发表机构
University of California, Irvine（加州大学尔湾分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文精确计数布尔阈值函数，将神经元容量误差从O(n)改进到O(n^{-99})，并确认记忆检索中r=n-1为尖锐阈值及相关猜想。
AI 中文摘要
单个神经元能记住多少信息？神经网络能检索多少记忆而不产生虚假记忆？这些问题与一个基本问题相关：存在多少个布尔阈值函数 $f(x)=\operatorname{sgn}(a_0+\langle a,x\rangle)$，其中 $x\in\{-1,1\}^n$？本文证明，不同的布尔阈值函数的数量 $T_n$ 为 \\[ T_n=2\binom{2^n-1}{n}\bigl(1+O(n^{-99})\bigr). \\] 等价地，单个阈值神经元的容量为 $n^2-\log_2(n!)+1+O(n^{-99})$ 比特，将 Kahn--Komlós--Szemerédi 结果中的 $O(n)$ 误差项改进为 $O(n^{-99})$。为证明此结果，我们证明，对于 $1\le r\le n-1$，且 $v_1,\ldots,v_r$ 从 $\{-1,1\}^n$ 中随机选取，\\[ \mathbb P\\!\left\{ \langle v_1,\ldots,v_r\rangle\cap\{-1,1\}^n =\{\pm v_1,\ldots,\pm v_r\} \right\} =1-O(n^{-99}). \\] 在 Kanter--Sompolinsky 哈密顿量用于记忆检索的背景下，这确定了 $r=n-1$ 为一个尖锐阈值，在该阈值处，对于几乎所有的 $r$ 个记忆的集合，唯一的基态就是这些记忆及其负向量，从而确认了 Kalai--Linial--Odlyzko 猜想的一个较弱形式。这也解决了 M. Anthony 最近提出的关于布尔阈值函数规范数的一个开放问题。此外，我们证明，对于每个 $1\le r\le n-1$，\\[ \mathbb P\{v_1,\ldots,v_r\text{ 线性相关}\} =2\binom r2\\,2^{-n}+O\\!\left(2^{-n}e^{-cn}\right), \\] 确认了 Kahn--Komlós--Szemerédi 的一个猜想。
英文摘要
How much information can a single neuron remember? How many memories can neural networks retrieve without creating false memories? These questions are related to a basic question: how many Boolean threshold functions $f(x)=\operatorname{sgn}(a_0+\langle a,x\rangle)$, $x\in\{-1,1\}^n$, are there? In this paper, we show that the number $T_n$ of distinct Boolean threshold functions is \[ T_n=2\binom{2^n-1}{n}\bigl(1+O(n^{-99})\bigr). \] Equivalently, the capacity of a single threshold neuron is $n^2-\log_2(n!)+1+O(n^{-99})$ bits, improving the $O(n)$ error term in the result of Kahn--Komlós--Szemerédi to $O(n^{-99})$. To prove this, we show that, for $1\le r\le n-1$, and $v_1,\ldots,v_r$ are chosen at random from $\{-1,1\}^n$, \[ \mathbb P\!\left\{ \langle v_1,\ldots,v_r\rangle\cap\{-1,1\}^n =\{\pm v_1,\ldots,\pm v_r\} \right\} =1-O(n^{-99}). \] In the context of the Kanter--Sompolinsky Hamiltonian for memory retrieval, this identifies $r=n-1$ as a sharp threshold, at which, for almost every collection of $r$ memories, the only ground states are these memories and their negatives, confirming a weaker form of the Kalai--Linial--Odlyzko conjecture. It also settles a recent open problem posed by M. Anthony on the specification number of Boolean threshold functions. In addition, we show that, for every $1\le r\le n-1$, \[ \mathbb P\{v_1,\ldots,v_r\text{ are linearly dependent}\} =2\binom r2\,2^{-n}+O\!\left(2^{-n}e^{-cn}\right), \] confirming a conjecture of Kahn--Komlós--Szemerédi.
