发表机构
University of New South Wales; University of Sydney; University of Cambridge（新南威尔士大学; 悉尼大学; 剑桥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对叠加态下的神经表征，推导了高斯随机字典阈值层计算布尔门的两种几何维度阈值，解释了误差差距来源，明确了合取等门的阈值且与模拟结果一致。
AI 中文摘要
神经表征可编码超出其维度的更多特征，这一现象被称为叠加态。我们研究了从这类表征计算布尔门所需的维度。对于具有高斯随机字典和均匀随机稀疏布尔输入的单个阈值层，我们在两种误差准则下推导了精确的维度阈值：消失的期望误差计数可能需要比高概率下每个输出都正确更多的维度。共享读取解释了这一差距：罕见的实现会一次性产生大量误差。期望计数阈值具有球几何结构，而当对每个特征元组评估门时，联合可靠性具有盒几何结构。优化共享读取权重和偏置，为合取、析取和多数门给出了明确的阈值。对于成对合取，该分析还描述了阈值附近的转变，与精确模拟结果一致。
英文摘要
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.
