Classification Methods for Remotely Sensed Data, Second Edition

Page 128

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 v_i from 0 to 1 will only occur at time n+1 if

ΔE will be negative and thus decrease the energy. The case for the state of v_i 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:

Equation (3.32) denotes the energy change due to neurone i, and its time derivative becomes:

Since

Equation (3.33) becomes:

Since f⁻¹(v_i) is a monotonically increasing function indicating f^−1′(v_i)>0, it follows that expression (3.35) will always be negative or equal to zero. Convergence is therefore guaranteed.