# Distribution Matching Review

This report collects the current theory-vs-empirical plots.

## 1. Multilayer Capacity Distribution

Theory:

\[
S_l=-\log P(Q_l'\ge Q_l)\sim \mathrm{Exp}(1),
\qquad
\sum_{l=1}^L S_l\sim \mathrm{Gamma}(L,1).
\]

Large run:

- dimensions: `64, 256, 1024, 4096`
- layers: `1, 2, 4, 8, 16`
- samples per pair: `100000`
- max KS statistic: `0.0042004`
- mean absolute mean error: `0.0042410`

Key plots:

![capacity KS heatmap](../outputs/multilayer_capacity_distribution/ks_heatmap.png)

![D4096 L16 histogram](../outputs/multilayer_capacity_distribution/hist_D4096_L16.png)

![D4096 L16 QQ](../outputs/multilayer_capacity_distribution/qq_D4096_L16.png)

Judgment: this is strong distribution matching. The empirical histograms and QQ plots should be close enough for a paper figure.

## 2. Initialization Coverage Distribution

Theory:

\[
A(a)=a^\top M_\mu a
=
\frac{\sum_i \lambda_i G_i}{\sum_i G_i},
\qquad
G_i\sim\chi^2_1.
\]

Large run:

- dimension: `128`
- target samples: `100000`
- feedback samples: `100000`
- max nondegenerate KS: `0.00470`

Key plots:

![coverage histograms](../outputs/initialization_distribution_matching/coverage_histograms.png)

![spectra](../outputs/initialization_distribution_matching/population_vs_empirical_spectra.png)

![worst best coverage](../outputs/initialization_distribution_matching/worst_best_coverage.png)

![subspace QQ](../outputs/initialization_distribution_matching/qq_subspace.png)

![axis QQ](../outputs/initialization_distribution_matching/qq_axis.png)

![geometric axis QQ](../outputs/initialization_distribution_matching/qq_geometric_axis.png)

Judgment: this is strong distribution matching for non-isotropic schemes. Isotropic/rademacher are theoretically point masses at \(1/D\), so their QQ/KS is not the right diagnostic; mean and finite-sample spectrum spread are the right diagnostics there.

## 3. Trajectory Gap Distribution

Bridge predictor:

\[
\widehat{\delta\theta}_T(B)
=
-\eta
\sum_{t<T}
\left[
g_{\mathrm{FA}}(\theta_t^{\mathrm{BP}};B)
-
g_{\mathrm{BP}}(\theta_t^{\mathrm{BP}})
\right],
\]

\[
\widehat{\Delta L}_T(B)
=
L(\theta_T^{\mathrm{BP}}+\widehat{\delta\theta}_T(B))
-
L(\theta_T^{\mathrm{BP}}).
\]

### h16x16, 1000 runs

- empirical gap mean: `0.120109`
- bridge gap mean: `0.318266`
- raw KS: `0.917`
- standardized KS: `0.056`
- standardized p-value: `0.0869`
- moment-matched Wasserstein: `0.00362`
- paired Spearman: `0.49288`

Raw scale:

![h16 raw histogram](../outputs/trajectory_gap_distribution/h16x16_1000/gap_histogram_overlay.png)

![h16 raw QQ](../outputs/trajectory_gap_distribution/h16x16_1000/gap_qq_plot.png)

Moment-matched / standardized shape:

![h16 moment matched](../outputs/trajectory_gap_distribution/h16x16_1000/gap_histogram_moment_matched.png)

![h16 standardized QQ](../outputs/trajectory_gap_distribution/h16x16_1000/gap_standardized_qq_plot.png)

Paired seed ranking:

![h16 paired scatter](../outputs/trajectory_gap_distribution/h16x16_1000/paired_gap_scatter.png)

### h24x24, 500 runs

- empirical gap mean: `0.145215`
- bridge gap mean: `0.432834`
- raw KS: `0.944`
- standardized KS: `0.056`
- standardized p-value: `0.4135`
- moment-matched Wasserstein: `0.00356`
- paired Spearman: `0.39455`

Raw scale:

![h24 raw histogram](../outputs/trajectory_gap_distribution/h24x24_500/gap_histogram_overlay.png)

![h24 raw QQ](../outputs/trajectory_gap_distribution/h24x24_500/gap_qq_plot.png)

Moment-matched / standardized shape:

![h24 moment matched](../outputs/trajectory_gap_distribution/h24x24_500/gap_histogram_moment_matched.png)

![h24 standardized QQ](../outputs/trajectory_gap_distribution/h24x24_500/gap_standardized_qq_plot.png)

Paired seed ranking:

![h24 paired scatter](../outputs/trajectory_gap_distribution/h24x24_500/paired_gap_scatter.png)

Judgment:

- Raw bridge distribution is not sufficiently consistent: it overestimates the absolute gap scale by roughly `2-3x`.
- Standardized/moment-matched distribution shape is much more consistent.
- Current trajectory claim should be: the BP-path bridge captures distributional shape and seed ranking, but needs a response/curvature/contraction factor to predict absolute scale.