Klr Hopfield Event Driven Retrieval

---
name: klr-hopfield-event-driven-retrieval
description: >
  Kernel Logistic Regression (KLR) Hopfield Network with asynchronous event-driven retrieval methodology.
  Enables high-capacity associative memory (P/N ≈ 30, vs classical 0.14N) with
  neuromorphic-compatible sparse computation. Use when: designing associative memory systems,
  event-driven neuromorphic hardware deployment, comparing Hopfield variants (KLR vs MHN),
  analyzing attractor landscapes, kernel-based neural networks, or studying asynchronous vs
  synchronous retrieval dynamics in neural computing.
  Triggers: KLR Hopfield, kernel associative memory, event-driven retrieval, asynchronous Hopfield,
  neuromorphic associative memory, kernel logistic regression memory, high-capacity Hopfield,
  margin-induced attractor, asynchronous neural dynamics, 核逻辑回归Hopfield.
---

# KLR Hopfield Network: Asynchronous Event-Driven Retrieval

## Overview

Kernel Logistic Regression (KLR) Hopfield networks achieve high storage capacity (P/N ≈ 30)
via margin maximization in kernel feature space, while supporting efficient asynchronous,
event-driven retrieval — making them suitable for neuromorphic hardware (Loihi, TrueNorth).

Source: Akira Tamamori, arXiv:2605.05978v1 [cs.NE] (May 2026).

## Core Mechanism

### KLR Hopfield Model

- Neurons: bipolar state s ∈ {-1, 1}^N, storing P patterns {ξ^μ}_{μ=1}^P
- Local field: h_i(s) = Σ_{μ} α^μ_i · K(s, ξ^μ)
- Kernel: RBF kernel K(x, y) = exp(-γ‖x - y‖²), locality parameter γ
- Learning: L2-regularized negative log-likelihood per neuron, yielding dual variables α ∈ R^{P×N}

### Pseudo-Energy Function

V(s) = -Σ_i s_i · h_i(s) = -Σ_i s_i · Σ_μ α^μ_i · K(s, ξ^μ)

KLR optimization creates **large-margin attractors** → smooth energy landscape.

### Two Update Schemes

| Scheme | Description | Pros | Cons |
|--------|-------------|------|------|
| Synchronous | All N neurons update simultaneously | GPU efficient | May oscillate; O(PN) per step |
| Asynchronous | Single neuron updates sequentially | Guaranteed fixed-point convergence; sparse events | Sequential order dependency |

Key finding: under optimal γ, trajectories are **statistically indistinguishable** between schemes.

## Key Results

### Storage Capacity
- Classical Hopfield: P ≈ 0.14N
- KLR Hopfield (async): P/N ≈ 30 with 100% recall (N=100, 200 tested)
- Capacity scales positively with network size

### Event Efficiency
- Async bit flips ≈ initial Hamming distance (near-optimal)
- 20% noise: ~10 events vs. 150 evaluations (synchronous, 3 steps × 50 neurons)
- **15× reduction** in state evaluations

### Kernel Parameter Trade-off
- γ = 0.02 (Ridge regime): sharp attractors, optimal for static memory, minimal noise
- γ = 0.1 (Robust regime): wider basins, robust to 10-20% noise, high storage load
- Smaller γ isolates patterns; larger γ enlarges basins but risks crosstalk

## Implementation Pattern

```python
import numpy as np
from scipy.special import expit  # sigmoid

class KLRHopfieldNetwork:
    def __init__(self, N, gamma=0.1, lr=0.1, weight_decay=0.01):
        self.N = N
        self.gamma = gamma
        self.lr = lr
        self.lam = weight_decay
        self.alpha = None  # dual variables, shape (P, N)
        self.patterns = None  # shape (P, N)
        self.K = None  # Gram matrix, shape (P, P)
    
    def rbf_kernel(self, x, y):
        return np.exp(-self.gamma * np.sum((x - y)**2))
    
    def gram_matrix(self, patterns):
        P = len(patterns)
        K = np.zeros((P, P))
        for i in range(P):
            for j in range(P):
                K[i, j] = self.rbf_kernel(patterns[i], patterns[j])
        return K
    
    def fit(self, patterns, epochs=500):
        """Learn dual variables via KLR."""
        self.patterns = np.array(patterns)
        P, N = self.patterns.shape
        self.alpha = np.zeros((P, N))
        self.K = self.gram_matrix(self.patterns)
        
        for epoch in range(epochs):
            for i in range(N):
                h = self.K @ self.alpha[:, i]  # (P,)
                sigma = expit(h)
                y = (self.patterns[:, i] + 1) / 2  # {0, 1}
                grad = self.K @ (sigma - y) + self.lam * self.alpha[:, i]
                self.alpha[:, i] -= self.lr * grad
    
    def local_field(self, s, i):
        """Compute local field for neuron i."""
        K_s = np.array([self.rbf_kernel(s, xi) for xi in self.patterns])
        return np.dot(self.alpha[:, i], K_s)
    
    def retrieve_sync(self, s_init, max_steps=50):
        """Synchronous retrieval."""
        s = s_init.copy()
        for _ in range(max_steps):
            h = np.array([self.local_field(s, i) for i in range(self.N)])
            s = np.sign(h)
        return s
    
    def retrieve_async(self, s_init, max_epochs=50):
        """Asynchronous event-driven retrieval."""
        s = s_init.copy()
        events = 0
        for _ in range(max_epochs):
            order = np.random.permutation(self.N)
            for i in order:
                old_s_i = s[i]
                s[i] = 1 if self.local_field(s, i) >= 0 else -1
                if s[i] != old_s_i:
                    events += 1
        return s, events
```

## Hardware Mapping

| Component | Software | Neuromorphic Hardware |
|-----------|----------|----------------------|
| Neuron state | Binary variable | SRAM cell / memristor |
| Kernel computation | Dense matrix-vector | Event-triggered MAC |
| Async update | Sequential loop | Spike-triggered neuron |
| Storage | α matrix in RAM | On-chip weight memory |

- Complexity: O(PN) synchronous vs. O(events × P) asynchronous
- Event count ≈ Hamming distance → substantial power savings on Loihi/TrueNorth

## When to Use

- **Need high capacity**: KLR Hopfield >> Classical Hopfield (30× vs 0.14×)
- **Deploy on neuromorphic hardware**: Async event-driven is native fit
- **Memory-constrained**: KLR stores patterns implicitly via dual variables
- **Compare with MHN**: KLR uses simpler binary architecture + margin optimization vs MHN's exponential energy

## Pitfalls

- **Kernel γ tuning is critical**: Too small → narrow basins, poor noise robustness; too large → crosstalk at high load
- **Limited to tested regime**: Results validated on random patterns; structured/correlated patterns may differ
- **No theoretical capacity bound**: P/N ≈ 30 is empirical lower bound; actual capacity unknown
- **Learning is per-neuron**: α_i learned independently, no inter-neuron weight sharing during training

## Related Skills

- `kernel-hopfield-attractor-geometry` — attractor boundaries and storage capacity analysis
- `kernel-hopfield-event-driven-retrieval` — event-driven retrieval in KLR Hopfield networks
- `klr-hopfield-associative-memory` — KLR Hopfield networks overview