Generative Anchored Fields (GAF)

• Generative Anchored Fields (GAF) is a framework that factorizes endpoint predictions into a base predictor and a data predictor, yielding an emergent velocity field.
• It leverages transport algebra to enable controllable interpolation, hybrid generation, and semantic morphing through compositional manipulation of endpoint heads.
• GAF employs a training objective with pair loss, residual penalty, and swap loss, achieving competitive performance in image synthesis using ODE integration.

Generative Anchored Fields (GAF) is a generative modeling framework that replaces the usual practice of learning a single time-dependent trajectory predictor with a factorized endpoint-prediction view. Instead of directly learning a score, denoising target, or velocity field, GAF learns two time-conditioned predictors—$J$, which predicts the base or noise endpoint, and $K$, which predicts the data endpoint—and defines the transport field as their difference, $v(x_t,t)=K(x_t,t)-J(x_t,t)$. The framework is designed so that sampling proceeds by integrating this induced field, while compositional control arises from algebraic operations on endpoint heads, a mechanism the paper calls Transport Algebra. In the formulation reported in "Generative Anchored Fields: Controlled Data Generation via Emergent Velocity Fields and Transport Algebra" (Deressa et al., 27 Nov 2025), this architecture is presented as a way to make controllable interpolation, hybrid generation, semantic morphing, and cyclic transport architectural primitives rather than external guidance procedures.

1. Conceptual identity and motivation

GAF is introduced against a background in which diffusion, score-based, flow-matching, and rectified-flow models typically learn a single monolithic time-conditioned predictor. In that family of methods, the predictor may estimate noise, score, or velocity, but control is usually applied externally through mechanisms such as classifier-free guidance, prompt engineering, latent editing, attention manipulation, or auxiliary control networks. GAF instead asks whether it is necessary to learn the trajectory itself, step by step, or whether it is sufficient to learn the origin and destination of the transport path (Deressa et al., 27 Nov 2025).

The framework answers that question by training two endpoint regressors and letting the trajectory field emerge afterwards as

$v = K - J.$

This means that the training target is endpoint accuracy, whereas sampling uses ODE integration over an emergent velocity field. The paper explicitly contrasts this with models that directly supervise a transport dynamic, such as score-based models for $\nabla_x \log p_t(x)$, DDPM/DDIM-style models for $\epsilon$ or related denoising targets, and flow-matching or rectified-flow models for $v_\theta(x_t,t)$ (Deressa et al., 27 Nov 2025).

A central practical claim is that predicting independent endpoints yields modularity. With a shared base predictor $J$ and class-specific $K_n$ heads, each class induces a transport operator

$v_n = K_n - J.$

Because those operators are represented separately, they can be mixed, interpolated, swapped, or chained. The paper treats this as the basis for compositional control and describes the resulting behavior as controllable interpolation, hybrid generation, and semantic morphing (Deressa et al., 27 Nov 2025).

2. Formal model and emergent transport field

The probabilistic setup uses two endpoint distributions. The data endpoint is

$K$0

and the base endpoint is

$K$1

Training samples one data point $K$2 and one Gaussian point $K$3, then constructs the linear bridge

$K$4

The paper states that it does not present an optimal transport coupling; it uses sampled endpoint pairs and a simple linear bridge (Deressa et al., 27 Nov 2025).

A shared trunk $K$5 processes the bridge state and time,

$K$6

or, with optional conditioning,

$K$7

From this shared representation, the model defines the twin endpoint predictors

$K$8

$K$9

with residual terms

$v(x_t,t)=K(x_t,t)-J(x_t,t)$0

The induced velocity field is

$v(x_t,t)=K(x_t,t)-J(x_t,t)$1

Sampling then integrates the transport ODE

$v(x_t,t)=K(x_t,t)-J(x_t,t)$2

The paper characterizes this as an emergent velocity field because the model never directly regresses a velocity target; instead, velocity appears as the disagreement between two separately trained endpoint estimators (Deressa et al., 27 Nov 2025).

The induced field can be expanded as

$v(x_t,t)=K(x_t,t)-J(x_t,t)$3

Letting

$v(x_t,t)=K(x_t,t)-J(x_t,t)$4

the paper analyzes the field as

$v(x_t,t)=K(x_t,t)-J(x_t,t)$5

This decomposition is used to argue that the emergent field carries information about both endpoints. The paper also states that no explicit trajectory supervision is required beyond endpoint construction and linear interpolation (Deressa et al., 27 Nov 2025).

3. Training objective, symmetry constraints, and architecture

The main training loss is the pair loss

$v(x_t,t)=K(x_t,t)-J(x_t,t)$6

This weights noise-endpoint regression more strongly near $v(x_t,t)=K(x_t,t)-J(x_t,t)$7 and data-endpoint regression more strongly near $v(x_t,t)=K(x_t,t)-J(x_t,t)$8 (Deressa et al., 27 Nov 2025).

GAF adds a residual penalty,

$v(x_t,t)=K(x_t,t)-J(x_t,t)$9

motivated by the desired endpoint boundary behavior: at $v = K - J.$0, $v = K - J.$1 so that $v = K - J.$2; at $v = K - J.$3, $v = K - J.$4 so that $v = K - J.$5. The paper states that endpoint-error expansions produce cross terms weighted by $v = K - J.$6, concentrating learning pressure around the midpoint of the bridge (Deressa et al., 27 Nov 2025).

A further regularizer imposes swap antisymmetry. The swap operator is

$v = K - J.$7

Under this transformation,

$v = K - J.$8

Swapped residual predictions are defined by

$v = K - J.$9

The ideal antisymmetry condition is

$\nabla_x \log p_t(x)$0

which implies

$\nabla_x \log p_t(x)$1

With

$\nabla_x \log p_t(x)$2

the swap loss becomes

$\nabla_x \log p_t(x)$3

The full objective is

$\nabla_x \log p_t(x)$4

The paper reports two hyperparameter settings: example values $\nabla_x \log p_t(x)$5 in the general method section, and

$\nabla_x \log p_t(x)$6

in the image-generation experiments (Deressa et al., 27 Nov 2025).

Training samples

$\nabla_x \log p_t(x)$7

forms

$\nabla_x \log p_t(x)$8

evaluates the main losses at $\nabla_x \log p_t(x)$9, and evaluates again at $\epsilon$0 for the swap loss. Architecturally, the model uses a shared Diffusion Transformer (DiT)-style trunk plus multiple heads: one $\epsilon$1 head and one or more $\epsilon$2 heads. The paper states that the trunk is conditioned by timestep and class embedding through adaLN, and that even a single linear projection for heads can already be sufficient, suggesting that the trunk captures most of the necessary information (Deressa et al., 27 Nov 2025).

4. Transport Algebra and compositional generation

Transport Algebra is the paper’s name for algebraic operations over learned endpoint heads $\epsilon$3, or equivalently over the induced velocity fields

$\epsilon$4

Its premise is that, because transports are represented explicitly by separate heads, inference can manipulate those learned components directly (Deressa et al., 27 Nov 2025).

The paper defines three bridge reconfiguration operators: $\epsilon$5

$\epsilon$6

$\epsilon$7

These are used to traverse or reverse bridges without leaving the same endpoint geometry (Deressa et al., 27 Nov 2025).

With a shared $\epsilon$8, single-class transport is

$\epsilon$9

If each modality has its own $v_\theta(x_t,t)$0, the paper also permits

$v_\theta(x_t,t)$1

For cross-modality transport with shared $v_\theta(x_t,t)$2, the paper describes a chained path through the base representation: $v_\theta(x_t,t)$3 summarized as

$v_\theta(x_t,t)$4

Pairwise interpolation in velocity space is

$v_\theta(x_t,t)$5

The multi-class extension is

$v_\theta(x_t,t)$6

and, for three-way mixing,

$v_\theta(x_t,t)$7

The paper presents AFHQ examples involving cat, dog, and wild classes, including pairwise interpolation, cyclic transport such as cat $v_\theta(x_t,t)$8 dog $v_\theta(x_t,t)$9 wild $J$0 cat, and barycentric blending. It explicitly notes, however, that it does not provide a formal closure theorem or guarantees for arbitrary algebraic combinations; the claims are empirical and architectural rather than mathematically proved for all possible operations (Deressa et al., 27 Nov 2025).

5. Sampling, empirical behavior, and limitations

Sampling starts from

$J$1

at $J$2, selects either a class-specific $J$3 head or a blended transport, defines

$J$4

and advances by Euler integration: $J$5 The paper states that inference is deterministic apart from the initial draw from $J$6. It reports $J$7 Euler steps for best-quality evaluation and $J$8 for faster sampling (Deressa et al., 27 Nov 2025).

The abstract reports FID $J$9 on CelebA-HQ $K_n$0, while the body text explicitly tabulates FID $K_n$1 on CelebA-HQ $K_n$2. For CelebA-HQ $K_n$3, the paper compares GAF with Glow $K_n$4, SDE $K_n$5, LSGM $K_n$6, and LDM-4 $K_n$7, and states that GAF is competitive but not state of the art against heavily engineered latent-diffusion baselines (Deressa et al., 27 Nov 2025).

The paper also studies step-count behavior on CelebA-HQ $K_n$8. It reports FID $K_n$9 at 20 steps, FID $v_n = K_n - J.$0 at 40 steps, and a progression from around $v_n = K_n - J.$1 to $v_n = K_n - J.$2 at 250 steps. The interpretation given is that GAF works with relatively few steps but still benefits from longer integration (Deressa et al., 27 Nov 2025).

For cyclic transport, the paper evaluates 3000 random samples on AFHQ through three cycles:

1. $v_n = K_n - J.$3,
2. $v_n = K_n - J.$4,
3. $v_n = K_n - J.$5.

For each pairwise leg it uses

$v_n = K_n - J.$6

with $v_n = K_n - J.$7 taking 10 uniform steps in $v_n = K_n - J.$8. The reported result is LPIPS $v_n = K_n - J.$9 between the initial state and the final state after the full cycle. The paper interprets this as empirical evidence of deterministic reversibility and cycle closure, while also making clear that it does not provide a theorem establishing global invertibility for all states (Deressa et al., 27 Nov 2025).

The stated limitations are correspondingly explicit. The paper notes that sample quality trails the best diffusion models on CelebA-HQ $K$00, that dependence on ODE step count remains nontrivial, that it provides no formal theory of algebraic validity for arbitrary transport combinations, and that its empirical scope is limited to image generation and relatively modest multi-class demonstrations. It also states that the strong cyclic claim is empirical rather than a general invertibility theorem (Deressa et al., 27 Nov 2025).

6. Antecedents and terminological disambiguation

The phrase "Generative Anchored Fields" is specific to the 2025 endpoint-prediction framework, but its use of "anchors" has a longer conceptual background. In "Modeling Gaussian Random Fields by Anchored Inversion and Monte Carlo Sampling" (Zhang, 2011), anchors are defined as linear functionals of a Gaussian random field,

$K$01

and the full field is generated conditionally on latent variables $K$02. In that formulation, the conditional field law remains Gaussian, and anchored inversion is explicitly described as a low-dimensional latent representation from which full spatial fields can be sampled (Zhang, 2011).

"Adaptive Anchored Inversion for Gaussian Random Fields Using Nonlinear Data" (Zhang, 2014) develops the same idea further with

$K$03

a posterior over anchors inferred from nonlinear observations, and a closed-form conditional generator

$K$04

Those papers do not use the term "Generative Anchored Fields," but they present anchors as interpretable latent variables and full-field generation as conditional sampling from an anchor-controlled model. This suggests a historical distinction between classical anchor parameterizations for Gaussian random fields and the later endpoint-factorized transport architecture that officially bears the GAF name (Zhang, 2014).

A separate source of confusion is that the acronym GAF is heavily overloaded. "Generalized Attention Flow" (Azarkhalili et al., 14 Feb 2025) uses GAF for a feature-attribution method for encoder-only Transformers; "GAF: Gaussian Action Field as a Dynamic World Model for Robotic Manipulation" (Chai et al., 17 Jun 2025) uses it for a motion-aware 4D Gaussian world model; "GAF: Gaussian Avatar Reconstruction from Monocular Videos via Multi-view Diffusion" (Tang et al., 2024) uses it for monocular head-avatar reconstruction; and "The heat flow, GAF, and SL(2;R)" (Hall et al., 2023) uses GAF strictly for "Gaussian analytic function." None of those papers define GAF as Generative Anchored Fields. In this sense, the 2025 GAF paper is both a distinct method and part of a broader terminological landscape in which the acronym cannot be interpreted without context (Deressa et al., 27 Nov 2025).