Stratonovich-Projection Vector Healing¶
A Google Colab session read Hu & Šverák's "Regularity of a stochastically perturbed Euler-Arnold equation" (arXiv:1510.05279) against ia_utils.vector_healing.enhanced_dense_healing_hybrid and proposed replacing the median-fallback correction step with a "Stratonovich projection" (local-mean baseline plus a damped drift term along the recent finite-difference velocity direction). A single-seed demo (seed=42, one spike corruption) reported a dramatic win: cosine phase alignment going from -0.16 to +0.98.
This page is the actual controlled test of that claim.
On the arXiv:1510.05279 citation¶
The paper is real (a genuine result in stochastic differential geometry on Lie groups), and its framing -- a "control mechanism" keeping some quantities close to constant while random perturbations induce drift along a constraint surface -- is legitimately quoted from Section 3.3 of the paper. But nothing in this healing code actually implements the paper's math: the paper's Euler-Arnold bracket q(z,z) requires a genuine finite-dimensional Lie algebra with known structure constants, and arbitrary hidden-state embedding vectors have no such structure. The q_force = ipg_vector * norm_A heuristic used here is not derivable from the paper's formalism -- it is a hand-rolled approximation only loosely inspired by the paper's shape, not its content.
Method¶
Holding the real production Phi-Trigger (dense_evolution.mitigation.healing.evaluate_phi_trigger) fixed and swapping only the correction step isolates the actual claim ("the replacement is better") from a separate question ("the trigger is better"), which the single-seed Colab demo conflated.
4 corruption scenarios (single spike, NaN string, scattered outliers, spike+NaN combined) x 40 seeds each, scored against the known-clean ideal trajectory (trend + IID Gaussian noise) on two metrics: L2 reconstruction error and flattened cosine phase alignment.
Results¶
Restricted to exactly the corrupted indices (the Colab's own comparison design, bypassing the trigger):
| Scenario | L2 error: median → Stratonovich | Win rate | Wilcoxon p |
|---|---|---|---|
| single_spike | 7.38 → 4.12 (-44%) | 100% | <10⁻¹¹ |
| scattered_outliers | 21.45 → 17.75 | 98% | <10⁻¹⁰ |
| combined | 19.47 → 17.09 | 100% | <10⁻¹¹ |
| nan_string | 18.36 → 19.37 (worse) | 0% | <10⁻¹⁰ |
Stratonovich correction beats the shipped median fallback on spike-type corruption, but loses on pure NaN-gap corruption. Even in the winning cases, absolute cosine alignment stays deeply negative (-0.72 to -0.04) -- nowhere near the demo's +0.98, which was a favorable artifact of that one seed, not the general behavior.
A separate finding: the trigger over-fires¶
The shipped production Phi-Trigger fires on 85-90% of every trajectory tested -- corrupted or not -- so in the real (non-oracle) pipeline, the correction-step choice barely moves total L2 error, because correction fires almost everywhere rather than selectively at real anomalies. This became its own investigation: see Healing Trigger False-Positive Audit.
Reproduce¶
Produces data/stratonovich_vector_healing.csv, data/stratonovich_vector_healing_summary.csv.