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diff --git a/notes/02_experiment_notes.md b/notes/02_experiment_notes.md index dcf797a..2fad18f 100644 --- a/notes/02_experiment_notes.md +++ b/notes/02_experiment_notes.md @@ -311,6 +311,55 @@ Interpretation: - Multilayer capacity accumulation matches the product law until the all-event becomes too rare for the finite sample budget. - Fixed-threshold high-dimensional tails quickly become too rare for direct Monte Carlo, which is itself consistent with the exponential/geometric volume-collapse interpretation. +## Multilayer Observed Capacity Distribution Run Log + +Script: + +```bash +python scripts/multilayer_capacity_distribution.py --dimensions 64 256 1024 4096 --layers 1 2 4 8 16 --samples 100000 --batch-size 8192 --seed 456 --plot +``` + +Theory: + +\[ +S_l=-\log P(Q_l'\ge Q_l)\sim \mathrm{Exp}(1) +\] + +and for independent layers: + +\[ +S_{1:L}=\sum_{l=1}^L S_l\sim \mathrm{Gamma}(L,1). +\] + +Setup: + +- dimensions \(D\): `64, 256, 1024, 4096` +- layers \(L\): `1, 2, 4, 8, 16` +- samples per \((D,L)\): `100000` +- total distribution-matching rows: `20` +- sampler: exact random-direction chi-square representation \(Q=X/(X+Y)\) + +Summary: + +- max KS statistic over all \((D,L)\): `0.0042004` +- mean absolute mean error: `0.0042410` +- mean absolute variance error: `0.0404870` + +Selected rows: + +| \(D\) | \(L\) | empirical mean | theory mean | empirical var | theory var | KS | +|---:|---:|---:|---:|---:|---:|---:| +| `64` | `1` | `1.00472` | `1` | `1.02000` | `1` | `0.00253` | +| `64` | `16` | `16.00773` | `16` | `15.87860` | `16` | `0.00246` | +| `4096` | `1` | `1.00041` | `1` | `1.00609` | `1` | `0.00194` | +| `4096` | `16` | `16.00106` | `16` | `16.04361` | `16` | `0.00265` | + +Interpretation: + +- The observed capacity surprisal distribution matches the predicted \(\mathrm{Exp}(1)\) and \(\mathrm{Gamma}(L,1)\) laws tightly. +- After the beta-tail transform, the null surprisal distribution is dimension-free; \(D\) controls the raw \(Q\) scale and fixed-threshold cost, while \(L\) controls the Gamma shape. +- This gives the cleanest “theory predicts a distribution, experiment recovers the same distribution” result for multilayer capacity. + ## Minimax Initialization Run Log Script: |