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The two 3x3 blocks read:

k,y := 0,1 ;

The stochastic SOM training algorithm is slow due to the updates of weights after each pattern presentation: all the weights are updated. Batch versions of the SOM training rule have been developed which update weight values only after all patterns have been presented. The first batch SOM training algorithm was developed by Kohonen, and is summarized as follows [Kohonen 1997]: 1. Initialize the codebook vectors by assigning to them the first KJ training patterns, where KJ is the total number of neurons in the map. 2. For each neuron, kj, collect a list of copies of all patterns Zp whose nearest codebook vector belongs to the topological neighborhood of that neuron. 3. Each codebook vector is then assigned the mean over the corresponding list of patterns. 4. If convergence is reached, then terminate training; otherwise return to step 2. Based on the batch learning approach above, Kaski et al. developed a faster version, as summarized below [Kaski et al. 2000]:

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where P and Q are the same as above, while R and 5 now read:

{ Invariant:

2 where k2, = k * x+ k2y is the in-plane wave vector. II y * Without going into detail, it should be noted that in order to make this transformation possible, the element R from equation (10.2) had to be approximated as given in equation (10.17). A simpler approximation, with 7o = (72 + 7a)/2 has previously been in use, but the one given above is much better. The important point to note is that the transformation in equation (10.13) does not depend on kz, and when this becomes an operator kz id/dz (for application to quantum wells) there are no difficulties in changing from one basis to another. To find the band energies and wave functions the block-diagonal form of the Hamiltonian may then be used. Solving one 3x3 block at a time, the wave function is now a vector of length 3, i.e.

1. Initialize the codebook vectors wkj, using any initialization approach. 2. For each neuron, kj, compute the mean over all patterns for which that neuron is the winner. Denote the average by 3. Adapt the weight values for each codebook vector, using

O ^ f e ^ M A y = Xk M-k }

Block-diagonalisation may also be applied to the 4x4 Hamiltonian of equation (10.7), using the basis states FI, F2, F4, and F5 defined in equation (10.13), which transforms the full 4x4 matrix into one which has two 2x2 blocks. Alternatively, one can simply cut out the third row and third column from the two blocks defined in equation (10.16) to obtain the same result. Taking for example the upper 2x2 block:

Bound function:

and evaluating det\H+ E\ = 0 gives:

where nm iterates over all neurons, Nnm is the number of patterns for which neuron nm is the winner, and hnm,kj is the neighborhood function which indicates if neuron nm is in the neighborhood of neuron kj, and to what degree. 4. Test convergence. If the algorithm did not converge, go to step 2. 4.5.3 Growing SOM

{ y = XM } .

which has two non-degenerate solutions:

Using Homer's Rule. Another method, which is commonly used for large values of M, makes use of Horner's rule. Suppose the binary representation of M is stored in array a. Specifically, we assume that M - (Zi\Q^i<N:a[i]x2) where , (13.14)

in which it can be immediately recognised that E- = EHH and E+ ELH from the previous section. The same values would be obtained from the lower 2x2 block. Putting these solutions (one at a time!) back into either of the two equations of the homogeneous linear system in equation (10.19), as convenient, and solving it, gives the eigenvectors for the two eigenenergies as:

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