Hopfield Nets
Setup
patterns
$$\boldsymbol{\xi}^\mu \in \{-1,+1\}^d, \quad \mu = 1,\ldots,N$$
i.i.d. Rademacher; load ratio $\alpha := N/d$
weight matrix (Hebbian)
$$W = \frac{1}{d} \sum_{\mu=1}^{N} \boldsymbol{\xi}^\mu (\boldsymbol{\xi}^\mu)^\top$$
update rule (sequential async)
$$s_i \leftarrow \mathrm{sign}(h_i), \quad h_i = \sum_\mu m^\mu \xi^\mu_i - \alpha s_i$$
one step = one full random-order sweep of all $d$ neurons; overlaps $m^\mu$ updated after each flip
masked query
$$\mathbf{s}(0)_i = \begin{cases} \xi^1_i & \text{w.p. } 1-p \\ 0 & \text{w.p. } p \end{cases}$$
fraction $p$ of entries zeroed out
Shared parameters
dimension
$d$
100
perturbation
$p$
0.20
fraction masked
max steps
$T$
30
update steps
trials
$K$
100
per point
converge after
steps
10
unchanged
Alpha sweep
$\alpha$ range
[0.01, 0.30]
40 points
idle
recovery
spurious
no convergence
Single parameter setting
load ratio
$\alpha$
0.10
($N = $ 10)
idle