Add initialization coverage distribution matching

1 files changed, 30 insertions, 0 deletions

diff --git a/scripts/README.md b/scripts/README.md
index efd4208..02b55b0 100644
--- a/scripts/README.md
+++ b/scripts/README.md
@@ -143,6 +143,36 @@ The prior-free minimax theorem says:
 
 with equality for isotropic feedback. Outputs are written under `outputs/minimax_initialization/`.
 
+## Initialization Coverage Distribution Matching
+
+Run:
+
+```bash
+python scripts/initialization_distribution_matching.py --dimension 128 --target-samples 100000 --feedback-samples 100000 --batch-size 8192 --seed 2026 --subspace-dim 8 --anisotropy 64 --plot
+```
+
+For an initialization distribution \(\mu\), define:
+
+\[
+M_\mu=\mathbb E[\hat b\hat b^\top].
+\]
+
+For a random target direction \(a\), theory predicts the coverage distribution:
+
+\[
+A(a)=a^\top M_\mu a
+=
+\frac{\sum_i \lambda_i G_i}{\sum_i G_i},
+\qquad
+G_i\sim\chi^2_1,
+\]
+
+where \(\lambda_i\) are eigenvalues of \(M_\mu\). The script compares this
+population-predicted distribution with the distribution induced by an empirical
+second-moment estimate from sampled feedback directions.
+
+Outputs are written under `outputs/initialization_distribution_matching/`.
+
 ## Functional Capacity Overlap
 
 Run: