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Optimizing the neighborhood If the Gaussian neighborhood function as given in equation (4.20) is used, all neurons will be in the neighborhood of the BMN, but to different degrees, due to the asymptotic characteristics of the function. Thus, all codebook vectors are updated, even if they are far from the BMN. This is strictly not necessary, since neurons far away from the BMN are dissimilar to the presented pattern, and will have negligible weight changes. Many calculations can therefore be saved by clipping the Gaussian neighborhood at a certain threshold without degrading the performance of the SOM. Additionally, the width of the neighborhood function can change dynamically during training. The initial width is large, with a gradual decrease in the variance of the Gaussian, which controls the neighborhood. For example, a(t) = a(0)e-t/Tl where r\ is a positive constant, and cr(0) is the initial, large variance. If the growing SOM (refer to Section 4.5.3) is used, the width of the Gaussian neighborhood function should increase with each increase in map size. Learning Rate A time-decaying learning rate may be used, where training starts with a large learning rate which gradually decreases. That is, rj(t) = r/COje-*/72 (4.22) (4.21)

Adopting a similar argument as above, the integral over the plane can then be written as:

Y x2xs + a [k--i]

and hence equation (6.33) gives the final expression for A as follows:

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where T2 is a positive constant and 7 (0) is the initial, large learning rate (refer to chapter 7 to read about the consequences of large and small learning rates). Shortcut Winner Search The shortcut winner search decreases the computational complexity by using a more efficient search for the BMN. The search is based on the premise that the BMN of a pattern is in the vicinity of the BMN for the previous epoch. The search for a BMN is therefore constrained to the current BMN and its neighborhood. In short, the search for a BMN for each pattern is summarized as 1. Retrieve the previous BMN. 2. Calculate the distance of the pattern to the codebook vector of the previous BMN.

exponentiation } y = (X ) xXa[k~l]

An analogous expression exists for B with the subscripts e and h, labelling electron and hole, respectively, being interchanged. Now consider C = (^\He-h\^) as defined in equation (6.24). As mentioned earlier, He-h is composed of two terms which represent the in-plane kinetic energy of the relative motion and the Coulombic potential energy between the electron and hole, i.e. By using the kinetic energy operator defined above in equation (6.11) then the first of these terms can then be written as:

So, to maintain invariant the property y = Xs, we must simultaneously perform the assignment y := y2xXa[k~1] . In this way, we have constructed the following algorithm:

where the in-plane reduced mass u, is given by:

3. Calculate the distance of the pattern to all direct neighbors of the previous BMN. 4. If the previous BMN is still the best, then terminate the search; otherwise, let the new BMN be the neuron (within the neighborhood) closest to that pattern. Shortcut winner search does not perform a search for the BMN over the entire map, but just within the neighborhood of the previous BMN, thereby substantially reducing computational complexity.

Again, whatever the functional form of the relative motion term wrr, it is apparent that the kinetic energy operator in CT will act only upon wr, as the one-particle electron and hole wave functions, we and wh, respectively, are not functions of a; or y. Furthermore, the integration over the plane will result in an entity which is a function of a only, i.e. in the spirit of equation (6.35):

k,s,y := N.0,1 ; { Invariant: CKfc^N A y = Xs A sx2k = (Zi:k^i<N:a[i]x2i)

since wr is real. In a similar manner, the potential energy term Cv can be written as:

Bound function: k }

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