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Learned Adaptive Multiresolution Diffusion Imaging

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

arXiv:2610.07884v1math.NAcs.LGcs.NAeess.IV

TL;DR

Adaptive multiresolution methods need to allocate fine-scale degrees of freedom without relying only on fixed local tree criteria. Learned AMDI replaces AMDI’s post-propagation selector with a trajectory-trained local policy while retaining its fixed-tree propagator and hierarchy constraints. It reduces reference error on the reported holdouts and improves metrics over deterministic AMDI at finer resolutions, although threshold controls remain competitive.

  • Problem

    Adaptive multiresolution methods concentrate fine-scale degrees of freedom where needed, but tree updates are usually governed by fixed local criteria.

    Source: §1 · p. 2

  • Method

    Learned AMDI trains a shared local policy with proximal policy optimization to select hierarchy-admissible post-propagation tree updates while preserving AMDI’s fixed-tree propagation.

    Source: §1 · p. 2, §3.3 · p. 9

  • Results

    Across nine holdouts, mean terminal reference error fell from 0.17496 to 0.13657; at 64×64 and 128×128, reference error, RMSE, and SSIM improved over deterministic AMDI.

    Source: §1 · p. 3, §6 · p. 29

  • Takeaways & Limitations

    The threshold rule remains competitive at comparable occupancy, and most benefit on the static benchmark comes from the initial allocation rather than repeated decisions.

    Source: §6 · pp. 28–29

    The learned policy’s local factorization need not reproduce the deterministic selector’s global candidate ranking, and its tree update does not inherit the deterministic energy inequality.

    Source: §3.6 · p. 11

Abstract

from arXiv · show

Adaptive multiresolution methods reduce representation cost by concentrating fine-scale degrees of freedom where needed, but their tree updates are usually governed by fixed local criteria. We introduce Learned Adaptive Multiresolution Diffusion Imaging (Learned AMDI), which preserves the AMDI fixed-tree propagator and hierarchy constraints while replacing the post-propagation selector with a shared local policy trained by proximal policy optimization. Regression tests reproduce deterministic AMDI trajectories to machine precision when identical trees are used. In the Haar implementation studied here, the deterministic one-step selector accepts no refinements in 54 decisions. Across nine held-out cases, Learned AMDI executes 393 refinements and reduces the mean terminal reference discrepancy from $0.17496$ to $0.13657$, while occupancy rises from $0.13737$ to $0.26660$. Step-resolved diagnostics reveal occasional small adaptation-energy increases; fixed-tree energy stability therefore does not guarantee monotonicity of the learned outer iteration. At comparable occupancy, a validation-tuned observed-detail threshold reaches a discrepancy of $0.13792$ with slightly better RMSE and SSIM, placing both methods on essentially the same accuracy--occupancy tradeoff. A decision-1-only control reaches $0.13742$, indicating that most of the improvement on this static benchmark arises from the initial allocation. The shared actor transfers without retraining to $64\times64$ and $128\times128$ images, improving reference discrepancy, RMSE, and SSIM relative to deterministic AMDI, while the frozen threshold rule remains competitive. Learned AMDI thus provides a hierarchy-constrained, resolution-transferable mechanism for adaptive allocation and clarifies the contribution of sequential decisions.

1. Introduction.

Learned AMDI replaces deterministic tree selection with a shared local policy trained on complete trajectories, while retaining AMDI propagation and hierarchy constraints. On the reported holdouts, it refined where the deterministic selector did not and reduced reference error, though threshold and first-decision controls qualify the benefit.

1. Introduction.

  • Learned AMDI preserves AMDI’s coefficient update and hierarchy constraints while replacing only the post-propagation tree-selection rule with a shared local policy trained by proximal policy optimization.The trajectory reward combines discrepancy from a uniformly resolved AMDI reference, representation occupancy, and hierarchy switching.Source: §1 · pp. 1–3
  • Refinements can increase immediate energy and representation cost yet make coefficients available for later propagation, motivating a policy trained on complete trajectories.The study tests whether repeated decisions outperform a single early adaptation.Source: §1 · p. 2
  • 393 refinements reduced mean terminal reference error from 0.17496 to 0.13657, while the deterministic selector accepted no refinements in 54 holdout decisions.The comparison is for the implemented Haar configuration across nine holdout cases.Source: §1 · p. 3
  • At nearby but not identical occupancy, the validation-tuned threshold achieved essentially the same accuracy–occupancy tradeoff, and the decision-1-only control showed that most static-benchmark benefit came from the initial allocation.The paper frames its contribution as objective-driven allocation under AMDI hierarchy constraints, with step-resolved analysis of actions, occupancy, reference accuracy, and energy change.Source: §1 · p. 3

2. Deterministic Adaptive Multiresolution Diffusion Imaging.

This section establishes the adaptive representation and fixed-tree diffusion update used by AMDI, then describes deterministic tree selection and the learned replacement. Both variants propagate on a fixed hierarchy before adapting, but the learned update does not inherit the deterministic energy argument.

2.1. Diffusion evolution.

  • The implemented energy combines data fidelity, graph interaction, coefficient sparsity, and tree complexity; fixed-tree updates use frozen interactions and a backtracking check against the original energy.The tree-complexity term is constant during fixed-tree minimization.Source: §2.1 · p. 4

2.2. Multiresolution representation.

  • Representation occupancy is the fraction of available finest-level degrees of freedom that are active, so it measures representation complexity rather than wall-clock time or memory use.Full occupancy corresponds to the uniformly resolved representation.Source: §2.2 · p. 4

2. Deterministic Adaptive Multiresolution Diffusion Imaging.

  • AMDI represents the image with active multiresolution degrees of freedom on a hierarchy and advances the field before adapting that representation.The full construction is inherited from deterministic AMDI [18].Source: §2.1 · p. 3, §2.2 · p. 4

2.3. Deterministic adaptive evolution.

  • The deterministic selector compares retention, refinement, and coarsening candidates, but zero-initialized new Haar details create a refinement barrier in the reported implementation.Every strict refinement candidate has a higher selection objective than the unchanged tree under the stated transfer and scoring construction.Source: §2.3 · p. 5

2.4. From deterministic to learned adaptive evolution.

  • Learned AMDI uses local policy proposals after the same fixed-tree propagation, then applies them through a hierarchy-constrained update that does not guarantee the deterministic energy decay.Once selected trees differ, later interaction graphs and propagated fields can also differ.Source: §2.4 · p. 6

3. Adaptive Multiresolution Diffusion as a Sequential Decision Prob-

The section formulates adaptive tree selection as a finite-horizon decision process while retaining AMDI’s propagation procedure. It specifies the decision state, admissible actions, trajectory objective, and conditions for matching deterministic AMDI.

3. Adaptive Multiresolution Diffusion as a Sequential Decision Prob-

  • The finite-horizon MDP models tree selection as sequential because each decision changes the representation available to later diffusion updates.The process specifies state, action space, transition, reward, initial-state distribution, discount factor, and horizon.Source: §3 · p. 6

3.1. State representation.

  • Each decision follows propagation on the current hierarchy; the state includes the propagated field, observed image, hierarchy, and decision stage.The actor receives local features rather than the complete field and image, so its observation is not itself a complete Markov state.Source: §3.1 · pp. 6–7

3.2. Adaptive action space.

  • At each candidate element, the policy proposes coarsening, retention, or refinement, while a deterministic update resolves proposals into an admissible hierarchy.Coarsening requires the required sibling decisions to agree; the realized tree is produced by UMR, not by the proposal distribution alone.Source: §3.2 · p. 7

3.3. Multiresolution update and dynamical transition.

  • Learned AMDI retains AMDI’s propagate–then–adapt ordering and changes only how the post-propagation adaptive update is selected.The final adaptive decision contributes to terminal occupancy and switching measures even though no additional propagation follows it.Source: §3.3 · pp. 8–9

3.4. Accuracy, representation occupancy, and adaptive switching.

  • Accuracy, active-representation occupancy, and hierarchy switching quantify distinct trajectory properties; occupancy and switching are algorithmic complexity measures, not runtime or memory measurements.The switching penalty need not produce a monotonic reduction in realized switching after retraining.Source: §3.4 · pp. 9–10, §3.3 · p. 9

3.5. Reward and trajectory objective.

  • The trajectory reward combines reference fidelity, occupancy, and switching, while its error measures agreement with the uniformly resolved AMDI computation rather than the clean image.In the reported experiments, γ = 1, so rewards from all six decision stages enter without discounting; PPO does not guarantee a global maximizer.Source: §3.5 · pp. 10–11

3.6. Consistency with deterministic AMDI.

  • Identical trajectories follow conditionally when initial states agree and both methods’ adaptive updates return the same field and tree at every decision.This consistency does not imply that the learned local policy reproduces the deterministic selector’s global candidate ranking or inherits its energy inequality.Source: §3.6 · p. 11

4. Learned Policy Architecture and Optimization.

The implementation uses a shared local policy trained with PPO and constructs its inputs after propagation and before adaptation. Local features describe multiresolution content, spatial variation, hierarchy, neighborhood occupancy, and decision timing.

4. Learned Policy Architecture and Optimization.

  • PPO optimizes a shared local actor–critic policy with an explicit admissibility mask, and the uniformly resolved reference trajectory informs training reward but is not an actor input.The implementation uses generalized advantage estimation for temporal credit assignment.Source: §4 · p. 11

4.1. Local feature construction.

  • A shared local policy acts after AMDI propagation and before adaptation, using local feature vectors rather than a flattened full-image representation.The feature vector captures local multiresolution content, spatial variation, hierarchy level, neighborhood occupancy, and position in the finite diffusion trajectory.Source: §4.1 · p. 12, §4.2 · p. 13
  • Local features include observed-image Haar detail magnitudes and post-propagation spatial features, while normalized level and decision-stage features encode hierarchy and timing.Observed-image Haar details remain fixed during an episode; propagated-field features, hierarchy, and decision stage can change over time.Source: §4.1 · pp. 12–13

4.2. Shared local policy.

A shared local network applies the same decision rule to each candidate element, supporting variable-size representations and use at unseen resolutions.

4.2. Shared local policy.

  • Weight sharing keeps trainable parameter count independent of active representation size and permits the same local rule at unseen resolutions.Resolution enters through the normalized level feature rather than the network dimensions.Source: §4.2 · p. 14

4.3. Admissibility masking.

The policy masks locally inadmissible proposals, while a deterministic update enforces global hierarchy admissibility and resolves proposals into the realized tree.

4.3. Admissibility masking.

  • An action mask removes locally inadmissible proposals, and deterministic UMR enforces global hierarchy admissibility after sampling.Coarsening occurs only when all four active leaf children propose it; refinement is then applied to eligible leaves that remain active.Source: §4.3 · p. 14
  • The policy distribution describes sampled proposals, which can map to the same realized hierarchy; rewards follow the update map, while log-probabilities correspond to sampled proposals.Retain remains available for every candidate element.Source: §4.3 · p. 14

4.4. Value function.

The value function uses pooled local features; mean and maximum pooling make the critic independent of the current candidate set’s cardinality.

4.4. Value function.

  • The pooling is permutation-invariant, so the critic’s representation does not depend on candidate ordering.Source: §4.4 · p. 15
  • The maximum used in pooling is taken componentwise.Source: §4.4 · p. 15
  • The value function is represented using pooled local features, with mean and maximum pooling making the critic independent of candidate-set cardinality.Source: §4.4 · p. 15

4.5. Advantage estimation.

Advantage estimation uses generalized advantage estimation and critic values fixed during rollout generation, while PPO treats the complete adaptive action as a joint action.

4.5. Advantage estimation.

  • Generalized advantage estimation constructs the advantages, with λGAE controlling the bias–variance tradeoff.Source: §4.5 · p. 15
  • Because each adaptive action contains all local decisions at a diffusion step, its PPO likelihood ratio is defined for the joint action.Source: §4.6 · p. 15

4.6. Clipped policy optimization.

PPO optimization uses a likelihood ratio for the complete sampled proposal and a clipped objective at the joint decision-step level. The protocol also specifies advantage scaling, entropy regularization, and rollout sampling details.

4.6. Clipped policy optimization.

  • The log-probability ratio sums over candidates for the complete sampled proposal without division by candidate-set size.Source: §4.6 · pp. 15–16
  • The actor uses advantages standardized across decision-step samples, while the value target retains the unstandardized advantage.Source: §4.6 · p. 16
  • A fixed joint clipping interval does not impose the same bound on each local action when candidate counts vary between steps.Source: §4.6 · p. 16
  • The training objective includes critic loss and entropy regularization, weighted by cV and cH, and optimization uses Adam.Source: §4.6 · p. 16
  • Each reported PPO update uses 16 trajectories and 96 decision-step samples, processed in four epochs for 384 Adam steps.Source: §4.6 · p. 16
  • Uniformly resolved reference trajectories construct the reward but are not supplied as actor or critic state information.Source: §4.7 · p. 16

4.7. Training protocol.

Training episodes sample initial images, propagate the current representation, then make post-propagation adaptive decisions. The resulting rollout supports reward construction and policy updates, with validation selecting the checkpoint for held-out evaluation.

4.7. Training protocol.

  • An episode samples an initial image, constructs its initial representation and field, and propagates the field on the current representation.Source: §4.7 · p. 16
  • After propagation, the protocol constructs a decision state, candidate set, local features, and admissibility masks before sampling actions and applying them with UMR.Source: §4.7 · pp. 16–17
  • The accepted state supplies approximation error, occupancy, and switching measures for reward construction; the episode continues through the rollout before advantage and actor–critic updates.Source: §4.7 · p. 17
  • Reward coefficients express relative tradeoffs, and the checkpoint is selected by greedy-action validation return while held-out images are reserved for final evaluation.Source: §4.7 · p. 17
  • Deployment preserves deterministic AMDI’s propagate-then-adapt ordering.Source: §4.7 · p. 17

4.8. Deployment and resolution transfer.

Deployment keeps AMDI propagation and hierarchy constraints while using a shared local actor to select tree actions. Resolution transfer is tested empirically because the architecture alone does not guarantee generalization to unseen resolutions.

4.8. Deployment and resolution transfer.

  • At deployment, deterministic features and admissibility masks guide local actor actions, which are applied through the constrained multiresolution update.The critic and uniformly resolved reference are not required.Source: §4.8 · pp. 17–18
  • The shared local parameterization keeps actor size independent of resolution and active-element count, while normalized level and time features permit evaluation across differently sized representations.This architectural property removes a fixed-dimensional restriction but does not guarantee generalization to unseen resolutions.Source: §4.8 · p. 17
  • The deployment sequence retains fixed-tree AMDI propagation, then constructs features and masks, evaluates the actor, and enforces hierarchy admissibility on the resulting tree.Different accepted trees can produce different interaction operators in later steps.Source: §4.8 · pp. 17–18
  • The evaluation explicitly tests transfer to finer resolutions alongside comparisons with deterministic AMDI, a uniform reference, and observed-detail controls.The study also assesses implementation reproduction, selector actions, occupancy tradeoffs, and reward and initialization sensitivity.Source: §5 · p. 18

5. Numerical validation.

Numerical validation compares post-propagation tree selectors under a shared AMDI propagator and hierarchy constraints, assessing holdout accuracy, adaptation behavior, accuracy–occupancy tradeoffs, and transfer without retuning. It also examines training-seed sensitivity, reward ablations, and qualitative localization diagnostics.

5.1. Synthetic data and evaluation sets.

  • The evaluation used nine holdout cases, distinguished clean-image RMSE and SSIM from uniform-AMDI reference error, and tested transfer on four image seeds at N = 32, 64, and 128 without retuning.Training was performed only at N = 32; the finer-resolution runs are genuine transfer calculations.Source: §5.1 · p. 19

5.2. Numerical protocol and implementation checks.

  • The regression checks reproduced deterministic AMDI tree and trajectory behavior, and the end-to-end test reproduced archived terminal metrics across all nine holdout cases.The low-level backend paths returned the same tree with 190 active degrees of freedom and maximum coefficient difference no larger than 2.2 × 10^-19; the integration test also exercised nonzero learned actions.Source: §5.2 · p. 20

5.3. Holdout controls and step-resolved action–energy analysis.

  • Deterministic AMDI accepted no refinements in 54 holdout decisions, while Learned AMDI executed 383 refinements at decision 1 and 10 at decision 2.Deterministic refinement candidates had higher AMDI energy than the unchanged candidate in all 54 decisions, even before the tree-change penalty.Source: §5.3 · pp. 21–22, Table 2 · p. 22
  • At nearby occupancies, the threshold control’s mean Eref was only 1.35×10^-3 above Learned AMDI, while RMSE was smaller and SSIM larger on average, improving both in eight of nine cases.The occupancy-matched threshold and Learned AMDI also had nearly indistinguishable mean holdout returns: −1.14540 and −1.14552, respectively; larger returns are better.Source: Table 3 · p. 22, §5.3 · p. 22
  • The decision-1-only control reached Eref = 0.13742 versus 0.13657 unrestricted, while later decisions executed no further changes and adaptation sometimes increased energy.At decision 1, learned adaptation raised energy in all nine cases, with a mean post-adaptation increment of 3.29 × 10^-5; the fixed-tree safeguard does not guarantee monotone decay after tree updates.Source: §5.3 · pp. 22–23

5.4. Accuracy–representation tradeoff.

  • On the four-image Pareto subset, the validation-tuned detail rule matched the learned policy at comparable occupancy and was slightly better on image metrics.The evidence supports a learned route to the allocation tradeoff, but not intrinsic PPO superiority to observed-detail ranking on these static synthetic images.Source: §5.4 · p. 24, Figure 3 · p. 25

5.5. Resolution transfer without retraining.

  • Without retraining, the actor transferred to 64 × 64 and 128 × 128 images, while the frozen threshold rule closely tracked its allocation and slightly improved all three accuracy measures.The fixed K = 36 rule executed 144 refinements at every resolution and became higher-occupancy than Learned AMDI on the finer grids, so it was not occupancy matched after transfer.Source: Table 4 · p. 25, §5.5 · pp. 25–26, Figure 4 · p. 26

5.6. Training robustness and reward ablation.

  • Across three policy-initialization seeds, reported Crel was 0.23608±0.05595, Eref was 0.14886±0.01476, and σsw was 0.01506±0.00956.Each retained checkpoint was evaluated on the same four image seeds and noise realizations.Source: §5.5 · p. 26, §5.6 · p. 26
  • Across three training seeds, image metrics vary less than representation occupancy and reference error, although learned allocation remains sensitive to policy initialization.The reported ± values are sample standard deviations of three policy-seed means, each averaged over four common evaluation cases.Source: Figure 5 · p. 27, §5.6 · p. 27
  • Removing the occupancy penalty drives representation to an almost uniform Crel = 0.99561, while removing switching raises Eref from 0.13769 to 0.15126 despite smaller representation and measured switching.The switching term is part of a coupled trajectory objective; in isolation it does not control observed switching monotonically.Source: §5.6 · p. 27

5.7. Independent multiresolution localization diagnostic.

  • Regional localization aligns only partly: VAMPyR assigns its largest effective level to edges, whereas Learned AMDI’s highest regional mean level is in textured regions and does not prioritize edges over gradients.Haar and VAMPyR effective levels have different definitions and are not quantitatively comparable; this is a qualitative diagnostic, not independent validation of learned dynamics. VAMPyR projects the clean analytic target, not the noisy image, AMDI state, or learned trajectory.Source: §5.6 · p. 27, §5.7 · p. 28, Figure 6 · p. 28
  • Learned AMDI makes 393 refinements versus deterministic AMDI’s zero in 54 holdout decisions, lowering mean terminal reference error from 0.17496 to 0.13657 as terminal occupancy rises from 0.13737 to 0.26660.Step-resolved data also show occasional small energy increases after learned tree changes, so fixed-tree energy stability does not guarantee monotonicity of the learned iteration.Source: §6 · p. 28

6. Conclusions.

At nearby but unequal occupancy, Learned AMDI and the tuned threshold occupy essentially the same accuracy–occupancy tradeoff; the shared actor also transfers to finer resolutions without retraining.

6. Conclusions.

  • At nearby but not identical occupancy, the threshold uses 0.26042 versus 0.26660 mean representation and gives 0.13792 versus 0.13657 reference error, with slightly better RMSE and SSIM.Source: §6 · p. 28
  • Without retraining, refinements fall from 154 at 32×32 to 24 at 64×64 and 12 at 128×128; at finer sizes, accuracy metrics improve over deterministic AMDI.Source: §6 · p. 29
  • The decision-1-only control indicates that most benefit on these static images comes from initial allocation, while moving or emerging localized structure is a natural test of sequential behavior.Source: §6 · p. 29
  • The repository openly provides code, a trained checkpoint, experiment scripts, result files, and the environment needed to reproduce the reported calculations.Source: Data Availability Statement · p. 29
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