# Graph Memory: Spectral Associative Memory via Dirichlet Energy

arXiv: https://arxiv.org/abs/2609.32365 · HTML: https://mlsanity.com/paper/2609.32365/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 26 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 21 of 21)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [From Local to Global: A Graph RAG Approach to Query-Focused Summarization](https://mlsanity.com/paper/2404.16130/) (2024)
- [Graph Contrastive Learning with Augmentations](https://mlsanity.com/paper/2010.13902/) (2020)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [Deep Graph Contrastive Representation Learning](https://mlsanity.com/paper/2006.04131/) (2020)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Rethinking Graph Transformers with Spectral Attention](https://mlsanity.com/paper/2106.03893/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Fully Decentralized Multi-Agent Reinforcement Learning with Networked Agents](https://mlsanity.com/paper/1802.08757/) (2018)
- [Position: Graph Foundation Models are Already Here](https://mlsanity.com/paper/2402.02216/) (2024)
- [Text-space Graph Foundation Models: Comprehensive Benchmarks and New Insights](https://mlsanity.com/paper/2406.10727/) (2024)
- [Efficient multivariate entropy estimation via k-nearest neighbour distances](https://mlsanity.com/paper/1606.00304/) (2016)
- [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024)
- [Modern Methods in Associative Memory](https://mlsanity.com/paper/2507.06211/) (2025)
- [Dense Associative Memory with Epanechnikov Energy](https://mlsanity.com/paper/2506.10801/) (2025)
- [Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression](https://mlsanity.com/paper/2311.00140/) (2023)
- [Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps](https://mlsanity.com/paper/2402.14985/) (2024)
- [Dense associative memory for Gaussian distributions](https://mlsanity.com/paper/2509.23162/) (2025)
- [A Flexible Approach for Normal Approximation of Geometric and Topological Statistics](https://mlsanity.com/paper/2210.10744/) (2022)

## Abstract (arXiv)

Dense associative memories have traditionally focused on storing and retrieving vector-valued patterns. Many modern machine learning problems, however, are naturally graph-structured, requiring memory mechanisms for relational patterns, graph diffusion geometries, community structures, and graph-based inductive biases. We propose a spectral dense associative memory for storage and retrieval of graph data, extending the classical vector-valued memories. Retrieval is performed through a log-sum-exp energy induced by Dirichlet energy with spectral norm distances, producing a softmax-weighted average of the stored Laplacians that remains a valid graph Laplacian. We prove exponential storage capacity and exponentially decaying retrieval error. Beyond graph retrieval, we establish theoretical guarantees for spectral quantities central to graph learning, including eigenvalues, eigenspaces, and diffusion operators. Experiments on synthetic graph data, real-world airline network, protein conformation data and wearable sensor data demonstrate robust graph retrieval while preserving the graph geometry of the data. Our framework provides a new associative memory paradigm for graph-structured data and bridges dense associative memory with modern graph learning and generative AI.