ADR-003: Anti-Degeneracy Layer for Velocity-Phase Model¶

Status:: Accepted
Date:: 2026-05-19
Deciders:: Core team

Context¶

The heterodyne two-component model fits the two-time correlation function at multiple azimuthal angles simultaneously:

\[c_2(\phi_k, t_1, t_2;\, \theta) = c_\mathrm{offset}(\phi_k) + \beta(\phi_k)\, \exp\!\Bigl(-q^2 \bigl[\mathcal{J}_r(t_1,t_2) + f_s\,\mathcal{J}_s(t_1,t_2)\bigr]\Bigr) \cos\!\Bigl(q\cos\phi_k \int_{t_1}^{t_2} v(t')\,dt'\Bigr)\]

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

If these angle-specific parameters are treated as free variables in the optimization (one pair per angle, \(2 \times n_\phi\) extra free parameters), the NLSQ landscape has a degenerate flat direction: any increase in \(D_{0,\mathrm{ref}}\) or \(D_{0,\mathrm{sample}}\) 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 \(D_{0,\mathrm{ref}} \approx 0\) 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.

The 14-parameter heterodyne model has additional degeneracies beyond the diffusion-contrast flat direction:

\(D_{0,\mathrm{ref}}\) / \(D_{0,\mathrm{sample}}\) trading (two diffusion prefactors)
\(f_0\) / \(f_3\) trading (fraction amplitude vs. baseline)
\(v_0\) / \(v_\mathrm{offset}\) trading (velocity magnitude vs. constant offset)
\(\alpha_\mathrm{ref}\) / \(D_{0,\mathrm{ref}}\) compensation (exponent-prefactor pair)
\(\alpha_\mathrm{sample}\) / \(D_{0,\mathrm{sample}}\) compensation

Decision¶

Heterodyne implements a four-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 with sufficient angles: the per-angle contrasts and offsets are expressed as a truncated Fourier series over \(\phi\), replacing \(2n_\phi\) independent values with \(2K+1\) Fourier coefficients (default \(K=2\)). With fewer angles, a single averaged pair \((\bar{\beta}, \bar{c}_\mathrm{offset})\) is used, initialized from quantile estimates and jointly optimized with the 14 physical parameters.

Layer 2: Hierarchical Optimization

First stage: fix scaling parameters at quantile estimates, optimize only the 14 physical parameters. Second stage: jointly optimize all 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

Tracks the ratio \(\|\nabla_\mathrm{physical}\| / \|\nabla_\mathrm{per\text{-}angle}\|\) during optimization. When this ratio falls below a threshold for several consecutive iterations (indicating gradient collapse), triggers a hierarchical restart.

Layer 5: Shear-Sensitivity Weighting — INTENTIONALLY ABSENT

Homodyne implements a fifth layer that upweights data points from angles near \(\phi = 0\) (parallel to flow), where the \(\operatorname{sinc}\) shear term is most sensitive to \(\dot{\gamma}_0\). This breaks the \((D_0, \dot{\gamma}_0)\) degeneracy specific to laminar-flow XPCS.

Heterodyne does not include Layer 5. The heterodyne velocity-phase model has no \(\operatorname{sinc}\) term: the flow contribution enters as \(\cos(q\cos\phi_k \int v\,dt')\), not as a sinc attenuation. The \((D_0, v_0)\) degeneracy is broken by the oscillatory angular dependence of the cosine term across multiple \(\phi_k\) values, without requiring explicit shear-sensitivity upweighting. This is a narrow physics exemption (spec §2).

The four layers are controlled by a single YAML setting:

optimization:
  nlsq:
    anti_degeneracy:
      per_angle_mode: "auto"   # "auto" activates all 4 layers

See Anti-Degeneracy Defense System for the mathematical details of each mode.

Rationale¶

Why reparameterization (Layer 1)?

Direct optimization of \(2n_\phi\) per-angle parameters creates a flat manifold in the loss landscape. Reparameterizing as Fourier coefficients exploits the physical prior that angular contrast variation is smooth (it arises from detector geometry), reducing the effective parameter count and eliminating the flat direction. The Fourier basis is model-agnostic: it captures smooth \(\phi\)-variation of any origin.

Why hierarchical (Layer 2)?

In the joint landscape, gradient descent for the 14 physical parameters is dominated by the per-angle scaling gradient. Fixing the scaling in the first stage forces the optimizer to explore the physically meaningful sub-space first, then fine-tune scaling in the second stage from a physically reasonable starting point.

Why CV regularization (Layer 3)?

A degenerate solution has large per-angle residuals with high variance across angles (high CV). A physical solution has small, roughly uniform residuals (low CV). The CV penalty provides a soft constraint that guides the optimizer away from the degenerate manifold without requiring hard constraints that might exclude physical solutions.

Why gradient monitoring (Layer 4)?

Gradient collapse (ratio < threshold for N consecutive iterations) is a reliable symptom of landing on the degenerate manifold. The response (hierarchical restart) is cheaper than abandoning the optimization and re-running from scratch.

Why not Layer 5 in heterodyne?

The shear-sensitivity weighting in homodyne upweights the sinc-dominated regime to break the \((D_0, \dot{\gamma}_0)\) degeneracy that is specific to laminar-flow XPCS. Heterodyne’s velocity-phase cosine term provides a different angular sensitivity structure: the amplitude of the cosine oscillation is \(q|\cos\phi_k|\), which is naturally largest near \(\phi_k = 0\) and vanishes near \(\phi_k = \pi/2\). This intrinsic angular sensitivity breaks the diffusion-velocity degeneracy without explicit upweighting. Adding Layer 5 would double-weight the \(\phi \approx 0\) region, biasing the residuals and increasing the effective noise floor.

Consequences¶

Positive:

No degenerate solutions in production use of auto mode.
Clear diagnostic logging when each layer activates (visible at INFO level).
All four layers are independently tested and can be individually disabled via config.
constant mode allows ablation studies (disabling the optimization of scaling).

Negative / Accepted trade-offs:

auto mode adds 2 parameters (16 vs 14 physical-only) and 2 optimization stages.
The 4-layer system adds code complexity: 4 modules, each with its own config class.
New contributors must read this ADR and Anti-Degeneracy Defense System to understand why the optimizer is structured this way.
The deliberate absence of Layer 5 must be documented to prevent well-intentioned re-addition by future contributors unfamiliar with homodyne’s design history.

Operational advice:

Always use auto mode unless there is a specific reason to use another.
If fitted \(D_{0,\mathrm{ref}} \approx 0\) or \(D_{0,\mathrm{sample}} \approx 0\) and contrasts are large: degeneracy was not fully resolved; switch to auto if not already, or verify initial parameter guesses.
Individual degenerate parameter pairs are logged at WARNING level during post-fit diagnosis.

Alternatives Considered¶

A. Single-mode only (auto)

Simpler user interface. Rejected because: limits the ability to perform model selection and may not be optimal for all experimental configurations.

B. Hierarchical Bayesian model for contrast

Model \(\beta(\phi_k) \sim \mathcal{N}(\mu_\beta, \sigma_\beta^2)\) with hyperpriors. Correct Bayesian treatment. Rejected because: this is only available in the CMC path; for the NLSQ point estimate (which runs first), a simple parameterization is needed. The CMC model could implement this hierarchy in a future version.

C. Automatic model selection (AIC/BIC)

Run all modes and select based on information criterion. Rejected because: would quadruple the optimization time; users who want model selection can run the modes manually.

D. Port homodyne Layer 5 without modification

Port shear-sensitivity weighting directly to heterodyne. Rejected because: heterodyne has no sinc term; the weighting would apply to the cosine oscillation amplitude, which already provides natural angular sensitivity. Explicit upweighting of \(\phi \approx 0\) would bias the fit toward data from that angular region and increase effective noise.