state, the coefficient-space operator is symmetric and positive semidefinite, and it annihilates the uniform coefficient vector.The function-space operator inherits self-adjointness and positive semidefiniteness; the coefficient- and function-space operators are uniformly bounded under the structural assumptions.Source: §3.5 · pp. 10–11

4. Variational Formulation.

The variational formulation defines an energy on the adaptive state space by combining fidelity, intrinsic diffusion, coefficient sparsity, and tree complexity. Its treatment accommodates state-dependent interactions and the discrete, potentially nonconvex character of changing trees.

4.1. Fidelity functional.

  • For inverse problems, a quadratic data-fidelity functional is continuous, nonnegative, and convex when the forward operator is bounded.The abstract variational formulation requires the fidelity functional to be proper, lower semicontinuous, and bounded below.Source: §4.1 · p. 11
  • The intrinsic diffusion regularizer is defined on the represented function in the active space and is nonnegative.Its coefficient-sensitive weights can make its full variation differ from the diffusion action at a frozen state.Source: §4.1 · p. 11, §4.2 · p. 11

4.2. Adaptive regularization energy.

  • Continuity of the interaction-weight mappings for each fixed admissible tree ensures continuity of the diffusion regularizer in the coefficient vector.The passage states that the coefficient-sensitive kernels satisfy this assumption.Source: §4.2 · pp. 11–12
  • Tree complexity penalizes the number of active indices and can add a cost for finer resolution levels.The stated level-dependent choice introduces the additional refinement-depth cost when γ > 0.Source: §4.2 · p. 12
  • Coefficient sparsity is included through a convex, lower-semicontinuous, coercive penalty with φ(0) = 0 and uniformly bounded coefficient weights.Typical penalty choices are introduced, but their formulas are not included in the supplied passage.Source: §4.2 · p. 12
  • The adaptive energy combines fidelity, intrinsic diffusion, coefficient sparsity, and tree complexity on the adaptive state space.Because the space and representation are determined by the state, no additional compatibility constraint between the function and coefficient vector is required.Source: §4.2 · p. 12
  • When interaction weights depend on coefficients, the frozen-state diffusion action is generally not the full gradient of the state-dependent regularization energy.The discrete variational evolution is therefore formulated directly through the total energy.Source: §4.2 · p. 13

5. Coupled Variational Evolution.

Because refinement and coarsening change the discrete adaptive tree, the coupled evolution is defined by time-discrete variational updates rather than a time derivative on a fixed space. Each update selects a tree and represented function together, while practical implementations may split the minimization or freeze interaction weights.

5.1. Motivation for a time-discrete formulation

  • Tree changes are discrete, so the coupled evolution uses successive variational updates instead of a formal time derivative of the tree variable.Coefficient variables can vary continuously on a fixed tree, whereas refinement and coarsening change the tree configuration.Source: §5.1 · p. 13

5.3. Joint variational time step

  • The update jointly selects an admissible tree and represented function by minimizing a transfer-change penalty together with the complete state-dependent energy.The current tree remains an admissible choice; candidate trees are drawn from a finite local neighborhood under finite-branching assumptions.Source: §5.2 · p. 13, §5.3 · p. 14

5.4. Split implementation

  • A split implementation first updates coefficients on the fixed tree, then selects a tree using a penalty controlled by ζ and a symmetric-difference distance.In the two-dimensional Haar experiments, this distance is computed from refined spatial cells rather than individual wavelet indices.Source: §5.4 · p. 14

5.5. Fixed-tree optimality and frozen-weight linearization

  • On a fixed tree, the exact coefficient minimizer uses a first-order condition, with a subdifferential inclusion when the sparsity term is nonsmooth.If the active basis is H-orthonormal, the frozen-weight linear system has a symmetric positive definite matrix and a unique solution for every positive time step.Source: §5.5 · p. 15
  • Freezing coefficient-dependent interaction weights simplifies the coefficient update, but LVc alone omits weight-variation terms in the full energy gradient.The frozen-weight update evaluates weights at the previous state and is distinguished from the exact fixed-tree minimizer.Source: §5.5 · p. 15

6. Existence, Stability, and Energy Decay.

Under the stated finite-neighborhood and energy assumptions, the discrete updates attain minimizers and dissipate energy. For a frozen adaptive state, diffusion is contractive and, on a connected interaction graph, converges to its coefficient-space null mode.

6.1. Existence of a discrete update

  • At least one minimizing tree–function pair exists when the candidate-tree neighborhood is finite, each fixed-tree functional is proper, and the stated Section 4 assumptions hold.The incremental H-norm term is coercive on each finite-dimensional candidate space, and the finite set of fixed-tree minima attains a smallest value.Source: §6.1 · p. 16

6.2. Discrete energy inequality

  • The exact joint update makes the adaptive energy nonincreasing and controls the accumulated transfer-corrected increments by the initial energy.The unchanged tree and current function provide an admissible competitor with zero incremental distance.Source: §6.2 · p. 16

6.3. Energy stability of the variational split scheme

  • The exact split scheme also satisfies an energy inequality and, when ζ > 0, changes its adaptive tree only finitely many times.Consequently, the tree eventually stabilizes; the stated estimate does not automatically extend to the frozen-weight approximation.Source: §6.3 · p. 17

6.4. Frozen-state intrinsic diffusion

  • With interaction weights fixed, the diffusion evolves through the coefficient representation of the H-space operator, whose coordinate generator is G_Λ^-1 L_V.The representation uses the Gram matrix unless the active basis is H-orthonormal.Source: §6.4 · p. 17

6.5. Conservation of the uniform-mode moment

  • Frozen-state diffusion conserves the H-inner-product moment associated with the synthesized uniform coefficient vector, not necessarily physical mass.Physical-constant preservation requires an additional constant-reproduction property or an interaction-operator null space containing that function's coefficient representation.Source: §6.5 · p. 18

6.6. Contractivity on a frozen adaptive state

  • For every t ≥ 0, the semigroup generated by −L_V is contractive on a frozen adaptive state.This follows from self-adjointness and positive semidefiniteness with respect to the H inner product.Source: §6.6 · p. 18

6.7. Long-time limit on a connected adaptive graph

  • On a connected positive-weight interaction graph, frozen-state diffusion converges exponentially to the H-orthogonal projection onto ker L_V, at a rate set by the generalized spectral gap λ2.The limit is the algebraic null mode of the frozen coefficient graph, not a physical mass-equilibration claim.Source: §6.7 · pp. 18–19

7. Adaptive Multiresolution Diffusion for Inverse Problems.

The inverse-problem formulation minimizes a state-dependent energy over adaptive states, selecting the reconstruction and its representation together. AMDI combines data fidelity, intrinsic diffusion, coefficient regularization, and tree complexity, while the numerical study focuses on denoising.

7.1. General inverse problem

  • The global inverse reconstruction minimizes over adaptive states to determine the reconstructed function, active tree, state-dependent diffusion operator, and active coefficient count together.Under finite branching and the stated assumptions, a minimizer exists when β > 0; this is a sufficient, not necessary, coercivity condition.Source: §7.1 · pp. 19–20

7.2. Adaptive multiresolution diffusion imaging

  • AMDI combines data consistency, intrinsic diffusion, coefficient regularization, and adaptive model complexity in one state-space minimization.The diffusion term is evaluated at the candidate state, retaining coefficient-dependent interaction weights rather than using the frozen-weight approximation.Source: §7.2 · pp. 20–21
  • The analysis treats fixed noise levels and regularization parameters; parameter-choice rules and regularization convergence as δ → 0 require additional assumptions.Deblurring, tomography, and Fourier-sampling models are included in the framework but are not numerically investigated; the experiments focus on denoising.Source: §7.2 · p. 21
  • The denoising implementation uses a data-driven bulk refinement and coarsening rule, while the general local indicator is presented as an alternative and not used for the reported numerical trees.The passage does not assert that this denoising marking rule transfers directly to the general forward operators.Source: §7.2 · p. 21

8. Adaptive Numerical Algorithm.

The algorithm uses split variational updates and candidate-tree selection, with an energy estimate for the exact split scheme. Its marking and implementation details distinguish the denoising prototype from possible general-purpose choices.

8. Adaptive Numerical Algorithm

  • The exact split algorithm inherits the energy estimate when both variational subproblems are solved according to the split formulation.The denoising prototype generates up to three candidates—unchanged, refined, and coarsened trees—and uses 0.10 refinement and coarsening fractions.Source: §8 · pp. 21–22

8.1. Local refinement and coarsening indicators

  • A local indicator can combine coefficient activity, projected data-fidelity gradient, and intrinsic-diffusion variation, but it is not asserted to be a reliable or efficient a posteriori error estimator.Marked indices propose local changes; the variational criterion selects the accepted admissible tree, with the unchanged tree retained as a candidate.Source: §8.1 · p. 22

8.2. Frozen-weight implementation

  • The frozen-weight linearization does not automatically satisfy the exact-step energy estimate, so the reported implementation enforces energy monotonicity by backtracking.The implementation tests at most twenty reductions and distinguishes this computational safeguard from the unconditional estimate for the exact variational split step.Source: §8.2 · p. 23

8.3. Stopping criteria

  • Stopping criteria are application-dependent and may use relative reconstruction change, energy decrease, tree stabilization, an iteration cap, or an available noise-level discrepancy.The experiments test operator identities, refinement compatibility, energy behavior, localization, reconstruction–representation tradeoffs, robustness, and the separate roles of tree adaptation and state-dependent interactions.Source: §8.3 · p. 23, §9 · p. 23

9. Numerical Validation.

The numerical validation checks the assembled intrinsic operator, tests refinement consistency and discrete energy stability, and examines how adaptive resolution localizes on heterogeneous data. Denoising is framed as a comparison between reconstruction quality and active representation size.

9. Numerical Validation.

  • The numerical experiments use orthonormal adaptive Haar so the Gram matrix is identity and coefficient and L2 norms coincide; this choice is not required by the theory.Image intensities are normalized to [0, 1] and the spatial domain is [0, 1]^2 unless stated otherwise.Source: §9 · p. 23

9.1. Intrinsic-operator structure and refinement consistency.

  • −1.16 × 10−16: the smallest computed eigenvalue is at roundoff scale, and the assembled operator satisfies the tested symmetry, positive-semidefiniteness, and coefficient-space kernel identities.The largest computed eigenvalue is 3.0812.Source: §9.1 · p. 24
  • More than two orders of magnitude: the state-dependent refinement defect falls across levels 2–6; the state-independent comparison has coarse-level nonmonotonicity but reaches 1.77 × 10−5.The diagnostic uses γr = 2 and supplies numerical evidence for this parameterized operator family, not proof that every γr satisfies (3.25).Source: §9.1 · p. 25, Figure 1 · p. 25

9.2. Discrete variational stability.

  • 6.61456 × 10−3 to 6.60468 × 10−3: fixed-tree energy fell over eight steps, and every residual was negative, ending at −1.36 × 10−8.This fixed-tree diagnostic directly tests the discrete variational inequality; fully adaptive runs instead enforce energy monotonicity with a frozen-weight backtracking safeguard.Source: §9.2 · pp. 25–26, Figure 2 · p. 26

9.3. Multiresolution localization on a heterogeneous target.

  • Adaptive refinement leaves the constant region coarse, uses moderate Haar resolution for the smooth gradient, and concentrates strongest refinement near the discontinuity and oscillatory texture.The resulting error is localized rather than distributed uniformly over the domain.Source: §9.3 · p. 26, §9.2 · p. 26

9. Numerical Validation.

  • The denoising benchmark compares reconstruction quality against active-representation fraction Crel, with conventional baselines at Crel = 1; wavelet Crel denotes the full representation.On a 64 × 64 synthetic image with Gaussian noise σ = 0.08, each baseline was independently tuned for RMSE and SSIM on the same calibration realization.Source: §9 · p. 24, §9.3 · p. 26

9.4. Denoising accuracy versus active representation size.

AMDI retains SSIM close to the full-resolution TV result at a compressed operating point, while its adaptive representation concentrates refinement in difficult image regions. Held-out and robustness tests show stable behavior within the tested settings, with accuracy and compression still sensitive to initialization and regularization choices.

9.4. Denoising accuracy versus active representation size.

  • AMDI’s adaptive refinement concentrates around discontinuities and oscillations, while the constant region stays coarse and the smooth gradient receives moderate resolution.The illustrated four-region target uses 436 of 16384 full-resolution degrees of freedom (Crel = 0.0266).Source: Figure 3 · p. 27
  • 5.97 × 10^-3 SSIM difference from the TV optimum accompanies an AMDI representation of 9.33% of full size: 382 active degrees of freedom versus 4096.The comparison supports a compression–quality trade-off, not a claim that AMDI universally outperforms TV in raw denoising accuracy.Source: §9.4 · p. 27
  • The AMDI sweep yields reconstruction–representation operating points rather than improving unrestricted reconstruction-error extrema; the reported minimum RMSE is 0.05134 at Crel = 0.2017.The SSIM-optimal calibration point occurs at β = 0, outside Theorem 7.1’s sufficient coercivity hypothesis β > 0.Source: §9.4 · p. 27

9.5. Held-out noise-realization assessment.

  • 1.93 × 10^-2 was the held-out SSIM gap between compressed AMDI and full-resolution TV; AMDI remained stable across unseen noise but below the strongest full-resolution and anisotropic-diffusion baselines.Parameters were frozen after calibration and evaluated on eight previously unseen noise realizations.Source: §9.5 · pp. 28–29, §9.4 · p. 28

9.6. Convergence robustness and standardized initialization.

  • All 16 robustness calculations remained energy-monotone, but across the 192-run calibration sweep the mean full-step acceptance rate was 0.891 and 34 runs required backtracking.This contrast shows that the safeguard was active in the broader sweep, even though every frozen-weight step was accepted in the robustness trajectories.Source: §9.6 · p. 29, Figure 5 · p. 29
  • One-at-a-time perturbations tested h, α, β, τ, and the initial coefficient threshold at half and twice their values; SSIM was most sensitive to β, τ, and the raw threshold.Doubling β, τ, and the threshold reduced SSIM to approximately 0.830, 0.794, and 0.649, respectively; changing h had negligible effect and α produced modest variation.Source: §9.6 · p. 29, §9.5 · p. 29
  • All 15 standardized-initialization runs remained energy-monotone with frozen-weight steps accepted; different initial complexity budgets nevertheless produced measurably different accuracy–compression balances.The prescribed relative-complexity budget determines a coefficient threshold automatically, using only the noisy input and respecting the discrete tree structure.Source: §9.6 · p. 30

9.7. Matched-complexity ablation.

At matched representation size, state-dependent interaction weights improve RMSE, while a higher-order multiwavelet cross-check recovers the qualitative localization pattern seen with Haar. The precision audit shows self-consistency against a reference projection, not convergence against an exact solution.

9.7. Matched-complexity ablation.

  • Removing intrinsic diffusion after tree selection slightly lowers RMSE and raises SSIM in all five realizations, but does not test diffusion’s role in selecting the tree.For this identity-denoising problem, the fixed-tree comparison provides no evidence of an additional denoising advantage from the diffusion contribution itself.Source: §9.7 · pp. 30–31
  • Replacing state-dependent weights with state-independent weights raises mean RMSE from 0.05362 to 0.06046, a deterioration of approximately 12.8%.The five-realization results do not show improvement on every metric: SSIM is slightly higher for the state-independent variant in all five paired realizations.Source: §9.7 · p. 30, Table 3 · p. 31

9.8. Higher-order multiwavelet cross-check with VAMPyR.

  • Haar and order-five VAMPyR both allocate more resolution to sharp and oscillatory regions than smooth ones, supporting qualitative localization rather than quantitative agreement.Different approximation orders and the VAMPyR depth cap preclude direct comparison of absolute refinement levels; the cross-check does not independently validate the AMDI diffusion operator.Source: §9.7 · p. 31, §9.8 · p. 32, Figure 7 · p. 32
  • As requested precision tightens from 10^-2 to 3 × 10^-5, end nodes increase from 28 to 328 and distance to the reference falls by approximately 451-fold.The audit measures self-consistency against a 10^-5 reference projection, not convergence against an exact analytic solution.Source: §9.8 · p. 32
  • The adaptive tree, approximation space, and coefficients generate a symmetric positive semidefinite intrinsic diffusion operator with a coefficient-space null mode.The construction derives the operator from the evolving adaptive state.Source: §10 · p. 32

10. Conclusions.

The conclusions present a time-discrete variational framework coupling an adaptive multiresolution state to its intrinsic diffusion operator. Numerical results support its structural properties and show a reconstruction–representation balance, while continuous-time limits and optimal-complexity results remain open.

10. Conclusions.

  • The precision audit tracks adaptive end-node count and L2 distance to the 10^-5 reference projection as requested precision changes.The conclusion describes monotone self-consistency as precision tightens.Source: Figure 8 · p. 33, §10 · p. 33
  • A Gram-matrix formulation preserves the function-space variational structure for general Riesz bases, without assuming an orthonormal basis.The coupled adaptive tree and represented function are updated by a time-discrete variational principle with existence of each update and a discrete energy inequality.Source: §10 · p. 33
  • Numerical results verify operator identities and energy dissipation, while denoising retains high structural fidelity with a small fraction of the full active representation.Held-out tests preserve the reconstruction–representation balance, though the accuracy gap to the strongest baselines is larger than on the calibration realization.Source: §10 · p. 33
  • The work establishes a time-discrete theory but not a continuous-time limit, asymptotic approximation rates, or optimal-complexity results.These are identified as directions for future analysis, alongside extensions to more general inverse problems and higher-order multiwavelet backends.Source: §10 · p. 34
  • The numerical data and reproduction workflow are openly available, and the study uses no proprietary or restricted datasets.The repository includes scripts, parameter settings, software environment, validation tests, and reproduction instructions.Source: Data Availability Statement · p. 34
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