The Ball and the Box: Two Geometries of Computation in Superposition
Neural representations can encode more features than they have dimensions, a phenomenon known as superposition. We study the dimension needed to compute Boolean gates from such representations. For a single threshold layer with a Gaussian random dictionary and uniformly random sparse Boolean inputs, we derive sharp dimension thresholds under two error criteria. A vanishing expected error count can require more dimensions than correctness of every output with high probability. Shared reads explain the gap: rare realizations can produce many errors at once. The expected-count threshold has ball geometry, while joint reliability has box geometry when a gate is evaluated on every feature tupl...
| Authors | Xiaoyu Li, Lequan Lin, Dai Shi et al. |
|---|---|
| Primary category | cs.LG |
| Published | 8 October 2026 |
| Last updated at source | 8 October 2026 |
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