silva_fixed_point_diffusion Reproduction Dossier

timestep-conditioned fixed-point denoising network. This dossier connects the source mechanism to its SILVA implementation, compact evidence, replaceable components, and source-scale route. Existing tests and notebooks remain the executable authority.

Evidence boundary

The mechanism is compact-verified in the package suite. The final source-scale stage remains planned until the cited data, complete optimization budget, checkpoints, and evaluation protocol have actually run.

Identity and Sources

Field	Value
Domain	vision and generation
Task contract	noisy latent, timestep, and optional condition -> denoised latent
Source relation	paper-adaptation
References	[74]
Repositories	https://lukemelas.github.io/fixed-point-diffusion-models/
Editable scale plan	experiments/reproduction/configs/silva_fixed_point_diffusion.json

Governing Equation

The domain-level state contract is

\[ z^\star=T_\theta(z^\star;\mathcal E(x),c),\qquad \widehat y=\mathcal D_\psi(z^\star). \]

The implementation registry specializes it operationally as

z_t_star=F_theta(z_t_star, P(x_t), t); epsilon_hat=Q(z_t_star, x_t, t)

Define the root residual

\[ R_\theta(z;x)=z-T_\theta(z;x). \]

At a regular equilibrium, differentiating \(R_\theta(z^\star;x)=0\) gives

\[ \frac{\partial z^\star}{\partial x} = \left(I-\frac{\partial T_\theta}{\partial z}\right)^{-1} \frac{\partial T_\theta}{\partial x}. \]

This identity explains why the forward residual, the conditioning derivative, and the adjoint linear solve must be diagnosed separately from the task metric.

What Is Preserved

• explicit pre-processing, input projection/injection, timestep-conditioned implicit block, and explicit post-processing
• sequential reverse diffusion with the previous timestep equilibrium reused as the next initialization
• per-timestep iteration allocation and stochastic Jacobian-free backpropagation through sampled unrolled steps

What Can Be Replaced

Each item below is an explicit control rather than an undocumented modification:

• replace pre/injection/equilibrium/post modules or reverse sampler
• change compute allocation, solution reuse, conditioning, or truncated-gradient policy
• latent resolution
• transition width
• iterations per timestep
• number of reverse timesteps

Constructor and Shape Contract

silva_fixed_point_diffusion(channels: 'int', *, preprocessor: 'nn.Module | None' = None, projection: 'nn.Module | None' = None, transition: 'nn.Module | None' = None, postprocessor: 'nn.Module | None' = None, config: 'SolverConfig | None' = None)

The transition must preserve the declared equilibrium-state shape even when the encoder, branch operators, constraints, solver, and readout are replaced. Test the transition by itself before testing the complete root solve.

Progressive Experiment Ladder

1. Equation and tensor contract

Objective: Make the state, conditioning variables, operator, and readout explicit.

Procedure:

• Write and evaluate the family equation: z_t_star=F_theta(z_t_star, P(x_t), t); epsilon_hat=Q(z_t_star, x_t, t)
• Declare every tensor axis, boundary, mask, graph, or physical unit.
• Check the transition output has exactly the same shape as the equilibrium state.

Acceptance checks:

• finite transition values
• shape-preserving state update
• all conditioning variables affect the intended branch

Evidence target: contract-verified.

2. Primitive mechanism reconstruction

Objective: Build the retained source mechanism from replaceable modules.

Procedure:

• explicit pre-processing, input projection/injection, timestep-conditioned implicit block, and explicit post-processing
• sequential reverse diffusion with the previous timestep equilibrium reused as the next initialization
• per-timestep iteration allocation and stochastic Jacobian-free backpropagation through sampled unrolled steps

Acceptance checks:

• primitive modules expose trainable parameters and gradients
• mechanism-specific invariance or constraint check passes
• direct transition evaluation is deterministic under a fixed seed

Evidence target: compact-verified.

3. Public abstraction equivalence

Objective: Verify that the assembled family evaluates the same transition as its primitives.

Procedure:

• Copy the primitive module parameters into the public family constructor.
• Evaluate one transition and one complete equilibrium with identical inputs.
• Compare outputs, residuals, and parameter gradients with declared tolerances.

Acceptance checks:

• transition outputs agree
• equilibrium residual is finite and decreases
• primitive and assembled gradients agree on the compact case

Evidence target: compact-verified.

4. Compact real or analytic task

Objective: Exercise training, evaluation, diagnostics, and serialization end to end.

Procedure:

• make_fixed_point_diffusion_dataset gives seeded latent fields and exact timestep-conditioned targets for allocation, reuse, and stochastic Jacobian-free checks.

Acceptance checks:

• record FID-50K or declared task metric
• record fixed-point block evaluations
• record per-timestep residual
• record sampling wall time
• checkpoint reload reproduces the recorded prediction
• result record contains data and configuration fingerprints

Evidence target: compact-verified.

5. Official-data subset

Objective: Validate the complete source data path before spending the full budget.

Procedure:

• Acquire one declared image task and reproduce its resize/crop, latent encoder, diffusion schedule, split, and evaluation preprocessing.
• Freeze preprocessing, split logic, metric code, and checkpoint format.
• Run a deterministic subset large enough to expose batching and memory failures.

Acceptance checks:

• dataset receipt and checksum are stored
• resume and evaluation paths reproduce the same subset metric
• memory and runtime are measured rather than estimated

Evidence target: subset-verified.

6. Source-scale reproduction or declared extension

Objective: Run the cited protocol, or change it explicitly as a SILVA extension.

Procedure:

• Acquire one declared image task and reproduce its resize/crop, latent encoder, diffusion schedule, split, and evaluation preprocessing.
• Configure explicit pre/projection/post blocks around the timestep-conditioned fixed point, then reproduce the source per-timestep iteration allocation and state reuse.
• Train with the declared stochastic Jacobian-free schedule and compare FID-50K, block evaluations, latency, memory, and residuals at equal sampling budgets.
• source latent encoder, image preprocessing, diffusion schedule, task split, and pretrained or jointly trained components
• reported timestep allocation, stochastic backward sampling, optimizer, precision, checkpoints, and generation budget
• FID-50K or task metric, block evaluations, equilibrium residual, wall time, memory, and source baselines

Acceptance checks:

• all required artifacts are archived
• reported metrics use the cited evaluation protocol
• every architectural or training deviation is listed
• claims match the achieved evidence status

Evidence target: planned.

Data, Access, and Storage

Candidate datasets:

• ImageNet 256x256 latent diffusion
• FFHQ, CelebA-HQ, or LSUN Church when configured from a matching source protocol
• compact synthetic latent denoising trajectories

Authoritative routes:

• https://arxiv.org/abs/2401.08741
• https://openaccess.thecvf.com/content/CVPR2024/html/Bai_Fixed-Point_Diffusion_Models_CVPR_2024_paper.html

Access obligations:

• ImageNet requires registration and its stated access terms; face and scene datasets each retain their own licenses and acquisition routes.
• Store dataset checksums separately from latent-encoder, diffusion-schedule, and checkpoint revisions so a source-scale claim is auditable.

Storage planning:

• Budget raw images, encoded latents, checkpoints, optimizer state, and generated evaluation samples separately.
• A standard FID-50K evaluation alone stores 50,000 decoded samples; latent trajectory caches scale again with timesteps and retained fixed-point states.

Preprocessing record:

• record dataset version, split, normalization, shape convention, and seed
• preserve masks, graph indices, boundaries, or physical units required by the domain

Metrics and Current Evidence

Required metrics:

• FID-50K or declared task metric
• fixed-point block evaluations
• per-timestep residual
• sampling wall time
• peak memory

This family is verified through its listed mechanism tests and executed notebook. It is not included in a same-task comparison when another family does not share its input, state, output, and loss contract. The absence of a comparison row is therefore a scope decision, not missing implementation evidence.

Executed notebook paths:

• notebooks/package_api/35_silva_fixed_point_diffusion.ipynb

Mechanism tests:

• tests/test_emerging_equilibria.py

Compact Defaults

Option	Value
tier	'smoke'
config	SolverConfig(solver='anderson', max_iter=12, tol=1e-05, alpha=1.0, history=3, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=1, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=20, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Full Defaults

Option	Value
tier	'full'
config	SolverConfig(solver='anderson', max_iter=60, tol=1e-05, alpha=1.0, history=6, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=1, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=80, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Defaults establish a starting budget; the cited source protocol takes precedence whenever reproduction is the claim.

Source-Scale Checklist

• Acquire one declared image task and reproduce its resize/crop, latent encoder, diffusion schedule, split, and evaluation preprocessing.
• Configure explicit pre/projection/post blocks around the timestep-conditioned fixed point, then reproduce the source per-timestep iteration allocation and state reuse.
• Train with the declared stochastic Jacobian-free schedule and compare FID-50K, block evaluations, latency, memory, and residuals at equal sampling budgets.

Benchmark-specific requirements:

• source latent encoder, image preprocessing, diffusion schedule, task split, and pretrained or jointly trained components
• reported timestep allocation, stochastic backward sampling, optimizer, precision, checkpoints, and generation budget
• FID-50K or task metric, block evaluations, equilibrium residual, wall time, memory, and source baselines

Required archived artifacts:

• machine-readable model and solver configuration
• dataset receipt with source revision, split, license, and checksum
• preprocessing and normalization record
• seeded training and evaluation log
• checkpoint and optimizer-resume state for trained experiments
• task metrics and equilibrium diagnostics in a machine-readable result
• runtime, peak-memory, device, precision, and dependency record
• declared deviations from the cited protocol

Reporting Rule

Report the achieved evidence status, not the intended one. A compact or subset run may validate the implementation and data path, but only a completed cited protocol supports a source-scale reproduction statement. Modified operators are valuable SILVA extensions when every deviation is named and measured.

Where to Go Next

Question	Page
Where are all family dossiers?	Family Dossier Index
How is a custom family assembled?	Advanced Extension Handbook
How are experiment stages represented in the API?	Research-Depth API
Which lab inspects every dossier?	Family Dossier Lab