# Bias-Induced Crossover in Absolute Capacity of Dense Associative Memory

arXiv: https://arxiv.org/abs/2609.17477 · HTML: https://mlsanity.com/paper/2609.17477/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 29 references that are in the map.
- First posted on arXiv 15 Sep 2026.

## Builds on (most cited first, 2 of 2)

- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)

## Abstract (arXiv)

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/Nlfloor 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 Nn-1/ln N capacity for fixed 0<q<1/2 within the conditioned-Gaussian approximation.