Generative Anchored Fields: Endpoint Learning for Generative Modeling

Deressa Wodajo Deressa; Hannes Mareen; Peter Lambert; Glenn Van Wallendael

cs.LGNov 27, 2025

Generative Anchored Fields: Endpoint Learning for Generative Modeling

Authors: Deressa Wodajo Deressa, Hannes Mareen, Peter Lambert, Glenn Van Wallendael

Organizations: Ghent University – imec, IDLab, Department of Electronics and Information Systems, Gent, Belgium

Abstract

We present Generative Anchored Fields (GAF), a generative model that learns independent endpoint predictors, $J$ (noise) and $K$ (data), from any point on a linear bridge. Instead of learning the local path along a trajectory, as in diffusion or flow-based models, GAF is trained to identify the endpoint coordinates of a bridge state directly via an endpoint regression objective. Central to our approach is the decoupling of time-dependent feature learning from endpoint prediction. Unlike the transport-based models that learn a monolithic time-conditioned predictor that maps noise to data via a unified field, GAF uses a shared trunk, $Φ$ , which produces a time-conditioned feature bank and a pair of linear predictors, $J$ and $K$ . This separation of feature learning from prediction enables Transport Algebra: algebraic operations on multiple $J$ and $K$ heads for compositional sampling. Additionally, with class-specific $K_n$ heads, where $n\in\{1,...,N\}$ for $N$ classes, GAF defines directed transport maps between a shared base noise distribution and multiple data domains, allowing controllable interpolation, multi-class composition, and semantic editing. This is achieved either directly on the predicted data coordinates ( $K$ ) using (i) Iterative Endpoint Refinement (IER), a novel sampler that achieves high-quality generation in $5-8$ forward steps, or (ii) on the derived velocity field $v=K-J$ . GAF achieves strong sample quality with an FID of 2.25 on ImageNet $256\times256$ while treating compositional generation as an architectural primitive. Code available at https://github.com/IDLabMedia/GAF

Figures & tables

Figure 1: GAF enables compositional generation through algebraic combinations of endpoint predictors ( $K$ ) and derived velocity fields ( $v=K-J$ ). (a) Endpoint blending across AFHQ classes via weighted $K$ heads. (b) Spatial scene composition on ImageNet via masked velocity fields, where $m_{1}$ and $m_{2}$ are spatial masks and $v_{p},v_{g},v_{c}$ are class-specific velocity fields for palace, gondola, and canola.

Figure 2: High-level overview of GAF architecture and its data flow.

Figure 3: Visualization of the swap operator

\mathcal{S}

acting on a bridge. (A) The initial configuration of the bridge ( 1 ). Subfigures (B) , (C) , and (D) illustrate the results of applying the operations described in equations ( 16 ), ( 17 ), and ( 11 ), respectively.

Panel of Figure 4 — Figure 4: Multi-class generation across two datasets. (a) AFHQ with three classes; (b) ImageNet with $1{,}000$ classes.

Panel of Figure 5 — Figure 5: Effect of the number of forward steps on sample quality using CelebA-HQ ( $256\times 256$ px).

Method	Params(M)	Training Steps	FID	FID(CFG) $\downarrow$	IS $\uparrow$	Prec. $\uparrow$	Rec. $\uparrow$
DiT-XL/2	675	7M	9.6	2.27	278.24	0.67	0.67
SiT-XL/2	675	7M	8.30	2.06	277.50	0.68	0.67
GAF-XL (ours)	694	7M	7.85	2.25	338.00	0.77	0.63
GAF-XL (retrunked, ours)	694	150k	7.51	2.63	338.00	0.77	0.62

Table 1: FID results on ImageNet

256\times 256

, both without and with classifier-free guidance (CFG). IS, precision, and recall are reported for the CFG settings.

Figure 6: Spatial composition on AFHQ using masked velocity blending.

Figure 7: Spatial composition on AFHQ using masked $K$ head blending (8-step IER).

Figure 8: Spatial composition on ImageNet using masked velocity blending.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9: Three

J/K

pairing topologies: (A) one-to-one, (B) star (one-to-many), and (C) clustered. Each topology shows different ways to map the noise endpoint

J

to a data endpoint

K

Figure 10: GAF architecture for image generation. The DiT-Trunk block follows the architecture of [ 22 ] . GAF replaces DiT’s single output projection with two independent heads,

H_{J}

and

H_{K}

(see Section 3.1 ), each producing endpoint residuals that are unpatchified separately and combined with scaled input to form the twins

J

and

K

	CIFAR-10	CelebA-HQ	AFHQ	ImageNet	ImageNet
Training	scratch	scratch	scratch	retrunked	scratch
Architecture	DiT-B/2	DiT-XL/2	DiT-XL/2	DiT-XL/2	DiT-XL/2
VAE	–	SD 1.5	SD 1.5	SD 1.5	SD 1.5
Batch size	256	32	32	128	256
Iterations	100k	100k	100k	150k	7M
Learning rate	2e-4	1e-4	1e-4	1e-4	1e-4

Appendix

Table 4: Training configurations for each dataset.

Figure 11: Single-class generated samples ( $256\times 256$ px) from GAF trained on the ImageNet school bus class.

Panel of Figure 12 — Figure 12: Pairwise interpolation samples ( $512\times 512$ px) from GAF trained with one $J$ and three independent $K$ heads on the AFHQ dataset.

Figure 13: Cyclic Transport Analysis. We visualize the cyclic transport maps across classes (Top images above the error graphs) and their corresponding convergence error rates (Bottom figures showing the error graphs across multi-class generation). The trajectories correspond to cyclic transport along the cycle $v_{i\to j\to k\to i}$ across 4 randomly chosen semantic cycles. The error difference between the first and the last image decreases monotonically, reaching machine precision ( $\approx 10^{-16}$ ) at $N=1,000$ , where $N$ is the number of forward steps.

Figure 14: Continuous cyclic interpolation transport preserving identity across three classes. Each row shows a complete cycle starting from one class (left) to two interpolated classes, and finally returning to the initial class. Distinctive features (coloring, facial marking) are semantically preserved throughout each cycle while class-specific structure transforms.

Figure 15: Spatial mixing results. Each column shows base images (left sub-column) and the corresponding generations (right sub-column) where structure is transferred according to one of four spatial masks: (a) face, (b) eyes, (c) nose, (d) ears.

Panel of Figure 16 — Figure 16: Barycentric Transport Algebra. Visualization of three-way velocity blending across the Dog, Cat, and Wild classes. (a and b) $7\times 7$ grids showing the semantic simplex. Corners represent pure classes; interior points show linear interpolations of the vector fields. (c) The corresponding weight map.

Figure 17: Effect of swap and residual losses under cyclic transport ( $z_{y}\rightarrow K_{0}\xrightarrow{J}K_{1}\xrightarrow{J}K_{2}\xrightarrow{J}K_{0}$ ).

Figure 18: Image-to-Image translation (8-step IER). (a) Visualization of the full cross-modality translation path ( $K_{\text{cat}}\xrightarrow{J}K_{\text{dog}}$ ). (b) Base images from the held-out AFHQ dataset. (c) Translations generated by GAF trained only with pair loss ( $\mathcal{L}_{\text{pair}}$ ), which exhibit significant drift and local artifacts. (d) Translations utilizing the full loss objective, demonstrating that the residual and swap losses are crucial for stabilizing the latent space and preserving semantic structure.

Figure 19: GAF comparison with Rectified Flow under identical compositions. The first three images in the grid are the base images, while the fourth is the weighted blend.

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