# Algorithmic Analysis of Dense Associative Memory: Finite-Size Guarantees and Adversarial Robustness

arXiv: https://arxiv.org/abs/2604.12811 · HTML: https://mlsanity.com/paper/2604.12811/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 26 references that are in the map.
- First posted on arXiv 14 Apr 2026.

## Builds on (most cited first, 12 of 12)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Certified Adversarial Robustness via Randomized Smoothing](https://mlsanity.com/paper/1902.02918/) (2019)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [Energy Transformer](https://mlsanity.com/paper/2302.07253/) (2023)
- [Modern Methods in Associative Memory](https://mlsanity.com/paper/2507.06211/) (2025)
- [A Biologically Plausible Dense Associative Memory with Exponential Capacity](https://mlsanity.com/paper/2601.00984/) (2026)
- [Dense Associative Memories with Analog Circuits](https://mlsanity.com/paper/2512.15002/) (2025)
- [Saddle Hierarchy in Dense Associative Memory](https://mlsanity.com/paper/2508.19151/) (2025)
- [On the Role of Hidden States of Modern Hopfield Network in Transformer](https://mlsanity.com/paper/2511.20698/) (2025)

## Abstract (arXiv)

Dense Associative Memory (DAM) generalizes Hopfield networks through higher-order interactions and achieves storage capacity that scales as O(Nn-1) under suitable pattern separation conditions. Existing dynamical analyses primarily study the thermodynamic limit N→∞ with randomly sampled patterns and therefore do not provide finite-size guarantees or explicit convergence rates. We develop an algorithmic analysis of DAM retrieval dynamics that yields finite-N guarantees under explicit, verifiable pattern conditions. Under a separation assumption and a bounded-interference condition at high loading, we prove geometric convergence of asynchronous retrieval dynamics, which implies O(log N) convergence time once the trajectory enters the basin of attraction. We further establish adversarial robustness bounds expressed through an explicit margin condition that quantifies the number of corrupted bits tolerable per sweep, and derive capacity guarantees that scale as Theta(Nn-1) up to polylogarithmic factors in the worst case, while recovering the classical Theta(Nn-1) scaling for random pattern ensembles. Finally, we show that DAM retrieval dynamics admit a potential-game interpretation that ensures convergence to pure Nash equilibria under asynchronous updates. Complete proofs are provided in the appendices, together with preliminary experiments that illustrate the predicted convergence, robustness, and capacity scaling behavior.