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Page 128
(3.30) |
According to Equation (3.26), one can ensure that the output described by Equation (3.30) will always be ΔE≤0. For example, a change in the state of vi from 0 to 1 will only occur at time n+1 if
(3.31) |
ΔE will be negative and thus decrease the energy. The case for the state of vi changing from 1 to 0 can be inferred in a similar way. One may therefore calculate how the change in neurone state can affect the energy level. In the continuous case, let λ> >1 in order to simplify the problem. Firstly, rewrite Equation (3.29) as:
(3.32) |
Equation (3.32) denotes the energy change due to neurone i, and its time derivative becomes:
(3.33) |
Since
(3.34) |
Equation (3.33) becomes:
(3.35) |
Since f−1(vi) is a monotonically increasing function indicating f−1′(vi)>0, it follows that expression (3.35) will always be negative or equal to zero. Convergence is therefore guaranteed.
(3.36) |
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