Bias-Induced Crossover in Absolute Capacity of Dense Associative Memory

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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.

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