On the Basin of Attraction and Capacity of Restricted Hopfield Network as an Auto-Associative Memory
Bibliographic record
Abstract
This paper introduces the eigenvalue interlacing theory and the concept of the condition number to analyze the structural characteristics and attraction basin radius generated by both the Hopfield Neural Network (HNN) and the Restricted Hopfield Network (RHN). Both networks can be viewed as higher-dimensional dynamical systems that store patterns as fixed points. By studying two sets of AUTS cases, involving 35 and 63 nodes respectively, memorized in both HNN and RHN, we introduce the concept of the “effective condition number” as an indicator of the models’ capacity. When the “effective condition number” surpasses a threshold, the HNN model becomes incapable of memorizing new patterns and forfeits all previously stored information when new patterns are added. In contrast, the RHN, when trained using Back-propagation Through Time (BPTT) or Subspace Rotation Algorithm (SRA) with appropriate weight initialization, consistently maintains an “effective condition number” close to one, thus, its capacity could increased with growing complexity of the model. For the sake of facilitating meaningful comparisons, the term “radius” is defined to objectively assess the performance of both HNN and RHN models. Experimental results demonstrate that the RHN model generally outperforms the HNN model in terms of both radius and the uniformity of attraction basins. Furthermore, this paper provides a brief discussion of how the capacity of the RHN tends to increase with a growing number of hidden nodes. Remarkably, within the capacity range of the RHN model, well-trained models with fewer hidden nodes exhibit larger attraction basin radii.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".