Graph Memory: Spectral Associative Memory via Dirichlet Energy

Zhaoyang Shi

cs.LGSep 26, 2026

Graph Memory: Spectral Associative Memory via Dirichlet Energy

Authors: Zhaoyang Shi

Organizations: Center for Applied Mathematics, Fudan University Shanghai, China

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.

Figures & tables

Figure 1: Energy landscape of the proposed graph LSE with $N=5$ stored two-community stochastic block model (SBM) graphs. Each graph has $n=60$ nodes with community sizes $(30,30)$ . Since the Laplacian space is high-dimensional, the landscape is visualized by restricting the query graph’s Laplacian to the two-dimensional subspace spanned by the first two principal components of the stored graphs’ Laplacians. Stars denote the projected locations of the stored graph memories.

G^{(t+1)}\leftarrow\mathrm{Decode}\!\left(\widetilde{L}^{(t+1)}\right)\ \text{via}\ W^{(t+1)}=\operatorname{diag}(\widetilde{L}^{(t+1)})-\widetilde{L}^{(t+1)}.

Algorithm 1 One-step graph LSE retrieval update (operator

\mathcal{T}

)

Panel of Figure 2 — Figure 2: SBM data retrieval performance. (a) Retrieval accuracy of Graph-DAM and Eu-DAM versus inverse temperature $\beta_{n}$ . (b)-(d) Fiedler vectors of the target memory, Graph-DAM retrieval, and Eu-DAM retrieval. Dashed lines indicate the four ground-truth communities.

Panel of Figure 3 — Figure 3: Airline-network retrieval performance. (a) Retrieval accuracy of Graph-DAM, Eu-DAM, and Raw Eu-DAM versus inverse temperature $\beta_{n}$ . (b) Target network. (c) Graph-DAM correctly retrieves the target memory. (d)-(e) Eu-DAM and Raw Eu-DAM retrieve an incorrect memory.

Panel of Figure 4 — Figure 4: Protein conformation retrieval. (a) Retrieval accuracy versus inverse temperature $\beta_{n}$ . (b) Target protein conformation. (c) Graph-DAM correctly retrieves the target memory. (d)–(e) Eu-DAM and Raw Eu-DAM retrieve incorrect protein memories.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: UCI HAR data retrieval performance. Retrieval accuracy of Graph-DAM, Eu-DAM, and Raw Eu-DAM versus inverse temperature $\beta_{n}$ .

Figure 6: UCI HAR retrieval example visualized in the original sensor space. The top and bottom rows show the three-axis accelerometer and gyroscope signals, respectively. From left to right, we show the corrupted query, its true target memory, and the memories retrieved by Graph-DAM, Eu-DAM, and Raw Eu-DAM. The query corresponds to a sitting recording from subject S23. Graph-DAM correctly retrieves the target S23 sitting memory, whereas Eu-DAM retrieves an S17 sit-to-lie recording and Raw Eu-DAM retrieves an S17 sitting recording. All signals are plotted over normalized time and standardized using the target-memory statistics to allow direct comparison of their temporal patterns and amplitudes.

Figure 7: Three-dimensional sensor trajectories for the UCI HAR retrieval example. The top and bottom rows visualize the three-dimensional accelerometer trajectories $(a_{x},a_{y},a_{z})$ and angular-velocity trajectories $(\omega_{x},\omega_{y},\omega_{z})$ , respectively, for the corrupted query, target memory, and memories retrieved by Graph-DAM, Eu-DAM, and Raw Eu-DAM. Graph-DAM recovers the true S23 sitting memory, while Eu-DAM retrieves an S17 sit-to-lie memory and Raw Eu-DAM retrieves an S17 sitting memory. The trajectories provide a direct visualization in the original sensor domain of the motion patterns associated with the retrieved graph memories. Circles and crosses indicate the starting and ending points of each trajectory, respectively.

Figure 8: UCI DSA retrieval performance. Retrieval accuracy of Graph-DAM, Eu-DAM, and Raw Eu-DAM as a function of inverse temperature $\beta_{n}$ . Graph-DAM rapidly approaches approximately $98\%$ retrieval accuracy and remains stable in the high-inverse-temperature regime, while Eu-DAM and Raw Eu-DAM plateau at approximately $26\%$ and $90\%$ , respectively. Error bars denote standard errors over five independent corruption trials.

Figure 9: UCI DSA retrieval example visualized in the original sensor space. The rows show the three-axis acceleration signals measured at the torso, right leg, and left leg, respectively. From left to right, we show the corrupted query, its true target memory, and the memories retrieved by Graph-DAM, Eu-DAM, and Raw Eu-DAM. The target is a jumping recording from subject S5. Graph-DAM correctly retrieves the S5 jumping memory, whereas Eu-DAM retrieves an S5 standing memory and Raw Eu-DAM retrieves an S7 jumping memory. Signals are standardized using the target-memory statistics to allow direct comparison of their temporal patterns and amplitudes.

Figure 10: Three-dimensional acceleration trajectories for the UCI DSA retrieval example. The rows visualize the three-dimensional acceleration trajectories $(a_{x},a_{y},a_{z})$ measured at the torso, right leg, and left leg, respectively, for the corrupted query, target memory, and memories retrieved by Graph-DAM, Eu-DAM, and Raw Eu-DAM. Graph-DAM recovers the true S5 jumping memory, Eu-DAM retrieves an S5 standing memory, and Raw Eu-DAM retrieves an S7 jumping memory. The trajectories provide a geometric visualization in the original sensor domain of the whole-body motion associated with each retrieved memory. Circles and crosses indicate the starting and ending points of each trajectory, respectively.

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We introduce Graph Hopfield Networks, whose energy function couples associative memory retrieval with graph Laplacian smoothing for node classification. Gradient descent on this joint energy yields an iterative update interleaving Hopfield retrieval with Laplacian propagation. Memory retrieval provides regime-dependent benefits: up to 2.0~pp on sparse citation networks and up to 5 pp additional robustness under feature masking; the iterative energy-descent architecture itself is a strong inductive bias, with all variants (including the memory-disabled NoMem ablation) outperforming standard baselines on Amazon co-purchase graphs. Tuning enables graph sharpening for heterophilous benchmarks without architectural changes.

Abstract

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