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Spatially-Aware Self-Organizing Graph Autoencoders via Localized Hebbian Learning

Robin Koç, Hazal Moğultay Özcan, Fatoş T. Yarman Vural

arXiv:2610.34346v1cs.CV

TL;DR

Existing approaches use global pooling and scalar connections that discard spatial relationships among feature maps, while abrupt message-passing transitions can cause representation shock. S-LooM combines localized spatial correlations, learnable spatial kernels, artifact-free decoding, and soft blending. On CIFAR-10, it reduces validation loss versus prior LooM at p = 0.05, though dense graphs can still entangle features.

  • Problem

    Early Hebbian graph autoencoders use global averaging and scalar connections, limiting how they represent spatial relationships among convolutional feature maps.

    Source: §I · p. 1

  • Method

    S-LooM uses localized spatial cross-correlation, learnable spatial kernels, artifact-free nearest-neighbor decoding, and a soft-blending curriculum.

    Source: §I · pp. 1–2, §VII · p. 6

  • Results

    At p = 0.05, validation loss was 0.0114 for S-LooM versus 0.0272 for prior LooM; validation loss stayed ∼0.010 across graph densities, though dense graphs entangled features.

    Source: A. Experimental Setup and Baseline… · p. 4, B. Stability Across Graph Densities… · p. 5

  • Takeaways & Limitations

    On CIFAR-10, sparse latent graphs compartmentalized into explainable functional modules isolating structural boundaries, high-frequency textures, and redundant connections.

    Source: §VII · p. 6

    At p = 0.50, qualitative analysis found severe feature entanglement, with single-node ablation causing widespread imagewide color and shape distortions.

    Source: B. Stability Across Graph Densities… · p. 5

Abstract

from arXiv · show

Traditional autoencoders rely on static network architectures to learn latent representations. Recently we suggested a dynamic architecture inspired by Hebbian plasticity, enabling topological self-organization among convolutional feature maps (R. Ko\c{c} et al., Proceedings of the SIU, 2026). Early implementations of this paradigm infer relationships among feature maps using global average pooling and model their interactions through scalar-valued connections. This approach falls short to fully exploit the underlying spatial information, embedded in input images. Furthermore, abrupt transitions between different loss functions and training dynamics can introduce representation shocks, leading to unstable feature learning. In this paper, we introduce a Spatially-Aware Self-Organizing Graph Autoencoder (S-LooM) to address these structural bottlenecks. By using spatial cross-correlation, and spatial kernels, S-LooM establishes dynamic connections among feature maps. We replace static scalar message passing of autoencoders with learnable spatial kernels and eliminate high-frequency deconvolutional artifacts observed in decoders using nearest-neighbor upsampling. Via a soft-blending training curriculum, the latent topology aligns smoothly during training. Our experiments show that S-LooM yields a more stable topology, compared to the state of the art dynamic architectures. Meaning, the graph structure that emerge has more resistance to feature entanglement and noise. It forms functional communities responsible for visual attributes, such as illumination, shape and texture even under high connection densities.

I. INTRODUCTION

S-LooM extends dynamic Hebbian graph learning to convolutional feature maps, addressing limitations in spatial co-activation, message passing, decoder artifacts, and training transitions. It builds on prior work on dynamic graphs and artifact-free decoding.

I. INTRODUCTION

  • Prior LooM dynamically optimized latent connectivity with a self-organizing graph, but its initial implementation had bottlenecks in translating the approach to convolutional features.The cited prior work is identified as.Source: §I · p. 1
  • Global averaging can merge features active in disjoint image regions, while scalar message passing uniformly amplifies or dims whole feature maps.Abrupt message-passing introduction can shift latent distributions and force the decoder to unlearn weights; transposed-convolution artifacts can also create false topological connections.Source: §I · p. 1
  • S-LooM uses localized spatial cross-correlation and learnable spatial kernels to connect feature maps, while artifact-free decoding and soft blending address structural noise and representation shock.The graph layer dynamically reorganizes feature maps based on co-activation relationships.Source: §I · pp. 1–2

II. RELATED WORK

  • Dynamic graph learning seeks efficient adaptive mechanisms for discovering latent feature relationships, in contrast to predefined adjacency matrices [3, 4].Source: §II · p. 2
  • Resize-convolutions—nearest-neighbor upsampling followed by standard convolution—are described as eliminating high-frequency checkerboard distortions.S-LooM integrates artifact-free decoding because structural noise can corrupt the Hebbian co-activation statistics guiding the model.Source: §II · p. 2

III. GENERAL ARCHITECTURE OF S-LOOM

S-LooM places a dual-graph layer between a convolutional encoder and decoder, preserving spatial feature maps through the pipeline. Its decoder uses resize-convolutions to avoid checkerboard artifacts.

III. GENERAL ARCHITECTURE OF S-LOOM

  • The encoder preserves K feature maps as two-dimensional matrices, and the graph layer reorganizes them without changing their dimensions before decoding.The encoder applies a sigmoid to normalize feature-map values.Source: §III · p. 2
  • The encoder’s feature maps branch into localized correlations for the Hebbian matrix and spatial maps for the learnable graph.The graphical abstract depicts the topological alignment loss between the two graph structures.Source: Figure 1 · p. 2, §III · p. 2
  • Uneven kernel overlaps in transposed convolutions can create checkerboard artifacts whose decoder noise produces false correlations and distorts Hebbian statistics.Source: §III · pp. 2–3
  • S-LooM replaces transposed convolutions with nearest-neighbor upsampling followed by standard convolution, separating spatial expansion from learnable feature processing.The uniform structural paths are described as preventing checkerboard patterns and keeping gradients clean for Hebbian learning.Source: §III · p. 3

IV. SPATIALLY-AWARE DUAL-GRAPH LAYER

The dual-graph layer self-organizes feature-map relationships without a predefined adjacency matrix. A Hebbian graph guides the learnable graph, and self-connections are masked to enforce inter-feature message passing.

IV. SPATIALLY-AWARE DUAL-GRAPH LAYER

  • The dual-graph layer comprises a learnable autoencoder graph and a Hebbian graph over the encoder’s spatial feature maps.The learnable graph is represented by a node set and edge-weight matrix.Source: §IV · p. 3
  • The Hebbian graph guides the learnable graph by pulling its weights toward co-activation relationships during backpropagation, without sacrificing reconstruction accuracy.Source: §IV · p. 3
  • Both graphs evolve during training, and zeroing each node’s self-connection enforces inter-feature message passing.The graph is described as evolving over shared nodes at each training step.Source: §IV · p. 3

A. Localized Hebbian Co-activation

S-LooM estimates Hebbian co-activation from spatially aligned feature maps and imposes sparsity by pruning weak edges. Surviving edge correlations are accumulated for the Hebbian graph.

A. Localized Hebbian Co-activation

  • Co-activation is computed from centered, normalized feature maps using their maximum spatial cross-correlation, allowing spatial shifts between maps.The cross-correlation operator uses two-dimensional spatial shifts.Source: A. Localized Hebbian Co-activation · p. 3
  • A target sparsity ratio p sets a quantile threshold over edges, dynamically pruning weak connections rather than applying activation thresholds to individual nodes.Source: A. Localized Hebbian Co-activation · p. 3
  • Surviving active-edge correlations are accumulated, and the Hebbian graph is normalized into a stable graph skeleton before guiding the learnable graph.The normalization finds the maximum edge weight across feature nodes.Source: A. Localized Hebbian Co-activation · p. 3

B. Spatial Message Passing

S-LooM replaces scalar-only message passing with learnable spatial kernels, allowing feature maps to exchange spatially transformed information. Incoming messages are combined with the original feature map and normalized to keep representations stable.

B. Spatial Message Passing

  • Spatial kernels let messages shift and recombine parts of co-activating feature maps, rather than uniformly amplifying or dimming an entire map.The kernels define spatial transformations between feature nodes and are initialized from a zero-mean normal distribution with standard deviation 0.02.Source: B. Spatial Message Passing · pp. 3–4
  • Each message convolves a source feature map with its spatial kernel, scaled by a continuous scalar edge weight.The aggregated message sums these weighted convolutions from source nodes.Source: B. Spatial Message Passing · p. 4
  • The update adds aggregated messages to the original feature map and normalizes by incoming edge weights, bounding outputs from sigmoid-constrained inputs.The stated normalization is intended to keep representations stable even with dense graphs.Source: B. Spatial Message Passing · p. 4

V. SOFT-BLENDING CURRICULUM AND OPTIMIZATION

S-LooM gradually introduces graph processing into the latent features and balances reconstruction with graph-topology alignment. This curriculum lets the decoder adapt gradually and contributes to stable validation performance.

V. SOFT-BLENDING CURRICULUM AND OPTIMIZATION

  • Soft blending interpolates between raw encoder features and graph-processed features while the graph loss coefficient scales up.The curriculum directly addresses the prior abrupt feature-map transition associated with representation shock.Source: §V · p. 4
  • Training begins with the graph layer bypassed, then activates it during topological alignment and soft blending.Source: §V · p. 4
  • The graph layer is frozen for a final feature-map fine-tuning stage, and the loss combines reconstruction with graph-topology similarity.The total loss dynamically balances the two objectives.Source: §V · p. 4
  • The smooth curriculum lets the decoder adapt gradually to changing latent topology, contributing to stable validation performance.Source: §V · p. 4

VI. EXPERIMENTS AND RESULTS

The evaluation is restricted to CIFAR-10 and compares S-LooM with the original LooM architecture, focusing on baseline fidelity, stability across graph densities, and visual explainability.

VI. EXPERIMENTS AND RESULTS

  • Evaluation is restricted to CIFAR-10 to isolate architectural effects, with the original LooM architecture as the benchmark.The evaluation focuses on baseline fidelity, stability across graph densities, and visual explainability.Source: §VI · p. 4

A. Experimental Setup and Baseline Performance

Under the same highly sparse target constraint, S-LooM reports lower validation loss than prior LooM. The evaluation also presents qualitative reconstructions and uses matched training hyperparameters.

A. Experimental Setup and Baseline Performance

B. Stability Across Graph Densities and Feature Entanglement

S-LooM maintains stable validation loss across graph densities, while topology quality varies: dense graphs become entangled, whereas sparse graphs align with the Hebbian teacher.

B. Stability Across Graph Densities and Feature Entanglement

  • Table I reports validation losses across target sparsity ratios, which determine the density of the topology dictated by the Hebbian teacher.The supplied table caption identifies its comparison as target sparsity versus validation loss.Source: B. Stability Across Graph Densities… · p. 5, Table I · p. 5
  • S-LooM maintains validation loss near 0.010 across densities, while the foundational architecture degrades as sparsity ratio p reaches 0.30 or higher.The paper attributes the foundational model’s degradation to structural noise from statistically weaker connections.Source: B. Stability Across Graph Densities… · p. 5
  • At p = 0.50, S-LooM’s latent space suffers severe feature entanglement, whereas at p = 0.05 its learnable graph strongly aligns with the localized Hebbian teacher.The p = 0.05 comparison is shown in Fig. 3.Source: B. Stability Across Graph Densities… · p. 5, Figure 3 · p. 5

C. Functional Specialization and Community Emergence

At p = 0.05, the learned graph forms functionally specialized communities, and individual connections can map to localized image details. Ablations also show that isolated nodes have little impact.

C. Functional Specialization and Community Emergence

  • Community ablations identify structural-body handlers (Comm 1), boundary and shadow mappers (Comm 2 and 3), and high-frequency texture handlers (Comm 4).The paper reports these functional roles as consistent across the dataset, despite Fig. 4 showing one selected validation sample.Source: C. Functional Specialization and… · p. 5, Figure 4 · p. 6
  • Ablating the 17-node isolated pool yields a nearly imperceptible difference map, suggesting the sparse graph quarantines redundant or uninformative filters.The isolated pool is attributed to the Hebbian teacher.Source: C. Functional Specialization and… · p. 5

D. Micro-Level Explainability via Single-Edge Lesioning

  • Severing edge 10 ↔ 31 disrupts only the truck’s lower-left headlight and bumper contour, rather than causing generalized blur or channel failure.Fig. 5 shows the localized disruption from lesioning the network’s strongest edge.Source: D. Micro-Level Explainability via… · p. 5, Figure 5 · p. 6

E. Computational Complexity

S-LooM adds spatial cross-correlation cost during training, but its inference uses message passing at a compressed latent bottleneck. The conclusion reports CIFAR-10 gains while noting the evaluation’s limited scope.

E. Computational Complexity

  • Training graph construction adds O(K^2 · (H′W′)^2) computation per batch; vectorized unfolding and parallel matrix multiplication trade increased VRAM use for faster GPU execution.This overhead comes from calculating pairwise spatial cross-correlations.Source: E. Computational Complexity · p. 6
  • Inference uses masked 2D convolution with complexity O(K^2 · s^2 · H′ · W′) at the compressed latent bottleneck, maintaining speed almost identical to a standard autoencoder.Hebbian statistics gathering is disabled after training.Source: E. Computational Complexity · p. 6

VII. CONCLUSION

  • The conclusion describes artifact-free nearest-neighbor decoding and soft blending as addressing structural noise and representation shock.The curriculum gradually introduces message passing to mitigate representation shock.Source: §VII · p. 6
  • The conclusion attributes reduced baseline reconstruction error and stability at high graph densities to S-LooM’s refinements, with evaluations limited to CIFAR-10.Future work is proposed to benchmark broader graph-based models and scale to higher-resolution, more complex datasets.Source: §VII · p. 6
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