Bias-Induced Crossover in Absolute Capacity of Dense Associative Memory
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 Perror=1/N, where Perror 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≤1/2. For polynomial interactions of order n, a signal-to-noise analysis gives an absolute capacity of order Nn−1/ln N at q=1/2. For fixed q<1/2, however, the capacity is O(Nn/2) for even n≥4 and O(N(n+1)/2) for odd n≥5. For n=3, both the unbiased and fixed-bias capacities remain O(N2/ln N). For n≥4, 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⌊ n/2⌋−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 Nn−1/ln N capacity for fixed 0<q<1/2 within the conditioned-Gaussian approximation.
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BibTeX
@misc{sakurai2026biasinduced,
title = {{Bias-Induced Crossover in Absolute Capacity of Dense Associative Memory}},
author = {Yuto Sakurai and Takeaki Shimokawa and Kazunori Iwata and Kazushi Mimura},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.17477},
eprint = {2609.17477},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17477},
}
RIS
TY - UNPB
TI - Bias-Induced Crossover in Absolute Capacity of Dense Associative Memory
AU - Yuto Sakurai
AU - Takeaki Shimokawa
AU - Kazunori Iwata
AU - Kazushi Mimura
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2609.17477
UR - https://arxiv.org/abs/2609.17477
AB - 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<q≤1/2. 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≥4 and O(N^(n+1)/2) for odd n≥5. For n=3, both the unbiased and fixed-bias capacities remain O(N^2/ln N). For n≥4, 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^⌊ n/2⌋−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.
ER -