Effective bits = total storage bits ÷ weight count, including scales, masks, group parameters and outlier indices.
Reported effective bits vs AF1
Table 1 numbers (bits). AF1's 1.0 applies to target linear weights only — not embeddings, output head or norms.
Product of two sign matrices
W ≈ scales · B₁ · B₂ · scales. Slide the inner rank r. Toy 16×16 layer with random signs; storage counted for sign bits only (scale vectors ignored).
Illustrative random signs, not trained factors. Each product entry is a sum of r values ±1, so it takes r+1 distinct values vs 2 for one sign matrix.
NABF: the four-stage chain
Green = inherited from DBF. Purple = new in AF1.
Why a null space makes errors free
Illustrative toy B = [[1,1],[1,1]] (rank 1). Unit error e at angle θ; output ‖B e‖ = 2|cos(θ−45°)|.
Layer budgets from Shapley scores (temperature softmax)
Illustrative scores are made up to show the mechanism. Total stays pinned: mean relative rank = 1.0. Low τ concentrates; high τ drifts uniform.
Monte Carlo permutation sampling
Illustrative running mean of random marginal contributions. Paper reports saturation near 60 samples (layer) and 40 (module).
Soft-OR of two distortions
Module score fuses direction and magnitude distortion so weakness in either counts. Shown as the generic probabilistic OR 1−(1−d)(1−m); the paper's exact normalization is not in the episode.