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arxiv: 2607.05319 · v1 · pith:P54DC26R · submitted 2026-07-06 · cs.CV · cs.AI

Steering Optimisation Trajectories in Diffusion Representation Learning

Rajat Rasal , Avinash Kori , Tian Xia , Ben Glocker This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-07 17:39 UTCglm-5.2pith:P54DC26Rrecord.jsonopen to challenge →

classification cs.CV cs.AI
keywords disentanglementqualitydiffusionearlyimagelearningoptimisationreconstruction
0
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The pith

Two training tricks steer diffusion autoencoders toward disentangled representations

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

This paper argues that diffusion autoencoders can produce equally good images while learning qualitatively different latent representations, and that which representation a model arrives at is determined early in training by two mechanisms: skip connections that let the U-Net bypass the semantic pathway (a reconstruction shortcut), and the range of noise levels the model sees in its first steps (which sets the relative pace of image fidelity versus semantic organization). The authors observe that training trajectories cluster into two regimes, a reconstruction regime where image fidelity is prioritized early and disentanglement lags, and a disentanglement regime where both improve gradually. They propose SteeringDRL, which replaces skip connections with gated residuals initialized near zero to suppress the reconstruction shortcut, and introduces a log SNR curriculum that starts training on a narrow band of noise levels and gradually widens it. The combined method improves attribute disentanglement metrics (DCI, FactorVAE, MIG) on Shapes3D, Cars3D, and MPI3D, improves spatial disentanglement (segmentation quality) on ClevrTex and PascalVOC, and reduces variance across random seeds.

Core claim

The central object is the optimization trajectory: a curve of reconstruction quality plotted against representation quality during training. The authors show these trajectories bifurcate into two regimes determined early, and that the regime can be steered by two concrete interventions. The first replaces U-Net skip connections with gated residuals initialized near zero, forcing the decoder to rely on semantic tokens rather than a high-bandwidth pixel bypass. The second is a log SNR curriculum that restricts the noise-level band at the start of training and linearly widens it, controlling when the model encounters reconstruction-favorable noise levels. Together these shift trajectories fromt

What carries the argument

Gated residual U-Net blocks with near-zero-initialized gates replacing skip connections; log SNR curriculum that linearly widens the sampled noise-level band from an initial range to the full training range over a fixed number of steps, with an importance correction to maintain the correct loss weighting.

If this is right

  • Diffusion autoencoders with identical objectives and architectures can converge to substantially different latent structures depending on early training dynamics, suggesting that random seed sensitivity in representation learning is partly a trajectory-selection problem rather than purely a capacity or data issue.
  • The log SNR curriculum is a lightweight intervention that can be applied to existing diffusion training pipelines without architectural changes, potentially improving representation quality in any diffusion model that uses a U-Net with skip connections.
  • The finding that skip connections act as reconstruction shortcuts that bypass semantic conditioning suggests that architectural choices in diffusion U-Nets have representation-learning consequences that are invisible when only measuring image fidelity.
  • The regime structure, if it generalizes to transformer-based diffusion architectures where skip pathways are already constrained, may interact differently with the curriculum and could require modified interventions.

Load-bearing premise

The two-regime structure and its causal mechanisms are identified and validated primarily in a 3M-parameter U-Net with an undertrained VQ-VAE checkpoint, where regime separation is most visible. The authors note that larger models mask these patterns and that better VQ-VAE training reduces regime separation. The extrapolation that the same mechanisms operate in the larger models used for main results rests on performance improvements rather than direct trajectory evidence at

What would settle it

If the gated residual and curriculum interventions improve disentanglement in larger models through a mechanism unrelated to the two-regime structure (for example, by regularizing the loss landscape or changing gradient noise), then the paper's causal story about skip shortcuts and noise-level exposure steering regime selection would not hold, even though the practical improvements would remain.

Figures

Figures reproduced from arXiv: 2607.05319 by Avinash Kori, Ben Glocker, Rajat Rasal, Tian Xia.

Figure 1
Figure 1. Figure 1: Optimisation trajectories organise around two distinct regimes. Reconstruction (LPIPS) vs. disentanglement (DCI left, FactorVAE centre, MIG right) trajectories for a low-capacity diffusion autoencoder on Shapes3D (25 runs). Curves are coloured by final disentanglement. For interpretabil￾ity, trajectories are smoothed, and colour bars are centred at the median final disentanglement. 4 Early Emergence of Opt… view at source ↗
Figure 2
Figure 2. Figure 2: Inductive biases in STEERINGDRL. (a) We define our U-Net recursively. We use gated residuals instead of skip connections, and gates in the SpatialTransformer. IBκ(·) uses an amortised semantic encoder with FiLM [60] for attribute disentanglement (c.f. Section C.1), and SlotAttention [52] for object-centric learning. (b) The log SNR curriculum linearly widens the sampled log SNR band from [0, 10] to [−5, 12… view at source ↗
Figure 3
Figure 3. Figure 3: STEERINGDRL steers optimisation trajectories. Trajectories for models in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: log SNR curriculum selects the optimisation regime. With architecture fixed (C1: Gated Residual U-Net + sigmoid(−λ) loss-weighting, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Latent feature discovery and controlled image editing. We manually select latent dimen￾sions for semantic attributes in a target and transfer them to a source. On Cars3D, STEERINGDRL often isolates the challenging Style/Object Type factor in a single variable, producing more faith￾ful edits of Azimuth and Elevation. On Shapes3D with N=10 tokens, STEERINGDRL successfully performs all attribute edits, while … view at source ↗
Figure 6
Figure 6. Figure 6: Optimisation trajectories support visual improvements. (a) STEERINGDRL discovers more objects with more complete segmentations than SlotDiffusion. (b) Compositions preserve object structure and background textures. (c) Optimisation trajectories are similar to attribute disentanglement ( [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: STEERINGDRL accelerates object-centric binding with pretrained features. (a) STEERINGDRL produces cleaner masks for dominant objects, while SlotDiffusion often underseg￾ments. (b) With the same pretrained Featψ(·), STEERINGDRL reaches higher segmentation and reconstruction quality earlier in training than SlotDiffusion. Training Efficiency with Pretrained Features (PascalVOC). We scale STEERINGDRL to a muc… view at source ↗
Figure 8
Figure 8. Figure 8: Architecture for Amortised Semantic Encoder [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: log SNR curriculum for object-centric learning on the Pascal VOC dataset. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: , we use the non-monotonic weighting w(λ) = sech(λ), which places most weight near intermediate log SNR values, akin to the EDM with shift −3 we used in Section 4. The same qualitative behaviour appears; runs separate into trajectories that prioritise reconstruction early and trajectories that maintain stronger disentanglement throughout training. This suggests that the regime structure is not specific to… view at source ↗
Figure 11
Figure 11. Figure 11: Cross-attention maps reveal semantic structure in the disentanglement regime. These are taken from the models used in [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Removing skip connections steers trajectories toward disentanglement. We follow the setup in [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: shows the corresponding optimisation trajectories for DCI, FactorVAE, and MIG. The trajectories reveal that curricula change the path through reconstruction-disentanglement space before final convergence. In particular, the best-performing curriculum follows a different trajectory, delaying early commitment to a reconstruction-dominated path and reaching high final disentanglement with lower variance acro… view at source ↗
Figure 14
Figure 14. Figure 14: Latent edits reveal interpretable factors across datasets. Each panel compares EncDiff and STEERINGDRL by replacing a single source slot with the corresponding target slot. On Cars3D, edits capture viewpoint (Azimuth, Elevation) and object identity (Object Type / Style). On Shapes3D with N = 10, we can more consistently perform all edits, whereas EncDiff often struggles with Floor Hue and Object Shape. 29… view at source ↗
Figure 15
Figure 15. Figure 15: STEERINGDRL produces more spatially coherent and compositional slot represen￾tations on ClevrTex with ResNet34 features. (a) Compared to SlotDiffusion, our slots are more spatially consistent: when read down each slot column, the same slot tends to capture objects in similar spatial locations across scenes, while fewer slots collapse to modelling only background. This yields cleaner segmentation masks and… view at source ↗
Figure 16
Figure 16. Figure 16: ClevrTex optimisation trajectories across segmentation metrics. We plot reconstruction–segmentation trajectories for FG-ARI, mIoU and mBO using ResNet34 features. STEERINGDRL learns semantic slot structure earlier than SlotDiffusion and converges to higher segmentation scores while maintaining strong reconstruction [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: ClevrTex optimisation trajectories with ResNet18 features. Using the weaker ResNet18 feature encoder, STEERINGDRL still improves segmentation earlier than SlotDiffusion and converges to higher FG-ARI, mIoU and mBO. E.8.2 PascalVOC GT Mask SlotDiffusion Ours [PITH_FULL_IMAGE:figures/full_fig_p031_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: PascalVOC qualitative masks with pretrained DINO features. STEERINGDRL better exploits the semantic features in DINO to segment dominant objects, whereas SlotDiffusion often merges objects with the background or nearby regions. However, both methods still exhibit object-part segmentation, such as separating the head and body of birds, suggesting that further gains may require explicit semantic alignment a… view at source ↗
Figure 19
Figure 19. Figure 19: Individual PascalVOC trajectories show faster and stronger convergence. We plot each metric against training progress for SlotDiffusion and STEERINGDRL. With the same pretrained Featψ(·), STEERINGDRL improves FG-ARI, mBO and mIoU earlier in training, while also reducing LPIPS faster. This supports the view that, with pretrained semantic features, our inductive biases improve the efficiency of object-centr… view at source ↗
Figure 20
Figure 20. Figure 20: Optimisation trajectories on PascalVOC differ from from-scratch OCL. We plot reconstruction–segmentation trajectories for FG-ARI, mBO and mIoU on PascalVOC. Unlike Clevr￾Tex, where representations must emerge from scratch, PascalVOC uses pretrained DINO features for Featψ(·). STEERINGDRL exploits these pretrained semantics more quickly than SlotDiffusion, reaching higher segmentation scores and better rec… view at source ↗
read the original abstract

We study why diffusion autoencoders can achieve similar image quality while learning substantially different latent structures. We trace this behaviour to optimisation dynamics; we analyse curves of image reconstruction against latent representation quality, revealing trajectories that organise around two distinct regimes early in training. Models in the reconstruction regime prioritise image fidelity early, whereas those in the disentanglement regime improve reconstruction and disentanglement more gradually. We hypothesise that this behaviour can be influenced by targeting shortcut pathways in the diffusion U-Net and controlling early noise-level exposure, thereby shaping the reconstruction-disentanglement trade-off during training. To steer optimisation toward stronger representations, we introduce SteeringDRL, combining gated residual U-Nets with a simple noise-level exposure curriculum for training. Across disentanglement benchmarks, SteeringDRL improves representation quality and reduces seed sensitivity. Our method further extends to spatial disentanglement in object-centric learning, improving segmentation quality on synthetic and real-world datasets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 8 minor

Summary. The paper studies optimisation dynamics in diffusion autoencoders for unsupervised representation learning, observing that training trajectories organise around two regimes (reconstruction vs. disentanglement) that are determined early in training. Two mechanisms are proposed: (1) gated residual U-Net blocks replacing skip connections to suppress reconstruction shortcuts (H1), and (2) a log SNR curriculum that gradually widens the noise-level band during training (H2). The combined method, SteeringDRL, is evaluated on attribute disentanglement (Shapes3D, Cars3D, MPI3D) and spatial disentanglement (ClevrTex, PascalVOC), showing improvements over EncDiff and SlotDiffusion baselines with reduced seed variance. The paper provides extensive ablations (Tables 1, 6–11) and trajectory visualisations (Figures 1, 3, 4, 12) to support the regime hypothesis and the individual contributions of each component.

Significance. The paper tackles a genuine question: why diffusion autoencoders with identical objectives and architectures can yield substantially different latent structures. The observation of two optimisation regimes is interesting and the proposed interventions (gated residuals, log SNR curriculum) are simple and well-motivated. The ablation study in Table 1 is thorough, testing each component independently and in combination across 10 seeds. The extension to spatial disentanglement with object-centric learning on both synthetic and real-world data (Tables 3, 4) demonstrates generality beyond the initial setting. The trajectory visualisations provide qualitative support for the regime hypothesis. The method is falsifiable: Figure 4 shows that different curricula produce different trajectories, and Table 11 ablates curriculum hyperparameters. The code release commitment would strengthen reproducibility.

major comments (3)
  1. Section 6.1 and Section E.5: The central causal claim—that two optimisation regimes exist and are driven by skip-connection shortcuts (H1) and early noise-level exposure (H2)—is validated only in a 3M-parameter U-Net with an undertrained 30K-step VQ-VAE checkpoint, where regime separation is 'most visible.' The authors acknowledge that 'larger models mask these patterns' (Section 4) and that the baseline 'no longer exhibits regime separation' at the scale used for main results (Section 6.1). This creates a gap between the explanatory framework (regime dynamics) and the models where improvements are reported. The improvements in larger models could potentially be explained by standard regularisation effects of gated residuals and curriculum learning without invoking regime dynamics. The paper should either (a) provide direct trajectory evidence of regime separation at the scale of the Enc
  2. Table 1: The component-wise ablation reveals that the gated residual U-Net alone does not improve DCI (0.832→0.826) and the curriculum alone applied to the standard U-Net hurts DCI (0.932→0.853). The improvement only appears when all three components (gated residual + sigmoid(−λ) weighting + curriculum) are combined (DCI 0.926). This three-way interaction is load-bearing for the claim that H1 and H2 are the operative mechanisms, yet the paper does not analyse why the components are individually ineffective or counterproductive but jointly beneficial. An alternative explanation is that the gated residual changes the optimisation landscape in a way that makes the curriculum effective for reasons unrelated to regime steering. The paper should discuss this interaction more carefully, ideally with trajectory plots for each two-component combination (gated+weighting, gated+curriculum, standard
  3. Section 6.1, Figure 3: The claim that SteeringDRL 'steers trajectories toward the disentanglement regime' in larger models is supported by trajectory plots, but the baseline in this setting 'no longer exhibits regime separation and instead follows the reconstruction regime' (Section 6.1). If there is no regime separation at this scale, it is unclear what 'toward the disentanglement regime' means operationally. The paper should clarify whether the trajectory shift in Figure 3 reflects the same regime dynamics observed in Figure 1, or whether it is simply an improvement in disentanglement metrics that is post-hoc labelled as regime steering. A more precise definition of what constitutes evidence of regime steering (as opposed to generic improvement) would strengthen the causal claim.
minor comments (8)
  1. Table 2: For MPI3D-toy at N=20, SteeringDRL achieves FactorVAE 0.870, which is lower than EncDiff (0.899) and DyGA (0.930). The text states the method 'remains competitive,' but the gap on FactorVAE is notable. This should be acknowledged more directly.
  2. Table 5: The curriculum hyperparameters (initial band, widening steps) vary substantially across datasets ([0,10] with 32K steps for attribute disentanglement vs. [−2,2] with 200K steps for OCL). The paper does not explain how these are selected. A brief note on the selection criterion would improve reproducibility.
  3. Section E.5: The VQ-VAE checkpoint choice (30K vs 150K steps) affects regime visibility, with the 30K checkpoint making regimes 'most visible.' Since the main results use the EncDiff setup (which appears to use a more trained VQ-VAE), the relationship between the analysis checkpoint and the main-results checkpoint should be stated more explicitly.
  4. Figure 4 caption: 'C5 beats C2 in disentanglement' — the DCI values in Table 11 show C2 (initial band [−5,0]) achieves DCI 0.914 while C5 (initial band [0,10], 32K) achieves 0.926, so the claim is correct but the margin is within one standard deviation. This should be noted.
  5. Section 3.3, Eq. (6): The derivative dλ/dt appears without explicit derivation. A brief note pointing to the noise schedule definition in Section A.1 would help readers.
  6. Table 3: The STEERINGDRL rows with 50-step DDIM and 200-step DDPM share the same segmentation metrics (indicated by quotation marks). A footnote or merged rows would be clearer.
  7. The paper would benefit from a brief discussion of computational overhead introduced by the gated residual U-Net and curriculum, particularly for the 138M-parameter PascalVOC model.
  8. Section C.1: The FiLM-based amortised encoder is described as reducing parameters by approximately O(N) relative to the split-MLP. A quantitative comparison of parameter counts would be informative, especially since Table 1 shows it matches the split-MLP baseline.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The core concern—the gap between the scale at which regime dynamics are most visible and the scale at which we report main results—is well-taken, and we agree the manuscript must do more to bridge it. Below we address each major comment in turn.

read point-by-point responses

  1. Referee: Section 6.1 and Section E.5: The central causal claim—that two optimisation regimes exist and are driven by skip-connection shortcuts (H1) and early noise-level exposure (H2)—is validated only in a 3M-parameter U-Net with an undertrained 30K-step VQ-VAE checkpoint, where regime separation is 'most visible.' The authors acknowledge that 'larger models mask these patterns' (Section 4) and that the baseline 'no longer exhibits regime separation' at the scale used for main results (Section 6.1). This creates a gap between the explanatory framework (regime dynamics) and the models where improvements are reported. The improvements in larger models could potentially be explained by standard regularisation effects of gated residuals and curriculum learning without invoking regime dynamics. The paper should either (a) provide direct trajectory evidence of regime separation at the scale of the Enc

    Authors: The referee correctly identifies a genuine gap in the manuscript: the regime phenomenon is most clearly visible in a low-capacity setting, while the main results use larger models where the baseline no longer exhibits clear two-regime separation. We agree this gap must be addressed more carefully, and we will revise the manuscript accordingly (revision_made = 'partial'). Here is our honest assessment of what we can and cannot do. First, we acknowledge that we cannot fully close the gap with the experiments currently in the paper. We do not have trajectory evidence showing clean two-regime separation at the scale of the EncDiff U-Net used in Table 1. At that scale, the baseline trajectories cluster in what we label the 'reconstruction regime' rather than separating into two distinct regimes. This is a limitation we will state more explicitly. Second, we can partially bridge the gap with evidence already in the paper but not sufficiently highlighted. Figure 4 (and the extended version in Figure 13) shows that, with the gated residual architecture fixed, different curricula produce qualitatively different trajectory shapes—some prioritise early reconstruction, others maintain gradual disentanglement improvement. This is direct evidence that the curriculum steers optimisation dynamics at the larger scale, even though the baseline does not exhibit two-regime separation. The zeroed-skip ablation in Table 6 and Figure 12 provides analogous evidence for H1 at the larger scale: removing skip connections shifts trajectories toward the disentanglement regime, and adding the curriculum on top shifts them further. Third, we agree that an alternative explanation—standard regularisation effects of gated residuals and curriculum—cannot be ruled out with the current evidence. We will re revision: no

  2. Referee: Table 1: The component-wise ablation reveals that the gated residual U-Net alone does not improve DCI (0.832→0.826) and the curriculum alone applied to the standard U-Net hurts DCI (0.932→0.853). The improvement only appears when all three components (gated residual + sigmoid(−λ) weighting + curriculum) are combined (DCI 0.926). This three-way interaction is load-bearing for the claim that H1 and H2 are the operative mechanisms, yet the paper does not analyse why the components are individually ineffective or counterproductive but jointly beneficial. An alternative explanation is that the gated residual changes the optimisation landscape in a way that makes the curriculum effective for reasons unrelated to regime steering. The paper should discuss this interaction more carefully, ideally with trajectory plots for each two-component combination (gated+weighting, gated+curriculum, standard

    Authors: The referee is correct that the three-way interaction in Table 1 is load-bearing and under-analysed. We will revise the manuscript to discuss this interaction explicitly and add trajectory plots for the two-component combinations (revision_made = 'yes'). Our current understanding of the interaction is as follows. The curriculum alone applied to the standard U-Net hurts DCI (0.932→0.853) because, when skip connections are present, restricting early noise-level exposure delays reconstruction without redirecting the optimisation away from the reconstruction shortcut. The skip pathway allows the decoder to reconstruct from encoder features without relying on the semantic tokens, so the curriculum's effect on noise-level exposure does not translate into improved representation—it simply slows reconstruction. The gated residual alone does not improve DCI (0.832→0.826) because, while it suppresses the skip shortcut, the model still sees the full noise-level band from the start. Without curriculum control, early exposure to reconstruction-favouring noise levels can still bias the model toward the reconstruction regime, even with a weaker shortcut pathway. The combination works because the gated residual removes the high-bandwidth bypass that would otherwise absorb the curriculum's effect, while the curriculum controls the noise-level exposure that drives regime selection. This is consistent with the regime-steering explanation, but the referee is right that it is also consistent with an alternative: the gated residual changes the optimisation landscape in a way that makes the curriculum effective for reasons unrelated to regime dynamics. We cannot fully distinguish between these explanations with the current evidence. However, the zeroed-skip + curriculum ablation in Table 6 is revision: no

  3. Referee: Section 6.1, Figure 3: The claim that SteeringDRL 'steers trajectories toward the disentanglement regime' in larger models is supported by trajectory plots, but the baseline in this setting 'no longer exhibits regime separation and instead follows the reconstruction regime' (Section 6.1). If there is no regime separation at this scale, it is unclear what 'toward the disentanglement regime' means operationally. The paper should clarify whether the trajectory shift in Figure 3 reflects the same regime dynamics observed in Figure 1, or whether it is simply an improvement in disentanglement metrics that is post-hoc labelled as regime steering. A more precise definition of what constitutes evidence of regime steering (as opposed to generic improvement) would strengthen the causal claim.

    Authors: The referee raises a valid conceptual point: if there is no two-regime separation at the larger scale, the phrase 'toward the disentanglement regime' is not well-defined. We will revise the manuscript to clarify this (revision_made = 'yes'). Our intended meaning is the following. In the low-capacity setting (Figure 1), the disentanglement regime is characterised by two properties: (1) reconstruction improves more gradually, and (2) disentanglement improves progressively throughout training rather than being deferred. In the larger-scale setting (Figure 3), the baseline follows what we call the reconstruction regime—fast early reconstruction with slower disentanglement improvement. SteeringDRL shifts the trajectory so that reconstruction is slower and disentanglement improves more progressively, which is qualitatively consistent with the disentanglement regime pattern from Figure 1. However, we agree that without two-regime separation at the larger scale, calling this 'regime steering' rather than 'trajectory improvement' involves an extrapolation from the low-capacity setting. We will make this explicit in the revision: the trajectory shift in Figure 3 is qualitatively consistent with the regime dynamics observed in Figure 1, but we cannot confirm it is the same phenomenon. We will also provide a more precise operational definition: evidence of regime steering (as opposed to generic improvement) requires that the trajectory shape changes in a way that matches the regime characteristics—specifically, delayed reconstruction with progressive disentanglement improvement, rather than uniform improvement on both axes. By this definition, Figure 3 shows trajectory shapes consistent with regime steering, but the causal link to the low-capacity regime dynamics remains an infer revision: no

Circularity Check

0 steps flagged

No significant circularity; one minor non-load-bearing self-citation

full rationale

The paper's derivation chain is self-contained. The two-regime observation (Section 4) is an empirical finding about trajectory shapes, not a definition that tautologically implies its own conclusion. Hypotheses H1 and H2 are tested via independent ablations (Tables 6–9): zeroing skip connections changes disentanglement/reconstruction tradeoffs, and restricting noise-level bands changes optimization trajectories. These are genuine experiments, not fitted-input-as-prediction. The method components (gated residuals from [29], sigmoid(-λ) weighting from [42], cross-attention from [88]) are drawn from external prior work, not from a self-citation chain. The one shared co-author citation is [66] (Ribeiro, Glocker et al.), a tutorial on VDM++ used for background formulation; it does not carry the central argument and is not invoked as a uniqueness theorem. Curriculum parameters are empirically ablated (Table 11) and tuned per-dataset (Table 5), not claimed as first-principles predictions. The fact that individual components sometimes fail in isolation (Table 1) while the combination succeeds raises correctness/generalization questions, but these are not circularity: the ablations are independent experiments with independent measurements. No step in the derivation reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

No new entities are invented. The method combines existing architectural components (gated residuals, cross-attention, slot attention) with a new training curriculum.

free parameters (5)
  • Initial curriculum band [λL(0), λU(0)] = [0,10] for attribute disentanglement; [-2,2] for OCL
    Chosen by ablation (Table 11, Figure 4); different initial bands produce different trajectories and final disentanglement scores.
  • Widening steps mmax = 32K for attribute; 200K for OCL
    Tuned hyperparameter controlling how quickly the full noise band is reached; ablated in Table 11.
  • Gate initialization ξ = Initialized such that Softplus(ξ) ≈ 0
    Stated in Section 5.1; the exact value is not given but the design choice (near-zero gates) is the key mechanism.
  • Full log SNR band [λmin, λmax] = [-5,12] for attribute; [-8.5,6.5] for ClevrTex; [-15,15] for PascalVOC
    Dataset-specific choices from Table 5; not independently derived.
  • Number of tokens/slots N = 10 or 20 for attribute; 11 for ClevrTex; 6 for PascalVOC
    Set to match prior work or dataset characteristics; not derived from first principles.
axioms (5)
  • standard math The VDM++ framework (variance-preserving diffusion in log SNR space) is a valid generative objective for representation learning.
    Section 3.3; follows from Kingma and Gao [42]. Standard diffusion framework.
  • domain assumption Cross-attention between semantic tokens and U-Net features induces disentanglement without additional regularisation.
    Section 1, Section 3.2; follows from EncDiff [88]. This is the inductive bias the paper builds on.
  • domain assumption Skip connections in U-Nets provide a high-bandwidth bypass around the semantic conditioning pathway.
    Section 4; cited from Jun et al. [33] and Ma et al. [53]. Central to H1.
  • domain assumption High noise levels in diffusion correspond to coarse semantic information; low noise levels correspond to fine-grained details.
    Section 1; cited from multiple prior works [80, 49, 78]. Central to H2.
  • domain assumption The cosine noise schedule is appropriate for all experiments.
    Section A.1; used throughout without comparison to alternatives.

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