From 78bb8016e4a779bfeddcfe1402c1bb9a79108a30 Mon Sep 17 00:00:00 2001
From: YurenHao0426 <Blackhao0426@gmail.com>
Date: Mon, 1 Jun 2026 16:54:15 -0500
Subject: Add distribution matching review report

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 reports/distribution_matching_review.md | 149 ++++++++++++++++++++++++++++++++
 1 file changed, 149 insertions(+)
 create mode 100644 reports/distribution_matching_review.md

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+# 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.
+
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