Anti-Degeneracy Defense System¶

This page describes the mathematical basis and operational details of the four-layer anti-degeneracy defense system implemented in heterodyne’s NLSQ optimizer. The system is coordinated by AntiDegeneracyController.

For the architectural decision to implement this system, see ADR-003: Anti-Degeneracy Layer for Velocity-Phase Model. For the per-angle scaling mode decision, see ADR-005: Per-Angle Scaling Modes.

The Degeneracy Problem¶

The heterodyne two-component correlation model evaluates

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

where:

\(\mathcal{J}_r(t_1,t_2) = \int_{t_1}^{t_2} J_r(t')\,dt'\) is the reference transport integral, \(J_r(t) = D_{0,\mathrm{ref}}\,t^{\alpha_\mathrm{ref}} + D_{\mathrm{offset,ref}}\)
\(\mathcal{J}_s(t_1,t_2) = \int_{t_1}^{t_2} J_s(t')\,dt'\) is the sample transport integral, \(J_s(t) = D_{0,\mathrm{sample}}\,t^{\alpha_\mathrm{sample}} + D_{\mathrm{offset,sample}}\)
\(f_s(t_1,t_2)\) is the sample fraction averaged over \([t_1, t_2]\)
\(v(t) = v_0\,t^\beta + v_\mathrm{offset}\) is the velocity field
\(\beta(\phi_k)\) and \(c_\mathrm{offset}(\phi_k)\) are per-angle scaling parameters

The model has 14 physical parameters and \(2n_\phi\) per-angle scaling parameters. If both sets are freely optimized simultaneously, the loss landscape has a flat direction: any multiplicative increase in \(D_{0,\mathrm{ref}}\) (or \(D_{0,\mathrm{sample}}\)) can be exactly compensated by dividing all \(\beta(\phi_k)\) by the same factor, leaving all residuals unchanged. The NLSQ optimizer cannot distinguish these solutions.

Beyond the diffusion-contrast flat direction, the 14-parameter space contains additional known degeneracies:

Parameter pair	Type	Mechanism
\(D_{0,\mathrm{ref}}\) / \(D_{0,\mathrm{sample}}\)	Trading	Two diffusion prefactors; identical effect for \(f_s = 0.5\)
\(f_0\) / \(f_3\)	Trading	Fraction amplitude vs. baseline; sum is nearly constant
\(v_0\) / \(v_\mathrm{offset}\)	Trading	Velocity magnitude vs. constant; degenerate at long lag times
\(\alpha_\mathrm{ref}\) / \(D_{0,\mathrm{ref}}\)	Compensation	Exponent changes the lag-time scaling; prefactor rescales the overall magnitude
\(\alpha_\mathrm{sample}\) / \(D_{0,\mathrm{sample}}\)	Compensation	Same as above for the sample component

Layer 1: Fourier / Constant Reparameterization¶

Implementation: heterodyne.optimization.nlsq.fourier_reparam

The first layer removes the per-angle scaling flat direction by constraining the \(2n_\phi\) parameters \(\{(\beta(\phi_k), c_\mathrm{offset}(\phi_k))\}\) to a lower-dimensional manifold.

Fourier mode (per_angle_mode = "fourier" or auto with \(n_\phi \geq 8\)):

The angular contrast and offset are expressed as truncated Fourier series:

\[\beta(\phi_k) = \sum_{m=0}^{K} a_m \cos(m\phi_k) + \sum_{m=1}^{K} b_m \sin(m\phi_k), \qquad c_\mathrm{offset}(\phi_k) = \sum_{m=0}^{K} c_m \cos(m\phi_k) + \sum_{m=1}^{K} d_m \sin(m\phi_k)\]

With \(K=2\) (the default), each series has \(2K+1 = 5\) coefficients, for a total of 10 parameters replacing \(2n_\phi\). For \(n_\phi \geq 8\), this is a reduction; for \(n_\phi = 3\), Fourier and individual modes have the same count.

The Fourier coefficients are initialized from the data using a discrete Fourier transform of the quantile-estimated per-angle contrasts. This provides a physically meaningful starting point close to the true solution.

Note

The Fourier basis is model-agnostic. In homodyne, the sinc-shear model motivates even-harmonic dominance (beam coherence is symmetric). In heterodyne, the cosine velocity-phase term also produces even-harmonic dominance in \(\beta(\phi)\), so the same Fourier basis is appropriate.

Auto-averaged mode (auto with \(3 \leq n_\phi < 8\)):

A single pair \((\bar{\beta}, \bar{c}_\mathrm{offset})\) replaces all per-angle values. The shared contrast and offset are initialized from the median of the quantile-estimated per-angle values:

\[\bar{\beta} = \mathrm{median}_k\,[\hat{\beta}(\phi_k)], \qquad \bar{c}_\mathrm{offset} = \mathrm{median}_k\,[\hat{c}_\mathrm{offset}(\phi_k)]\]

This adds only 2 parameters to the 14-parameter physical model.

Individual mode (per_angle_mode = "individual" or auto with \(n_\phi < 3\)):

All \(2n_\phi\) per-angle parameters are freely optimized. For small \(n_\phi\) (1–2 angles), this is not degenerate because the angular coverage is insufficient to create a flat direction. For \(n_\phi \geq 3\), this mode is high-risk and should be used only for ablation studies.

Constant mode (per_angle_mode = "constant"):

Per-angle scaling parameters are fixed at quantile estimates and not optimized. The 14 physical parameters are optimized in isolation. This is useful for:

Ablation studies (testing whether optimized scaling improves the fit)
Cases where independent calibration of contrast is available
Fast approximate fits

Layer 2: Hierarchical Optimization¶

Implementation: heterodyne.optimization.nlsq.hierarchical

The second layer separates the optimization into two stages to avoid gradient entanglement between physical parameters and per-angle scaling.

Stage 1 — Physical-only optimization:

Fix the per-angle scaling at quantile estimates and optimize only the 14 physical parameters:

\[\hat{\theta}_\mathrm{phys} = \arg\min_{\theta_\mathrm{phys}} \sum_{k,i,j} \bigl[c_2^\mathrm{data}(\phi_k, t_i, t_j) - c_2^\mathrm{model}(\phi_k, t_i, t_j;\, \theta_\mathrm{phys},\, \bar{\beta},\, \bar{c}_\mathrm{offset})\bigr]^2\]

Stage 2 — Joint optimization:

Starting from \((\hat{\theta}_\mathrm{phys}, \bar{\beta}, \bar{c}_\mathrm{offset})\), optimize all parameters jointly:

\[(\hat{\theta}_\mathrm{phys}, \hat{\beta}, \hat{c}_\mathrm{offset}) = \arg\min_{\theta_\mathrm{phys},\, \beta,\, c_\mathrm{offset}} \sum_{k,i,j} \bigl[c_2^\mathrm{data} - c_2^\mathrm{model}\bigr]^2\]

The hierarchical approach ensures that Stage 1 explores the physical parameter space without interference from scaling, providing a physically meaningful initialization for Stage 2.

Activation condition: Hierarchical optimization is active when hierarchical_enable = True (default) and the controller is in any mode other than fixed_constant.

Layer 3: Adaptive CV-Based Regularization¶

Implementation: heterodyne.optimization.nlsq.adaptive_regularization

The third layer adds a regularization penalty based on the per-angle residual coefficient of variation (CV):

\[\mathrm{CV}(\phi_k) = \frac{\sigma_k}{\mu_k}\]

where \(\sigma_k\) and \(\mu_k\) are the standard deviation and mean of the residuals \(r_{ij}(\phi_k) = c_2^\mathrm{data}(\phi_k, t_i, t_j) - c_2^\mathrm{model}(\phi_k, t_i, t_j)\) for all \((t_i, t_j)\) pairs at angle \(\phi_k\).

A physically correct fit has small, roughly uniform residuals across angles (low CV). A degenerate fit has large residuals with high angular variance (high CV). The augmented loss is:

\[L(\theta) = \sum_{k,i,j} r_{ij}^2(\phi_k) + \lambda \sum_k \max\!\bigl(0,\, \mathrm{CV}(\phi_k) - \mathrm{CV}_\mathrm{target}\bigr)^2\]

The regularization weight \(\lambda\) is adapted dynamically: it is increased when the current CV exceeds regularization_max_cv, and decreased when the fit is already within the target. This prevents the regularization from dominating the physics residual after convergence.

Activation condition: Layer 3 is active when regularization_mode != "disabled" (default: "relative").

Layer 4: Gradient Collapse Monitoring¶

Implementation: heterodyne.optimization.nlsq.gradient_monitor

The fourth layer monitors the Jacobian during optimization. Gradient collapse — where the gradient with respect to physical parameters becomes negligible compared to the gradient with respect to per-angle scaling — is a diagnostic symptom of the degenerate manifold.

The monitored ratio is:

\[\rho = \frac{\|\nabla_{\theta_\mathrm{phys}} L\|}{\|\nabla_{\beta, c_\mathrm{offset}} L\|}\]

When \(\rho < \rho_\mathrm{threshold}\) (default: 0.01) for gradient_consecutive_triggers consecutive iterations (default: 5), the controller triggers the configured response action:

Response mode	Action
`"warn"`	Log a warning; continue optimization
`"hierarchical"`	Trigger a hierarchical restart (Stage 1 from current physical params)
`"reset"`	Reset per-angle scaling to quantile estimates; restart joint optimization
`"abort"`	Raise an error and abort the optimization

The default response is "hierarchical".

Activation condition: Layer 4 is active when gradient_monitoring_enable = True (default) and the optimizer is running in any mode other than constant.

Layer 5: Absent — Shear-Sensitivity Weighting¶

Homodyne implements a fifth layer that upweights data points near \(\phi = 0\) (parallel to flow), where the laminar-flow sinc term

\[\operatorname{sinc}^2\!\bigl(q\cos\phi_k \cdot \dot{\gamma}_0 \cdot \Delta t / 2\bigr)\]

is most sensitive to the shear rate \(\dot{\gamma}_0\). This breaks the \((D_0, \dot{\gamma}_0)\) degeneracy specific to the homodyne laminar-flow model.

Heterodyne does not include Layer 5. The heterodyne velocity-phase model has no sinc term. The flow contribution enters as:

\[\cos\!\Bigl(q\cos\phi_k \int_{t_1}^{t_2} v(t')\,dt'\Bigr)\]

The oscillation amplitude is \(q|\cos\phi_k|\), which is naturally largest near \(\phi_k = 0\) and vanishes near \(\phi_k = \pi/2\). This intrinsic angular sensitivity provides sufficient differentiation between the diffusion exponential and the velocity cosine term without requiring explicit upweighting.

Adding Layer 5 to heterodyne would double-weight the \(\phi \approx 0\) data region, biasing the fit and increasing the effective noise floor. This is a narrow physics exemption documented in spec §2 and tracked as a D3 divergence from homodyne.

The use_shear_weighting property of AntiDegeneracyController always returns False.

Per-Angle Mode Summary¶

Mode	Layer 1	Layer 2	Layer 3	Layer 4	Notes
`constant`	Quantile	Disabled	Optional	Optional	Scaling fixed; no per-angle optimization
`auto`	Averaged	Active	Active	Active	2 shared scaling params; recommended default
`fourier`	Fourier	Active	Active	Active	K=2 Fourier coefficients; best for smooth variation
`individual`	Per-angle	Optional	Optional	Optional	\(2n_\phi\) free params; degeneracy risk for \(n_\phi \geq 3\)

Configuration Reference¶

All anti-degeneracy settings are nested under optimization.nlsq.anti_degeneracy:

optimization:
  nlsq:
    anti_degeneracy:
      enable: true                        # Master switch
      per_angle_mode: "auto"              # "auto" | "constant" | "individual" | "fourier"
      fourier_order: 2                    # K in the Fourier series (2K+1 coefficients)
      fourier_auto_threshold: 8           # n_phi threshold for auto -> fourier
      constant_scaling_threshold: 3       # n_phi threshold for auto -> auto_averaged

      # Layer 2: Hierarchical
      hierarchical_enable: true
      hierarchical_max_outer_iterations: 5
      hierarchical_outer_tolerance: 1.0e-4
      hierarchical_physical_max_iterations: 100
      hierarchical_per_angle_max_iterations: 50

      # Layer 3: Adaptive CV regularization
      regularization_mode: "relative"     # "relative" | "absolute" | "disabled"
      regularization_lambda: 1.0
      regularization_target_cv: 0.10
      regularization_target_contribution: 0.10
      regularization_max_cv: 0.20

      # Layer 4: Gradient monitoring
      gradient_monitoring_enable: true
      gradient_ratio_threshold: 0.01
      gradient_consecutive_triggers: 5
      gradient_response_mode: "hierarchical"  # "warn"|"hierarchical"|"reset"|"abort"

Diagnostic Output¶

The controller logs each layer’s activation at INFO level, framed with separator lines for visibility:

============================================================
ANTI-DEGENERACY: Auto-selected 'auto_averaged' mode
  Reason: n_phi (7) >= constant_scaling_threshold (3)
  Behavior: Quantile estimates -> AVERAGED -> OPTIMIZED
  Parameters: 14 physical + 2 averaged scaling = 16 total
============================================================
ANTI-DEGENERACY: Layer 4 - Gradient Collapse Monitor
  Enabled: True
  Ratio threshold: 0.01
  Response mode: hierarchical
============================================================

Layer 4 triggers are logged at WARNING level:

WARNING: Gradient collapse detected (ratio=0.003 < threshold=0.01, 5 consecutive)
Triggering hierarchical restart from current physical parameters.

Post-fit, any identified degenerate parameter pairs are logged at WARNING level, enabling downstream diagnosis.