发表机构
Hiroshima City University; RIKEN Center for Advanced Intelligence Project; The University of Tokyo（广岛市立大学; 理化学研究所先进智能项目中心; 东京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究密集联想记忆中偏置对绝对容量的影响，发现偏置诱导的容量交叉，并提出活动依赖控制势以恢复无偏容量。
AI 中文摘要
密集联想记忆的绝对容量主要是在无偏模式条件下进行分析的。本文研究了在Krotov-Hopfield单点准则$P_{\mathrm{error}}=1/N$下，中心化二元模式中偏置的影响，其中$P_{\mathrm{error}}$是单点翻转降低存储模式能量的概率，$N$是神经元数量。每个模式分量以概率$q$取$1-q$，否则取$-q$，其中$0<q\le1/2$。对于$n$阶多项式相互作用，信噪比分析给出在$q=1/2$时绝对容量为$N^{n-1}/\ln N$量级。然而，对于固定的$q<1/2$，当$n\ge4$为偶数时容量为$O(N^{n/2})$，当$n\ge5$为奇数时容量为$O(N^{(n+1)/2})$。对于$n=3$，无偏和固定偏置容量均保持为$O(N^2/\ln N)$。对于$n\ge4$，这些不同的渐近形式意味着在$q=1/2$附近存在非均匀的大$N$极限。渐近匹配预测在$1-2q=O(\ln N/N^{\lfloor n/2\rfloor-1})$区域出现偏置诱导的交叉。该交叉源于偏置相关的串扰均值，它降低了携带更频繁值$-q$的站点的稳定性。将计算机模拟与有限尺寸条件高斯预测进行了比较。在条件高斯近似下，一种消除条件串扰均值的活动依赖控制势恢复了固定$0<q<1/2$时的$N^{n-1}/\ln N$容量。
英文摘要
The absolute capacity of dense associative memory has mainly been analyzed for unbiased patterns. Here we examine the effect of bias in centered binary patterns under the Krotov-Hopfield single-site criterion $P_{\mathrm{error}}=1/N$, where $P_{\mathrm{error}}$ is the probability that a single-site flip lowers the energy of a stored pattern and $N$ is the number of neurons. Each pattern component takes $1-q$ with probability $q$ and $-q$ otherwise, where $0<q\le1/2$. For polynomial interactions of order $n$, a signal-to-noise analysis gives an absolute capacity of order $N^{n-1}/\ln N$ at $q=1/2$. For fixed $q<1/2$, however, the capacity is $O(N^{n/2})$ for even $n\ge4$ and $O(N^{(n+1)/2})$ for odd $n\ge5$. For $n=3$, both the unbiased and fixed-bias capacities remain $O(N^2/\ln N)$. For $n\ge4$, these different asymptotic forms imply a nonuniform large-$N$ limit near $q=1/2$. Asymptotic matching predicts a bias-induced crossover in the region $1-2q=O(\ln N/N^{\lfloor n/2\rfloor-1})$. The crossover originates from a bias-dependent crosstalk mean that reduces the stability of sites carrying the more frequent value $-q$. Computer simulations are compared with the finite-size conditioned-Gaussian predictions. An activity-dependent control potential that cancels the conditional crosstalk mean restores the $N^{n-1}/\ln N$ capacity for fixed $0<q<1/2$ within the conditioned-Gaussian approximation.
Comments19 pages, 6 figures
