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This research introduces a spectral dense associative memory specifically designed for graph-structured data, addressing the limitations of traditional vector-valued models in handling relational patterns. By leveraging Dirichlet energy and spectral norms, the framework achieves exponential storage capacity while preserving essential graph geometries like diffusion operators and community structures. The work establishes a critical bridge between classical associative memory paradigms and modern graph learning, offering new theoretical guarantees for eigenvalues and eigenspaces relevant to generative AI tasks.
Graph Memory: Spectral Associative Memory via Dirichlet Energy
arXiv:2609.32365v1 Announce Type: new Abstract: 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.
Abstract: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.
Machine Learning (cs.LG); Information Retrieval (cs.IR)
arXiv:2609.32365 [cs.LG]
(or arXiv:2609.32365v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.32365 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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