kernel-hopfield-associative-memory

name: kernel-hopfield-associative-memory description: "Kernel Hopfield networks: geometric analysis of attractor boundaries and storage capacity limits. KLR-trained associative memories with P/N ~16 for random sequences and ~20 for structured data. Trigger words: kernel Hopfield, associative memory, KLR, attractor basin, storage capacity, kernel logistic regression." category: neuroscience

Kernel Hopfield Associative Memory Networks

Skill based on arXiv:2605.00366v1 - Geometric analysis of attractor boundaries and storage capacity limits in kernel Hopfield networks.

Core Concept

High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit strong storage capabilities. This paper investigates the global geometry of attractor basins and the physical determinants of storage limits in KLR-trained Hopfield networks.

Key Findings

Storage Capacity

• Random sequences: P/N ≈ 16 (patterns per neuron)
• Structured data (CIFAR-10): Effective load near P/N ≈ 20
• Significantly exceeds classical Hopfield limit of P/N ≈ 0.14

Attractor Geometry

• "Ridge of Optimization": Attractors separated by sharp, phase-transition-like boundaries
• Morphing analysis reveals:
  • Well-defined basin boundaries
  • Transition regions between attractors
  • Capacity limit corresponds to boundary collapse

Analysis Methods

1. Empirical evaluation: Random sequences + real-world image embeddings
2. Phenomenological morphing: Interpolate between stored patterns
3. Statistical SNR analysis: Signal-to-Noise Ratio of pattern retrieval

Mathematical Framework

Kernel Logistic Regression Training

E(x) = -Σᵢ αᵢ K(x, ξᵢ)

where:

• K(x, ξᵢ): Kernel function
• ξᵢ: Stored patterns (memories)
• αᵢ: Learned weights from logistic regression

Dynamics

dx/dt = -∇E(x)

• Energy landscape defined by kernel expansion
• Gradient flow converges to attractors
• Attractors correspond to stored patterns

Kernel Choice

• RBF/Gaussian kernels for smooth landscapes
• Polynomial kernels for structured data
• Kernel bandwidth controls basin size

Attractor Basin Analysis

Morphing Experiments

• Create interpolated states between two stored patterns
• Run dynamics from interpolated states
• Observe which attractor captures the state
• Map basin boundaries in pattern space

Phase Transition at Capacity Limit

• Below capacity: Sharp basin boundaries
• Near capacity: Boundaries become irregular
• Above capacity: Boundaries collapse, retrieval fails

SNR Analysis

• Signal: Component aligned with stored pattern
• Noise: Interference from other stored patterns
• Capacity limit: SNR drops below retrieval threshold

Implementation

Training Procedure

# Store patterns using KLR
def store_patterns(patterns, kernel='rbf', gamma=1.0):
    # Compute kernel matrix
    K = kernel_matrix(patterns, patterns, kernel, gamma)
    # Train logistic regression for each pattern
    weights = train_klr(K, targets)
    return weights, patterns, kernel_params

# Retrieve pattern from cue
def retrieve(cue, weights, stored_patterns, kernel_params, steps=100):
    x = cue.copy()
    for _ in range(steps):
        # Compute energy gradient
        grad = compute_gradient(x, weights, stored_patterns, kernel_params)
        # Gradient descent
        x -= learning_rate * grad
    return x

Parameters

• Kernel type: RBF, polynomial, linear
• Kernel bandwidth: Controls interaction range
• Regularization: Prevents overfitting
• Steps: Number of retrieval iterations

Applications

Associative Memory

• Pattern completion from partial cues
• Error correction in noisy inputs
• Content-addressable memory

Machine Learning

• Kernel-based classification
• Memory-augmented neural networks
• Few-shot learning via pattern storage

Neuroscience Modeling

• Memory storage in biological networks
• Attractor dynamics in cortex
• Capacity limits of neural systems

Comparison with Classical Hopfield

References

• Paper: Geometric analysis of attractor boundaries and storage capacity limits in kernel Hopfield networks
• Author: Akira Tamamori
• arXiv: 2605.00366v1 [cs.NE]
• Categories: Neural and Evolutionary Computing (cs.NE)
• Date: May 1, 2026

Related Skills

• hippocampal-replay-credit-assignment
• neuro-attractor-landscape-working-memory
• attractor-metadynamics-neural
• hippi-hippocampal-inspired-memory