SILVA Snapshot Compressive Equilibrium¶
Reconstruct a video cube from one coded snapshot while keeping the analytic projection and learned spatiotemporal prior independently replaceable. This lab adapts the cited mechanism into explicit SILVA components [[104]], runs a deterministic compact check, and separates that evidence from a source-scale reproduction claim.
Numbered literature: [1], [4], [104]. 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¶
Snapshot compressive imaging observes
$$y=\sum_{t=1}^{T}M_t\odot x_t+\varepsilon.$$
For $\mathcal A(x)=\sum_tM_t\odot x_t$, the data correction is
$$ \Pi_y(x)=x+\eta\,M\odot \frac{y-\mathcal A(x)}{\sum_t M_t^2+\epsilon}. $$
SILVA solves $x^\star=\Pi_y(x^\star)+\lambda D_\theta(\Pi_y(x^\star))$. With $\eta=1$ and $\lambda=0$, the compact check isolates exact measurement consistency.
2. SILVA State and Shape Contract¶
Snapshot: (batch, height, width). Masks and video state: (batch, frames, height, width). The static measure method implements the sensing equation.
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 SILVASnapshotCompressiveEquilibrium
height = width = 20
frames = 4
yy, xx = torch.meshgrid(torch.linspace(-1, 1, height), torch.linspace(-1, 1, width), indexing="ij")
video = torch.stack([torch.exp(-12 * ((xx - 0.3 * t) ** 2 + yy ** 2)) for t in torch.linspace(-1, 1, frames)])
video = video.unsqueeze(0)
masks = (torch.rand_like(video) > 0.35).float().clamp_min(0.15)
measurement = SILVASnapshotCompressiveEquilibrium.measure(video, masks).requires_grad_()
model = SILVASnapshotCompressiveEquilibrium(
frames, prior=nn.Identity(), step_size=1.0, prior_scale=0.0, config=config
)
result = model(measurement, masks, return_result=True)
remeasured = model.measure(result.output, masks)
result.output.square().mean().backward()
summary = {
"video_shape": tuple(result.output.shape),
"measurement_error": float((remeasured - measurement).abs().max().detach()),
"residual": result.solver_result.residual,
"measurement_grad_norm": float(measurement.grad.norm()),
}
summary
{'video_shape': (1, 4, 20, 20),
'measurement_error': 1.1920928955078125e-07,
'residual': 1.9419381658281054e-07,
'measurement_grad_norm': 0.004739257972687483}
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))
axes[0].imshow(measurement[0].detach(), cmap="magma")
axes[0].set_title("coded snapshot")
axes[1].imshow(video[0, 1], cmap="magma")
axes[1].set_title("source frame")
axes[2].imshow(result.output[0, 1].detach(), cmap="magma")
axes[2].set_title("equilibrium frame")
for axis in axes:
axis.axis("off")
fig.tight_layout()
plt.show()
4. Inspect and Replace the Internals¶
Replace prior with a 3-D convolutional, recurrent, transformer, or operator prior. Keep measure and the analytic correction tied to the calibrated masks; alter them only when the camera model changes.
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: 0 prior -> Identity
5. Compact, Workstation, and Source Scale¶
Full experiments require the article's coded-mask construction, benchmark videos, frame counts, noise assumptions, crop policy, training schedule, and PSNR/SSIM evaluation. The compact cell verifies sensing and gradients rather than video quality.
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_snapshot_compressive_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 coded video | less than 100 MB source: generated://silva/snapshot-compressive split: fixed frames, masks, and seed run: python experiments/reproduction/run_family_protocol.py --family silva_snapshot_compressive_equilibrium --tier smoke --work-dir runs/silva_snapshot_compressive_equilibrium/smoke workstation | DEQSCI benchmark subset | 1-20 GB source: https://github.com/IndigoPurple/DEQSCI split: recorded sequence, masks, and frame group run: python experiments/reproduction/run_family_protocol.py --family silva_snapshot_compressive_equilibrium --tier workstation --work-dir runs/silva_snapshot_compressive_equilibrium/workstation full | DEQSCI video benchmarks and real measurements | 20-500 GB with checkpoints source: https://github.com/IndigoPurple/DEQSCI split: source masks, crop protocol, and metrics run: python experiments/reproduction/run_family_protocol.py --family silva_snapshot_compressive_equilibrium --tier full --work-dir runs/silva_snapshot_compressive_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 66 Silva Snapshot Compressive 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 | the tensor solved to equilibrium |
| Condition | the observed input or source tensor |
| Repeated computation | the state-preserving transition evaluated by the root solver |
| Required invariants | shape, device, dtype, finiteness, and differentiability |
| Replaceable components | initializer, source encoder, transition, readout, 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 fixed-point residual and task error against a deterministic target. 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 state width, batch size, and data volume. 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": '66_silva_snapshot_compressive_equilibrium.ipynb',
"state": 'the tensor solved to equilibrium',
"condition": 'the observed input or source tensor',
"transition": 'the state-preserving transition evaluated by the root solver',
"invariants": 'shape, device, dtype, finiteness, and differentiability',
"compact_metric": 'fixed-point residual and task error against a deterministic target',
"scale_axis": 'state width, batch size, and data volume',
}
assert all(notebook_reproduction_record.values())
notebook_reproduction_record
{'notebook': '66_silva_snapshot_compressive_equilibrium.ipynb',
'state': 'the tensor solved to equilibrium',
'condition': 'the observed input or source tensor',
'transition': 'the state-preserving transition evaluated by the root solver',
'invariants': 'shape, device, dtype, finiteness, and differentiability',
'compact_metric': 'fixed-point residual and task error against a deterministic target',
'scale_axis': 'state width, batch size, and data volume'}
Worked Convergence and Sensitivity Study¶
The preceding example demonstrates one configured solve. This additional study changes the transition 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, the tensor solved to equilibrium, and its repeated map, the state-preserving transition evaluated by the root solver. 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('transition 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)
transition 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('transition 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 transition 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 | fixed-point residual and task error against a deterministic target |
| 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 | shape, device, dtype, finiteness, and differentiability |
| Scale sweep | Change one of state width, batch size, and data volume 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.