Fusion Surface Models in Diffusion Systems

• Fusion surface models are conditional generative systems that fuse external signals with a latent denoising trajectory using mechanisms like concatenation, feature addition, and gated fusion.
• They optimize the interface between the conditioning signal and the denoising state to improve controllability, convergence, and maintain privacy in various diffusion tasks.
• Empirical results demonstrate that these models achieve lower FID scores and faster convergence in tasks such as mammogram synthesis, subject-driven generation, and scientific data compression.

Searching arXiv for the papers on arXiv and related diffusion conditioning/fusion terminology. “Fusion surface models” is not a standardized designation in the recent diffusion literature considered here. A plausible interpretation is an umbrella label for conditional generative systems in which external signals are fused with a latent denoising state or trajectory through concatenation, feature addition, cross-attention, token-wise gating, or predictive gradient transport. Under that reading, the relevant literature includes MPC-like guide approximation for sparse conditional sampling (Shen et al., 2022), unified gated condition injection for linear-attention diffusion transformers (Liu et al., 29 Mar 2026), gated radiomics-geometric fusion for lesion-controllable mammogram synthesis (Li et al., 25 Jul 2025), and slice-wise latent conditioning for scientific data compression with deterministic decoding and error guarantees (Lee et al., 18 Feb 2025). The unifying issue across these systems is not merely the existence of conditioning, but the locus and mechanism of fusion inside the reverse process.

1. Conceptual scope and generative formulations

Recent work shows that conditional fusion can be attached to several generative objectives rather than a single canonical diffusion loss. In the denoising setting, a diffusion model $\hat{x}_\theta$ may be trained with the weighted reconstruction objective

$$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$

with the associated noise-prediction parameterization

$$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$

and sampling update

$$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$

Closely related DDPM-style conditional formulations appear in mammogram synthesis and scientific compression, where the model is trained by a noise-prediction loss and the reverse process is conditioned on masks, lesion descriptors, or slice-wise latent codes (Shen et al., 2022, Li et al., 25 Jul 2025, Lee et al., 18 Feb 2025).

A different backbone appears in linear-attention diffusion transformers built on SANA, where controllable generation is trained with the flow-matching objective

$$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$

There, the fusion problem is not how to modify a U-Net denoiser at every step, but how to preserve control signals inside a linear-attention stack whose information compression can weaken conditional pathways (Liu et al., 29 Mar 2026).

This suggests that the central design variable is the conditional interface itself. The same broad problem—how to combine global structure, local detail, and external control—recurs across denoising diffusion, classifier-free guidance, flow matching, and deterministic conditional decoding.

2. Recurrent fusion operators

Across the surveyed systems, five recurrent fusion operators appear: direct feature addition for aligned conditions, concatenation of latent and condition embeddings, token-sequence interaction through attention, token-wise gating after interaction, and predictive gradient transport through the denoising trajectory. Slice-wise conditioning in scientific compression is a sixth pattern, but it acts more as structured conditioning than as gating in the usual neural-network sense (Liu et al., 29 Mar 2026, Li et al., 25 Jul 2025, Shen et al., 2022, Lee et al., 18 Feb 2025).

Mechanism	Representative formulation	Reported role
Feature addition	$h_x \leftarrow h_x + h_c$	Spatially aligned conditioning
Token interaction plus gating	$[X; C_T; C_I]$; ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$	Flexible multi-type control with token preservation
Soft-mask concatenation	$$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$	Global anatomical control
Predictive guide transport	$$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$	Approximate missing current-step guidance
Slice-wise latent conditioning	$$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$0	3D-to-2D conditional reconstruction

The principal distinction is between fusion by correspondence and fusion by selection. Direct addition assumes that the condition and latent occupy compatible spatial coordinates. Gated fusion, top-$$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$1 selection, and predictive gradient transport instead attempt to decide which conditional components should survive, which should be suppressed, and how control should be propagated when explicit alignment is weak or unavailable.

3. Predictive fusion along the denoising trajectory

A particularly explicit trajectory-level formulation appears in "Conditional Diffusion with Less Explicit Guidance via Model Predictive Control" (Shen et al., 2022). The paper addresses conditional sampling when explicit guidance is available only at a small number of diffusion steps, rather than at every denoising step. Standard baselines such as classifier guidance, classifier-free guidance, and conditional diffusion models normally require an explicit conditional guide throughout the trajectory. The proposed alternative treats diffusion sampling as a control problem with state $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$2, control $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$3, dynamics $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$4 given by denoising, and a future conditional objective $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$5.

The central mechanism is lookahead guidance. The model simulates unconditional denoising from $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$6 to a future latent $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$7, evaluates explicit guidance there, and backpropagates that signal to the current latent: $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$8 For a noised classifier, the objective is based on $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$9; for a conditional diffusion model, the objective is a squared error around a target defined from $$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$0 with gradients through the target blocked. The paper frames this as diffusion-analogue receding-horizon control: optimize using a short simulated future, apply only the first action, and repeat at the next step.

The reported approximation quality is unusually high. The approximated gradient has cosine similarity above $$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$1 even when the approximation spans $$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$2 of $$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$3 diffusion steps, and similarity remains above $$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$4 even at $$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$5. By contrast, a CLIP spherical loss applied after denoising to image space yields a gradient with mean cosine similarity around $$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$6, described as nearly orthogonal to the true diffusion guide. This supports a strong methodological distinction: guidance should be propagated through the diffusion latent dynamics, not through a clean-image objective.

The main experimental regime uses Stable Diffusion, a PLMS scheduler, classifier-free guidance with $$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$7, and MS-COCO prompts. Explicit guidance is restricted to five time steps, within an eight-step schedule

$$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$8

Explicit guidance is applied at $$\hat{x}_\theta(z_t,c) = \frac{z_t - \sigma_t \epsilon_\theta(z_t,c)}{\alpha_t},$$9, while MPC guidance is inserted at $$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$0. Under this regime, the baseline with $$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$1 reports FID to Reference $$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$2 and FID to Gold Standard $$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$3, whereas adding MPC with $$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$4 yields $$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$5 and $$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$6. The paper also notes qualitative cases in which MPC samples appeared closer to the gold standard than the reference itself.

4. Token-wise gated injection in linear-attention transformers

"Gated Condition Injection without Multimodal Attention: Towards Controllable Linear-Attention Transformers" develops a fusion architecture tailored to linear-attention diffusion transformers such as SANA (Liu et al., 29 Mar 2026). The motivation is partly systems-oriented: controllable diffusion is usually deployed on cloud models, raising privacy concerns because users must upload sensitive images, sketches, or identity cues, while linear-attention backbones are described as scalable and memory-efficient enough for edge or on-device use.

The paper diagnoses two transfer failures. ControlNet-style feature addition,

$$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$7

works when the condition is spatially aligned with the latent, such as edges, depth, or pose maps, but breaks down for non-aligned conditions such as subject-driven generation. OminiControl-style multimodal attention is more general, since it converts conditions into tokens and mixes them with noisy latent tokens, but on linear-attention backbones it converges slowly, especially for spatially aligned tasks. The stated reason is that linear attention compresses information more aggressively than softmax attention, so important control signals may be suppressed unless they are explicitly preserved.

The proposed framework, described as GCDM or GateControl, adopts a shared-module, dual-path pipeline. The image condition and noisy latent are transformed through the same VAE encoder and then passed through an identical model structure, with LoRA fine-tuning instead of full finetuning. Path 1 performs internal interaction by concatenating latent tokens $$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$8, text tokens $$z_{t-1} = \frac{z_t - \sigma_t \tilde{\epsilon}_\theta}{\alpha_t}.$$9, and image-condition tokens $$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$0 into a single token sequence $$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$1. Path 2 applies token-wise gating after the interaction: $$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$2 followed by

$$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$3

The gate is explicitly token-wise rather than full element-wise, so each token receives its own score.

The framework is trained on SANA-1.0 with LoRA rank $$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$4, using Prodigy with safeguard warmup and bias correction, weight decay $$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$5, learning rate $$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$6, batch size $$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$7 per GPU, on $$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$8 NVIDIA H200 GPUs. Subject-driven generation is trained on the $$\mathcal {L}_{F M}:=\mathbb{E}_{t, p_t(x)}\left[\left\|v_t(x) - u_t(x)\right\|_2^2\right].$$9 subset of Subject200K for $h_x \leftarrow h_x + h_c$0 steps, while spatially aligned tasks are trained on $h_x \leftarrow h_x + h_c$1 images from Text-to-Image-2M for $h_x \leftarrow h_x + h_c$2 steps. The convergence claim is central: spatial tasks in OminiControl require roughly $h_x \leftarrow h_x + h_c$3 steps versus $h_x \leftarrow h_x + h_c$4 for subject-driven tasks in the original setup, whereas the gated version for canny-to-image surpasses a baseline without gated spatial alignment using only $h_x \leftarrow h_x + h_c$5 steps, and the baseline needs more than $h_x \leftarrow h_x + h_c$6 steps to reach comparable positional correspondence.

The quantitative results are reported across aligned and non-aligned control. On Canny, the gated method reports F1 $h_x \leftarrow h_x + h_c$7, FID $h_x \leftarrow h_x + h_c$8, SSIM $h_x \leftarrow h_x + h_c$9, MUSIQ $[X; C_T; C_I]$0, and CLIP-Image $[X; C_T; C_I]$1, compared with OminiControl’s F1 $[X; C_T; C_I]$2, FID $[X; C_T; C_I]$3, SSIM $[X; C_T; C_I]$4, MUSIQ $[X; C_T; C_I]$5, and CLIP-Image $[X; C_T; C_I]$6. On another aligned benchmark at $[X; C_T; C_I]$7, it reports MSE $[X; C_T; C_I]$8, FID $[X; C_T; C_I]$9, MUSIQ ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$0, and CLIP-Image ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$1, compared with OminiControl’s MSE ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$2, FID ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$3, MUSIQ ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$4, and CLIP-Image ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$5. On deblurring, it reports ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$6, FID ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$7, SSIM ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$8, MUSIQ ${h_X}^{\prime}= \sigma(XW_{g1}) \odot h_X$9, and CLIP-Image $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$0, compared with OminiControl’s $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$1, FID $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$2, SSIM $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$3, MUSIQ $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$4, and CLIP-Image $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$5. In subject-driven generation on DreamBooth, the GPT-4o-based score increases from $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$6 for IP-Adapter on SANA to $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$7. Parameter overhead is also reported as small: the gating module adds only $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$8 parameters, about $$\epsilon_\theta\!\left(\mathtt{concat}(z_t,\mathcal{E}(\mathcal{G}(M))), t, \mathbf{c}\right)$$9 of SANA’s size, and the total additional trainable parameters with LoRA are $$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$0, versus $$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$1 extra parameters for ControlNet on SANA-scale models.

5. Hybrid global-local fusion in mammogram synthesis

A domain-specific but technically revealing instance appears in "Joint Holistic and Lesion Controllable Mammogram Synthesis via Gated Conditional Diffusion Model" (Li et al., 25 Jul 2025). The problem is not only to synthesize realistic mammograms, but to preserve lesion shape, texture, radiomic appearance, and anatomical blending with surrounding tissue. The model therefore combines two conditioning pathways: a holistic mask-conditioning path for breast layout and lesion placement, and a gated lesion-control branch for lesion-specific clinical descriptors.

The backbone is a Stable Diffusion v1.5-style latent denoising diffusion model. A real mammogram $$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$2 is encoded by a VAE encoder $$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$3 into latent $$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$4, and diffusion proceeds in latent space. The global branch uses a three-channel mask $$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$5 for background, breast tissue, and lesion or mass region. A Gaussian blur operator $$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$6 is applied to the lesion channel to create a soft mask with a transitional region, and the denoiser receives the concatenated condition

$$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$7

The role of the soft mask is explicit: it introduces a transitional zone between lesion and surrounding tissue, so the model can learn more realistic anatomical blending and avoid hard-edged artifacts.

The lesion-control branch supplies what the paper treats as clinically meaningful lesion descriptors. A $$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$8-dimensional radiomic vector $$\boldsymbol{\xi}_t = - \nabla_{z_t}\ell(z_{t-\delta})$$9 is extracted with PyRadiomics, consisting of $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$00 shape features, $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$01 histogram or first-order features, $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$02 GLSZM features, and $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$03 GLCM features. In parallel, the lesion mask is embedded using CLIP, producing geometric candidate features $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$04. The model forms a cross-combination set

$$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$05

computes a relevance score

$$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$06

and retains the most relevant fused candidates through

$$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$07

This output $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$08 is injected into the diffusion U-Net via cross-attention. The paper’s interpretation is that naive concatenation can introduce noisy or conflicting control signals, whereas gating dynamically emphasizes the most informative geometry-radiomics combinations.

The implementation is specific: images are resized to $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$09, the latent has size $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$10, AdamW is used with learning rate $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$11, batch size $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$12 per GPU, and training runs for $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$13 epochs on $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$14 NVIDIA A100 80GB GPUs. Diffusion uses $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$15 steps, inference uses $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$16 steps, the noise schedule $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$17 increases linearly from $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$18 to $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$19, classifier-free guidance masks conditional inputs with probability $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$20 during training and uses guidance scale $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$21 at inference, Gaussian blur uses $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$22, and the gate-fusion hyperparameters are $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$23.

Experiments are conducted on VinDr-Mammo using only cranial-caudal oblique images. The split is $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$24 training images, $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$25 validation images, and $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$26 test images, with $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$27, $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$28, and $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$29 mass-containing images in the respective partitions. Against SPADE, pSp, SR3, ControlNet, and Seg-Diff, the model reports FID $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$30, Mass IoU $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$31, Breast IoU $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$32, and Pixel Accuracy $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$33. Relative to the second-best result, the paper reports a $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$34 improvement in FID and a $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$35 improvement in Mass IoU, with statistically significant $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$36-values of $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$37 and $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$38. Ablations show a progression from a mask-only baseline with FID $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$39 and Mass IoU $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$40, to adding the lesion control branch with $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$41 and $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$42, then adding radiomics features with $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$43 and $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$44, and finally the full model with gated fusion. The soft-mask study further indicates a trade-off: increasing softness beyond $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$45 improves FID but reduces Mass IoU.

6. Slice-wise conditional fusion in scientific compression and recurring limitations

"Guaranteed Conditional Diffusion: 3D Block-based Models for Scientific Data Compression" is an important boundary case because it does not define a model literally called a gated conditional diffusion model and does not introduce a gate layer in the standard architectural sense (Lee et al., 18 Feb 2025). Instead, it presents Guaranteed Conditional Diffusion with Tensor Correction, or GCDTC. The front end partitions scientific data into $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$46 3D tensor blocks, compresses them into latent variables $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$47, transforms them into a 3D embedding $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$48, quantizes each value by

$$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$49

with $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$50 and $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$51, and entropy-codes the result with Huffman coding. The diffusion decoder is a 2D denoising U-Net conditioned slice-by-slice on latent embeddings $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$52: $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$53 Inference is deterministic because reconstruction starts from zero noise, $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$54. A tensor correction network and a PCA-based error guarantee stage then enforce a maximum distortion target $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$55. On E3SM and S3D, the framework outperforms a standard convolutional autoencoder, achieves at least $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$56 higher compression ratios above $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$57 NRMSE in E3SM, and is competitive with SZ3 on S3D, but decoding is slow: GCDTC reports $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$58, compared with GCAE at $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$59 and SZ at $$\mathbb{E}_{x, c, \boldsymbol{\epsilon}, t} \left[ w_t \left\| \hat{x}_\theta (\alpha_t x + \sigma_t \boldsymbol{\epsilon}, c) - x \right\|_2^2 \right],$$60.

Taken together, these works also delimit the main misconceptions surrounding fusion-centric conditional models. Fusion is not equivalent to direct addition: ControlNet-style feature addition assumes spatial correspondence and is not a general solution for non-aligned conditions (Liu et al., 29 Mar 2026). Fusion is not necessarily gating: slice-wise latent conditioning in GCDTC is conditioning rather than gate-based selection (Lee et al., 18 Feb 2025). Image-space proxy objectives are not reliable substitutes for trajectory-aware guidance: CLIP spherical loss can be nearly orthogonal to the true diffusion guide (Shen et al., 2022). Nor is more fusion always better: increasing the number of approximate MPC steps can accumulate error and worsen divergence, larger guidance weights can amplify small guide differences, multi-condition control can produce conflicts such as slight subject distortion under subject-plus-depth constraints, and excessive soft-mask blur improves realism at the expense of lesion localization (Shen et al., 2022, Liu et al., 29 Mar 2026, Li et al., 25 Jul 2025).

Within this literature, the most defensible interpretation of “fusion surface models” is therefore methodological rather than taxonomic. The term would denote models whose behavior is determined by the interface surface between condition and denoising state: whether that interface is a concatenated latent tensor, a token sequence, a gated modulation path, a future-state gradient, or a slice-wise latent code. The surveyed papers show that this interface governs controllability, convergence, parameter overhead, privacy-oriented deployment, lesion fidelity, and even distortion guarantees, making fusion not an auxiliary detail but a primary design axis of modern conditional diffusion systems.