When the Gradient Sees Rank: Provable Necessity, Causal Recruitment, and Composition in Trained Matrix Memories
Can gradient-based training learn the rank needed to store and compose associations in a matrix memory? In our earlier study, we used a matrix-augmented reasoner on a task that admits a rank-1 solution, leaving this question open. We train matrix memories on $K$ fresh key-value bindings whose exact linear recovery requires $\mathrm{rank}(Z) \geq K$. A fixed linear readout queries a single matrix state without access to the original bindings. Experiments measure recovery by cosine similarity greater than 0.9, a threshold distinct from mathematical equality. Learned effective rank increases with $K$ across the tested grid (Spearman $ρ= 1.0$ at $d = 16$). Training-time rank caps produce a re...
| Authors | Samuel Larson |
|---|---|
| Primary category | cs.LG |
| Published | 12 September 2026 |
| Last updated at source | 12 September 2026 |
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