الملخص
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 tuple. Optimizing shared readout weights and biases gives explicit thresholds for conjunction, disjunction, and majority. For pairwise conjunction, the analysis also describes the transition near the threshold, in agreement with exact simulations.
الكلمات المفتاحية
الموضوع
بيانات النشر
- المجلة
- غير متاح
- وصول مفتوح
- وصول مفتوح أخضر
اقتبس هذه المقالة
APA 7
Li, X., Lin, L., Shi, D., Jiang, J., Gao, J., & Han, A. (2026). The Ball and the Box: Two Geometries of Computation in Superposition. https://omanscience.com/ar/articles/the-ball-and-the-box-two-geometries-of-computation-in-superposition
MLA 9
Li, Xiaoyu, et al. "The Ball and the Box: Two Geometries of Computation in Superposition." https://omanscience.com/ar/articles/the-ball-and-the-box-two-geometries-of-computation-in-superposition.
شيكاغو (المؤلف–التاريخ)
Li, Xiaoyu, Lequan Lin, Dai Shi, Jiaojiao Jiang, Junbin Gao, and Andi Han. 2026. "The Ball and the Box: Two Geometries of Computation in Superposition." https://omanscience.com/ar/articles/the-ball-and-the-box-two-geometries-of-computation-in-superposition.
هارفارد
Li, X., Lin, L., Shi, D., Jiang, J., Gao, J. and Han, A. (2026) 'The Ball and the Box: Two Geometries of Computation in Superposition', Available at: https://omanscience.com/ar/articles/the-ball-and-the-box-two-geometries-of-computation-in-superposition.
فانكوفر
Li X, Lin L, Shi D, Jiang J, Gao J, Han A. The Ball and the Box: Two Geometries of Computation in Superposition. https://omanscience.com/ar/articles/the-ball-and-the-box-two-geometries-of-computation-in-superposition
IEEE
X. Li, L. Lin, D. Shi, J. Jiang, J. Gao, and A. Han, "The Ball and the Box: Two Geometries of Computation in Superposition," https://omanscience.com/ar/articles/the-ball-and-the-box-two-geometries-of-computation-in-superposition.