Bias-Induced Crossover in Absolute Capacity of Dense Associative Memory

arXiv:2609.17477v1 Announce Type: cross Abstract: 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_{error}=1/N, where P_{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<qle1/2. 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 nge4 and O(N^{(n+1)/2}) for odd nge5. For n=3, both the unbiased and fixed-bias capacities remain O(N^2/ln N). For nge4, 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/2rfloor-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.

Source: arXiv cs.LG | 2026-09-16