How to prove the Energy function of a Hopfield net is monotonically decreasing?
$E = -1/2 \sum_{i,j} {w_{ij}}{s_i}{s_j} + \sum_{i}^{}s_{i} \theta_i$
I'll assume a proof involves the standard definition(s) of the activation function:
(1) $a_i \leftarrow \left\{ \begin{matrix} 1 & \mbox {if }\sum_{j}{w_{ij}s_j}>\theta_i, \\ -1 & \mbox {otherwise.}\end{matrix} \right\}$
(2) $a_i \leftarrow \left\{ \begin{matrix} 1 & \mbox {if }\sum_{j}{w_{ij}s_j}>\theta_i, \\ 0 & \mbox {otherwise.}\end{matrix} \right\}$
And the normal rules for weights:
$w_{ij}=0, \forall i=j$ (not reflexive)
$w_{ij} = w_{ji}, \forall i,j$ (symmetric)
If someone could set me on the right path, it would be appreciated.