研究揭示偏置对密集关联记忆绝对容量的影响
Bias-Induced Crossover in Absolute Capacity of Dense Associative Memory
这篇论文研究的是偏置对密集关联记忆绝对容量的影响，对于理解关联记忆模型在处理有偏数据时的性能很有价值。
本文分析了中心化二元模式偏置对密集关联记忆绝对容量的影响。在Krotov-Hopfield单位点准则下，当每个模式分量以概率q取1-q时，对于n次多项式相互作用，当q=1/2时，绝对容量为N^(n-1)/ln N。对于固定的q<1/2，当n为偶数且≥4时，容量为O(N^(n/2))，当n为奇数且≥5时，容量为O(N^((n+1)/2))。这种不同渐近形式意味着在q=1/2附近存在非均匀的大N极限。渐近匹配预测在区域1-2q=O(ln N/N^⌊n/2⌋-1)处发生偏置诱导的交叉。计算机模拟与有限尺寸条件高斯预测进行了比较。一种活动依赖的控制势能可以抵消条件交叉项均值，在条件高斯近似下恢复固定0<q<1/2时的N^(n-1)/ln N容量。
Bias-Induced Crossover in Absolute Capacity of Dense Associative Memory
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.