SILVA Diffusion Restoration Equilibrium¶
Solve all restoration-time variables jointly while projecting observed pixels at every equilibrium transition. This lab adapts the cited mechanism into explicit SILVA components [[108]], runs a deterministic compact check, and separates that evidence from a source-scale reproduction claim.
Numbered literature: [1], [4], [108]. Each number opens the complete citation and its primary external source.
from pathlib import Path
import sys
root = Path.cwd()
while root != root.parent and not (root / "src" / "silva_networks").exists():
root = root.parent
if not (root / "src" / "silva_networks").exists():
root = Path("/content/silva-networks")
sys.path.insert(0, str(root / "src"))
import matplotlib.pyplot as plt
import torch
from torch import nn
from silva_networks import SolverConfig, silva_family_experiment_protocol
plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
torch.manual_seed(121)
config = SolverConfig(
solver="picard",
max_iter=30,
tol=1e-6,
backward_mode="unrolled",
anderson_batch_dims=1,
return_best=True,
)
1. Mechanism and Derivation¶
Stack the restoration trajectory as $Z=(x_0,\ldots,x_{T-1})$. For $t\geq1$,
$$ \widetilde x_t=(1-\eta)D_\theta(x_{t-1})+\eta x_t, \qquad x_t' = M\odot y+(1-M)\odot\widetilde x_t. $$
The multivariate fixed point is $Z^\star=F_\theta(Z^\star;y,M,x_0)$. Hard projection makes observed-pixel consistency exact throughout the solved trajectory; the denoiser controls only unobserved content.
2. SILVA State and Shape Contract¶
Measurement, mask, and noise: (batch, channels, height, width). Joint state: (batch, timesteps, channels, height, width). Output: final trajectory slice.
The transition remains a named callable, the numerical method is selected by
SolverConfig, and the result exposes both the solved state and solver record.
This makes architecture equivalence, numerical equivalence, and task quality
three separate questions.
from silva_networks import SILVADiffusionRestorationEquilibrium
grid = torch.linspace(-1, 1, 20)
truth = torch.exp(-7 * (grid[:, None] ** 2 + grid[None, :] ** 2)).unsqueeze(0).unsqueeze(0)
mask = torch.zeros_like(truth)
mask[..., ::3, :] = 1
measurement = (mask * truth).requires_grad_()
initial_noise = 0.15 * torch.randn_like(truth)
model = SILVADiffusionRestorationEquilibrium(1, 5, eta=0.15, config=config)
result = model(measurement, mask=mask, initial_noise=initial_noise, return_result=True)
observed_error = ((result.state[:, 1:] - measurement.unsqueeze(1)) * mask.unsqueeze(1)).abs().max()
result.output.square().mean().backward()
summary = {
"trajectory_shape": tuple(result.state.shape),
"observed_pixel_error": float(observed_error.detach()),
"residual": result.solver_result.residual,
"measurement_grad_norm": float(measurement.grad.norm()),
}
summary
{'trajectory_shape': (1, 5, 1, 20, 20),
'observed_pixel_error': 0.0,
'residual': 3.8420688497353694e-07,
'measurement_grad_norm': 0.013004198670387268}
3. Read the Compact Evidence¶
The preceding output is a measured contract check: shapes, constraints, residuals, and gradients were produced by this notebook. It does not imply that the cited source benchmark has been reproduced. The figure below makes one family-specific state or diagnostic visible.
fig, axes = plt.subplots(1, 3, figsize=(9, 2.8))
for axis, image, title in zip(
axes,
[truth[0, 0], measurement[0, 0].detach(), result.output[0, 0].detach()],
["compact truth", "observed pixels", "joint equilibrium output"],
):
axis.imshow(image, cmap="magma")
axis.set_title(title)
axis.axis("off")
fig.tight_layout()
plt.show()
4. Inspect and Replace the Internals¶
Replace denoiser with a trained diffusion prior or restoration network. Supply the true observation projection through the mask or subclass the transition for a general sensing operator while preserving the joint trajectory state.
The following inventory is deliberately mechanical: an advanced experiment can replace a child module without changing the solver or reporting contract.
print("trainable parameters:", sum(p.numel() for p in model.parameters() if p.requires_grad))
for name, child in model.named_children():
print(f"{name:24s} -> {child.__class__.__name__}")
trainable parameters: 305 denoiser -> SILVAResidualImagePrior
5. Compact, Workstation, and Source Scale¶
Full DeqIR-style studies require the pretrained diffusion model, degradation operators, noise schedule, datasets, sampling settings, baselines, FID/LPIPS/PSNR/SSIM protocol, and runtime comparison. The compact lab establishes joint-state and hard-projection contracts only.
SILVA stores all three execution routes in the family protocol. Resource figures are planning ranges; measured hardware, runtime, peak memory, data revision, split, seed, and deviations belong in the completed result record.
protocol = silva_family_experiment_protocol("silva_diffusion_restoration_equilibrium")
for tier in protocol.tiers:
print(f"{tier.tier:11s} | {tier.dataset.name} | {tier.dataset.expected_storage}")
print(" source:", tier.dataset.source_url)
print(" split: ", tier.dataset.split)
print(" run: ", tier.command)
smoke | generated masked-image trajectory | less than 100 MB source: generated://silva/diffusion-restoration split: fixed image, mask, schedule, and seed run: python experiments/reproduction/run_family_protocol.py --family silva_diffusion_restoration_equilibrium --tier smoke --work-dir runs/silva_diffusion_restoration_equilibrium/smoke workstation | DeqIR source image subset | 10-100 GB source: https://github.com/caojiezhang/DeqIR split: recorded images, degradation, and checkpoint run: python experiments/reproduction/run_family_protocol.py --family silva_diffusion_restoration_equilibrium --tier workstation --work-dir runs/silva_diffusion_restoration_equilibrium/workstation full | DeqIR restoration benchmarks | 100 GB to multiple TB source: https://github.com/caojiezhang/DeqIR split: source degradation, schedule, initialization, and metrics run: python experiments/reproduction/run_family_protocol.py --family silva_diffusion_restoration_equilibrium --tier full --work-dir runs/silva_diffusion_restoration_equilibrium/full
6. Reproduction Checklist¶
Before labeling a result as source-scale reproduced, preserve the cited equation and architecture choices, use the declared source data and split, match preprocessing and evaluation, run the required seeds, and report task metrics beside equilibrium residuals, iterations, failures, runtime, and peak memory. Compact and subset runs remain valuable, but keep their evidence level explicit.
7. Build the Next Variant¶
- Replace one named component and keep its tensor contract fixed.
- Verify the transition on a deterministic fixture before solving it.
- Compare finite iteration and converged outputs at the same weights.
- Add a task loss only after constraints, invariances, and gradients pass.
- Scale the data and architecture independently so the cause of each change is visible.
- Record the exact source relation: reproduced, adapted, or newly extended.
From 70 Silva Diffusion Restoration Equilibrium to a Custom SILVA Family¶
The construction in this notebook can be separated into the universal conditioned-equilibrium contract
$$ z_0=I_\eta(x),\qquad z^\star=T_\theta(z^\star,x),\qquad \widehat y=Q_\psi(z^\star). $$
For this topic:
| Part | Concrete interpretation |
|---|---|
| Equilibrium state | a joint trajectory or equilibrium token/image state |
| Condition | noise, timestep, condition, one-time injection, or teacher target |
| Repeated computation | a tied denoising or injected-attention transition |
| Required invariants | trajectory ordering, token/image shape, conditioning, and deterministic noise |
| Replaceable components | patch/noise source, injection blocks, transition, decoder, schedule, and solver |
The initializer and source path are evaluated outside or alongside the root solve. Only the state-preserving transition is repeated. Replacing an internal architecture does not change this equation, provided the transition still maps the same state space into itself.
import torch as silva_extension_torch
from torch import nn as silva_extension_nn
from silva_networks import (
SILVAConditionedEquilibrium,
SILVAZeroInitializer,
SolverConfig,
validate_silva_transition,
)
class NotebookExtensionTransition(silva_extension_nn.Module):
def __init__(self, condition_dim=2, state_dim=3):
super().__init__()
self.source = silva_extension_nn.Linear(condition_dim, state_dim)
self.state_field = silva_extension_nn.Sequential(
silva_extension_nn.Linear(state_dim, 2 * state_dim),
silva_extension_nn.Tanh(),
silva_extension_nn.Linear(2 * state_dim, state_dim),
)
def forward(self, state, condition):
return silva_extension_torch.tanh(
self.source(condition) + 0.15 * self.state_field(state)
)
silva_extension_torch.manual_seed(610)
notebook_condition = silva_extension_torch.linspace(-1.0, 1.0, 8).reshape(4, 2)
notebook_state0 = silva_extension_torch.zeros(4, 3)
notebook_transition = NotebookExtensionTransition()
notebook_report = validate_silva_transition(
notebook_transition,
notebook_state0,
notebook_condition,
)
assert notebook_report.valid
with silva_extension_torch.no_grad():
notebook_reference_step = silva_extension_torch.tanh(
notebook_transition.source(notebook_condition)
+ 0.15 * notebook_transition.state_field(notebook_state0)
)
silva_extension_torch.testing.assert_close(
notebook_transition(notebook_state0, notebook_condition),
notebook_reference_step,
)
notebook_custom_model = SILVAConditionedEquilibrium(
notebook_transition,
SILVAZeroInitializer(3),
readout=silva_extension_nn.Linear(3, 1),
config=SolverConfig(
solver="picard",
max_iter=40,
tol=1e-7,
backward_mode="implicit",
backward_solver="gmres",
anderson_batch_dims=1,
),
)
notebook_custom_result = notebook_custom_model(
notebook_condition,
return_result=True,
)
assert notebook_custom_result.output.shape == (4, 1)
assert notebook_custom_result.solver_result.residual < 1e-5
notebook_custom_result.output.square().mean().backward()
assert all(
parameter.grad is not None and silva_extension_torch.isfinite(parameter.grad).all()
for parameter in notebook_custom_model.parameters()
)
print("custom transition:", notebook_report)
print("equilibrium residual:", notebook_custom_result.solver_result.residual)
custom transition: SILVATransitionReport(state_shape=(4, 3), output_shape=(4, 3), preserves_shape=True, preserves_device=True, preserves_dtype=True, finite=True, differentiable=True, parameter_count=54) equilibrium residual: 5.960464477539063e-08
Numerical Equivalence, Compact Reproduction, and Scale¶
Before training, compare one packaged transition with an independently written update:
$$ e_{\mathrm{step}} =\frac{\|T_\theta(z,x)-T_{\mathrm{ref}}(z,x)\|_2} {\|T_{\mathrm{ref}}(z,x)\|_2+\varepsilon}. $$
After solving, report the fixed-point residual separately:
$$ e_{\mathrm{fp}} =\frac{\|T_\theta(z^\star,x)-z^\star\|_2} {\|z^\star\|_2+\varepsilon}. $$
For this notebook, a compact reproduction must declare and assert teacher error, reconstruction/generation metric, and equilibrium residual. A full experiment must additionally record the source dataset version and split, preprocessing, architecture widths, solver and optimizer schedules, random seeds, baseline configuration, checkpoints, and every deviation from the cited protocol.
The principal scaling axes are image size, token count, hidden width, heads, and sampling schedule. Increase one axis at a time, retain the compact deterministic case as a regression test, and record task error, domain-specific residual, forward residual, backward linear residual, memory use, and runtime independently.
Extension Exercises¶
- Replace one component from this notebook while preserving its state and domain invariants.
- Write the replacement first as an independent reference function, then as a module, and assert one-step equivalence.
- Compare two solver configurations on the identical trained transition.
- Add a compact baseline and a predeclared metric threshold.
- Create a full-scale configuration without weakening the compact tests.
The complete authoring protocol is documented in Extending SILVA.
notebook_reproduction_record = {
"notebook": '70_silva_diffusion_restoration_equilibrium.ipynb',
"state": 'a joint trajectory or equilibrium token/image state',
"condition": 'noise, timestep, condition, one-time injection, or teacher target',
"transition": 'a tied denoising or injected-attention transition',
"invariants": 'trajectory ordering, token/image shape, conditioning, and deterministic noise',
"compact_metric": 'teacher error, reconstruction/generation metric, and equilibrium residual',
"scale_axis": 'image size, token count, hidden width, heads, and sampling schedule',
}
assert all(notebook_reproduction_record.values())
notebook_reproduction_record
{'notebook': '70_silva_diffusion_restoration_equilibrium.ipynb',
'state': 'a joint trajectory or equilibrium token/image state',
'condition': 'noise, timestep, condition, one-time injection, or teacher target',
'transition': 'a tied denoising or injected-attention transition',
'invariants': 'trajectory ordering, token/image shape, conditioning, and deterministic noise',
'compact_metric': 'teacher error, reconstruction/generation metric, and equilibrium residual',
'scale_axis': 'image size, token count, hidden width, heads, and sampling schedule'}
Worked Convergence and Sensitivity Study¶
The preceding example demonstrates one configured solve. This additional study changes the refinement feedback factor while keeping the source fixed, so solver effort and implicit sensitivity can be read separately from task behavior. Locally, one eigendirection of a nonlinear transition can be represented by
$$ z_{k+1} = \rho z_k + u, \qquad 0 \leq \rho < 1. $$
Its equilibrium is
$$ z^\star = \frac{u}{1-\rho}. $$
Subtracting the fixed-point equation from the iteration gives the exact error recursion
$$ e_{k+1} = \rho e_k, \qquad |e_k| = \rho^k |e_0|. $$
For a requested absolute tolerance $\tau$, the idealized iteration estimate is
$$ k \geq \frac{\log(\tau/|e_0|)}{\log \rho}. $$
The same factor controls sensitivity. Differentiating the equilibrium with respect to the source gives
$$ \frac{\partial z^\star}{\partial u} =\frac{1}{1-\rho}. $$
Thus a transition can remain contractive while becoming expensive and highly sensitive as $\rho$ approaches one. The table and figure below measure this effect rather than merely stating it. They provide a reference envelope for the notebook's actual state, a joint trajectory or equilibrium token/image state, and its repeated map, a tied denoising or injected-attention transition. The scalar study does not replace the domain model; it supplies a result whose convergence rate and derivative are known exactly, so the same reporting code can be trusted before it is applied to the larger transition.
import math as silva_deepening_math
import torch as silva_deepening_torch
silva_deepening_rates = (0.20, 0.45, 0.70, 0.85)
silva_deepening_source = 0.35
silva_deepening_tolerance = 1e-8
silva_deepening_histories = {}
silva_deepening_rows = []
for silva_deepening_rho in silva_deepening_rates:
silva_deepening_state = silva_deepening_torch.tensor(0.0)
silva_deepening_exact = silva_deepening_source / (1.0 - silva_deepening_rho)
silva_deepening_history = []
for silva_deepening_iteration in range(1, 241):
silva_deepening_next = (
silva_deepening_rho * silva_deepening_state + silva_deepening_source
)
silva_deepening_residual = abs(
float(silva_deepening_next - silva_deepening_state)
)
silva_deepening_history.append(silva_deepening_residual)
silva_deepening_state = silva_deepening_next
if silva_deepening_residual < silva_deepening_tolerance:
break
silva_deepening_u = silva_deepening_torch.tensor(
silva_deepening_source, requires_grad=True
)
silva_deepening_solution = silva_deepening_u / (1.0 - silva_deepening_rho)
silva_deepening_solution.backward()
silva_deepening_expected_sensitivity = 1.0 / (1.0 - silva_deepening_rho)
silva_deepening_gradient_error = abs(
float(silva_deepening_u.grad) - silva_deepening_expected_sensitivity
)
silva_deepening_histories[silva_deepening_rho] = silva_deepening_history
silva_deepening_rows.append(
(
silva_deepening_rho,
silva_deepening_iteration,
silva_deepening_history[-1],
abs(float(silva_deepening_state) - silva_deepening_exact),
float(silva_deepening_u.grad),
silva_deepening_gradient_error,
)
)
print('refinement feedback factor')
print("rho | iterations | final residual | exact-state error | sensitivity | gradient error")
for silva_deepening_row in silva_deepening_rows:
print(
f"{silva_deepening_row[0]:.2f} | {silva_deepening_row[1]:3d} | "
f"{silva_deepening_row[2]:.3e} | {silva_deepening_row[3]:.3e} | "
f"{silva_deepening_row[4]:.4f} | {silva_deepening_row[5]:.3e}"
)
assert all(row[2] < silva_deepening_tolerance for row in silva_deepening_rows)
assert all(row[3] < 1e-6 for row in silva_deepening_rows)
assert all(row[5] < 1e-6 for row in silva_deepening_rows)
refinement feedback factor rho | iterations | final residual | exact-state error | sensitivity | gradient error 0.20 | 12 | 0.000e+00 | 5.551e-17 | 1.2500 | 0.000e+00 0.45 | 23 | 0.000e+00 | 1.084e-08 | 1.8182 | 6.502e-08 0.70 | 45 | 0.000e+00 | 1.589e-07 | 3.3333 | 7.947e-08 0.85 | 93 | 0.000e+00 | 5.563e-07 | 6.6667 | 1.589e-07
import matplotlib.pyplot as silva_deepening_plt
silva_deepening_plt.rcParams.update({"figure.dpi": 300, "savefig.dpi": 300})
silva_deepening_figure, silva_deepening_axes = silva_deepening_plt.subplots(
1, 2, figsize=(8.6, 3.2)
)
for silva_deepening_rho, silva_deepening_history in silva_deepening_histories.items():
silva_deepening_axes[0].semilogy(
range(1, len(silva_deepening_history) + 1),
silva_deepening_history,
marker="o",
markersize=2,
linewidth=1.2,
label=f"rho={silva_deepening_rho:.2f}",
)
silva_deepening_axes[0].axhline(
silva_deepening_tolerance, color="black", linestyle="--", linewidth=0.9
)
silva_deepening_axes[0].set_xlabel("iteration")
silva_deepening_axes[0].set_ylabel("absolute residual")
silva_deepening_axes[0].set_title("Residual trajectories")
silva_deepening_axes[0].legend(fontsize=7)
silva_deepening_axes[1].plot(
[row[0] for row in silva_deepening_rows],
[row[1] for row in silva_deepening_rows],
marker="o",
label="iterations",
)
silva_deepening_sensitivity_axis = silva_deepening_axes[1].twinx()
silva_deepening_sensitivity_axis.plot(
[row[0] for row in silva_deepening_rows],
[row[4] for row in silva_deepening_rows],
color="tab:red",
marker="s",
label="sensitivity",
)
silva_deepening_axes[1].set_xlabel('refinement feedback factor')
silva_deepening_axes[1].set_ylabel("iterations")
silva_deepening_sensitivity_axis.set_ylabel("implicit sensitivity", color="tab:red")
silva_deepening_axes[1].set_title("Cost and sensitivity")
silva_deepening_figure.tight_layout()
silva_deepening_plt.show()
Reading and Extending the Result¶
The measured residual curves become flatter as the refinement feedback factor increases. The iteration count and the exact sensitivity rise together, but they answer different questions: iterations measure numerical work, while sensitivity describes how strongly the equilibrium reacts to the source. The gradient-error column verifies the differentiation path against the analytic derivative.
Apply the same separation to this notebook's full model:
| Report | Notebook-specific interpretation |
|---|---|
| Task evidence | teacher error, reconstruction/generation metric, and equilibrium residual |
| Forward residual | Re-evaluate the complete transition at the returned state |
| Empirical rate | Compare consecutive residuals only after the transient regime |
| Backward residual | Record the linear-adjoint stopping value independently |
| Sensitivity | Perturb one declared source field while preserving all other inputs |
| Structural checks | trajectory ordering, token/image shape, conditioning, and deterministic noise |
| Scale sweep | Change one of image size, token count, hidden width, heads, and sampling schedule at a time |
A richer experiment should now repeat the sweep with at least two forward solvers, two tolerances, and multiple seeds. Keep model parameters and data identical when comparing solvers. Then change one architecture or data-scale axis, retain the compact analytic study as a regression test, and report task quality, residuals, iterations, runtime, memory, gradient norms, and failed convergence cases together.