ADR-003: Anti-Degeneracy Layer for Laminar Flow

Status:

Accepted

Date:

2024–2025

Deciders:

Core team

Context

The laminar-flow model fits the two-time correlation function at multiple azimuthal angles simultaneously:

\[c_2(\phi_k, t_1, t_2; \theta) = c_\mathrm{offset}(\phi_k) + \beta(\phi_k) \exp(-q^2\mathcal{D}) \operatorname{sinc}^2(\ldots)\]

Each angle \(\phi_k\) has its own contrast \(\beta(\phi_k)\) and offset \(c_\mathrm{offset}(\phi_k)\) arising from the detector geometry and beam coherence.

If these angle-specific parameters are treated as free variables in the optimization (one pair per angle, \(2 \times 23 = 46\) extra free parameters for 23 angles), the NLSQ landscape has a degenerate flat direction: any increase in \(D_0\) can be compensated by increasing all \(\beta(\phi_k)\) by a corresponding factor, without changing the residual.

This degeneracy causes optimizers to return unphysical solutions with \(D_0 \approx 0\) and inflated contrasts, or vice versa. It is not a numerical artifact — it is a genuine identifiability issue rooted in the functional form of the model.

Decision

Homodyne implements a five-layer anti-degeneracy defense system coordinated by AntiDegeneracyController:

Layer 1: Fourier/Constant Reparameterization

Transforms the parameter space to remove the flat direction. In auto mode: the per-angle contrasts and offsets are summarized by a single pair \((\bar{\beta}, \bar{c}_\mathrm{offset})\), initialized from quantile estimates and jointly optimized with the 7 physical parameters.

Layer 2: Hierarchical Optimization

First stage: fix scaling parameters at quantile estimates, optimize only the 7 physical parameters. Second stage: jointly optimize all 9 parameters from the Layer-1 solution. This avoids the flat landscape during the physically important first phase.

Layer 3: Adaptive CV-based Regularization

Penalizes solutions where the per-angle residual coefficient of variation (CV) is anomalously large, which signals that the optimizer is “fitting noise” in the degeneracy direction rather than the physics signal.

Layer 4: Gradient Collapse Monitoring

Detects when gradient norms approach zero (flat landscape symptom). On detection: perturbs the current iterate and restarts from the new point.

Layer 5: Shear-Sensitivity Weighting

Upweights data points from angles near \(\phi = 0\) (parallel to flow), where the sinc term is most sensitive to \(\dot{\gamma}_0\). This provides additional information to break the \((D_0, \dot{\gamma}_0)\) degeneracy.

The five layers are controlled by a single YAML setting:

optimization:
  nlsq:
    anti_degeneracy:
      per_angle_mode: "auto"   # "auto" activates all 5 layers

See Per-Angle Scaling and Anti-Degeneracy for the mathematical details of each mode.

Rationale

1. The degeneracy is generic and unavoidable without intervention

The degeneracy arises from the Siegert relation: \(c_2 = 1 + \beta |c_1|^2\). If \(|c_1|^2 \to |c_1|^2 / k\) and \(\beta \to k\beta\), the model is unchanged. This scale invariance is broken only when:

  1. The functional form of \(c_1\) depends on \(\beta\) (it does not in the standard Siegert model), or

  2. The per-angle contrasts are constrained to share structure (as in auto/fourier mode).

Any NLSQ implementation without constraint (b) will find the degenerate manifold.

2. Five layers are needed because each addresses a different failure mode

  • Layer 1 is the primary fix (algebraically reduces degrees of freedom).

  • Layer 2 ensures the optimizer reaches the physical minimum before the contrast parameters are released.

  • Layer 3 catches cases where Layer 1 is bypassed by large initial steps.

  • Layer 4 handles restarts when the optimizer stalls in the degenerate region.

  • Layer 5 improves conditioning so that the physical minimum is more distinct.

Testing showed that any single layer alone was insufficient for all dataset types.

3. ``auto`` mode is the safe default

Auto mode uses quantile-based initialization and only 2 extra parameters (not 46). It is robust to noisy data, handles well-conditioned datasets efficiently (converges in 1 stage), and prevents the degeneracy in all tested configurations.

4. Extensibility

The AntiDegeneracyController is a thin orchestrator: each layer is an independently testable module. Adding a new defense strategy requires only a new layer class and a small update to the controller.

Consequences

Positive:

  • No degenerate solutions in production use of auto mode.

  • Clear error messages and diagnostics when degeneracy is detected.

  • All five layers are independently tested.

  • constant mode allows ablation studies (disabling the optimization of scaling).

Negative / Accepted trade-offs:

  • auto mode adds 2 parameters (9 vs 7) and 2 optimization stages.

  • The 5-layer system adds code complexity: 5 modules, each with its own config class.

  • New contributors must read this ADR and Per-Angle Scaling and Anti-Degeneracy to understand why the optimizer is structured this way.

Operational advice:

  • Always use auto mode unless there is a specific reason to use another.

  • If fitted \(D_0 \approx 0\) and contrasts are large: degeneracy was not fully resolved; switch to auto if not already, or verify initial parameter guesses.

Alternatives Considered

A. Constrain beta to a fixed value

Fix \(\beta(\phi_k) = \beta_\mathrm{fixed}\) for all angles. Simple but introduces systematic bias when true contrast varies across angles (which it always does at some level).

B. Regularize beta toward equality

Add an L2 penalty \(\lambda \sum_k (\beta_k - \bar{\beta})^2\) to the objective. Requires tuning \(\lambda\); hard to set automatically without domain knowledge.

C. Sequential fitting (per-angle then joint)

Fit each angle independently to get \(\beta(\phi_k)\), then fix contrasts and fit physical parameters jointly. Fails when per-angle data is insufficient for independent fits (small datasets, few time points).

D. Bayesian priors on beta

In the CMC model, use a hyperprior \(\beta_k \sim \mathcal{N}(\mu_\beta, \sigma_\beta)\). This is the correct Bayesian approach but does not help the NLSQ point estimate, which is needed first. CMC with NLSQ warm-start already benefits from having resolved the degeneracy.

See also