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Finding the outputs

We shall now return back to the reconstruction mapping. We derived the winning ratios tex2html_wrap_inline1687 by starting with two neurons and the reconstruction mapping in equation 3.3. The optimal outputs were then defined by equation 2.1. The relation between the winning strength tex2html_wrap_inline1659 and the output y was defined in equation 3.5 to be tex2html_wrap_inline1741 . The reconstruction mapping was thus


There are no guarantees, however, that we could use the tex2html_wrap_inline1737 defined in equation 3.7 in place of tex2html_wrap_inline1663 . There is no reason why the reconstruction defined in equation 3.8 would be even close to the optimal. It turns out, however, that we are already quite close and can get even closer by making a few corrections to tex2html_wrap_inline1737 . The corrections are such that they approximately preserve the relations between the winning strengths, but correct their overall magnitudes so that we get an agreement with the reconstruction equation 3.8.

Again it should be emphasised that we want to get an agreement with the reconstruction mapping only in order to be able to derive the learning rule and to make sure that the outputs contain information about the inputs. For these purposes even an approximation will suffice.

We would now like to find a mapping from tex2html_wrap_inline1737 to tex2html_wrap_inline1663 . It should be such that equation 3.8 yields a fairly good reconstruction. We shall make two corrections to the preliminary winning strengths tex2html_wrap_inline1737 . We obtain the first one by considering n neurons, which have nearly parallel weight vectors and which are equally far from the input tex2html_wrap_inline1467 , that is, tex2html_wrap_inline1799 and tex2html_wrap_inline1801 for all i, j. Then it follows from the definition of the winning ratios in equation 3.6 that tex2html_wrap_inline1805 . By symmetry, the solution to equation 3.7 has to be tex2html_wrap_inline1807 . From these initial conditions we get


On the other hand, we know that for nearly parallel weight vectors the sum of tex2html_wrap_inline1663 should be near to one in order to minimise the reconstruction error in equation 2.1 if we are using the reconstruction mapping in equation 3.8. Thus the optimal solution is tex2html_wrap_inline1811 . Solving 1/n from equation 3.9 gives the first correction to tex2html_wrap_inline1737 .


After the first correction the winning strengths are tex2html_wrap_inline1817 . The correction function defined in equation 3.10 is monotonically increasing and it will therefore leave the order of the outputs unchanged although the relations may change.

We get the second correction when we notice that if the tex2html_wrap_inline1817 were optimal in terms of equation 3.8, then tex2html_wrap_inline1821 in the following equation


should equal tex2html_wrap_inline1629 . If tex2html_wrap_inline1825 it means that tex2html_wrap_inline1827 should be multiplied by tex2html_wrap_inline1829 to make the equation 3.8 hold. However, for those neurons j that most contribute to tex2html_wrap_inline1821 the correlation tex2html_wrap_inline1835 is biggest, which means that for them tex2html_wrap_inline1837 . We shall therefore make the correction by tex2html_wrap_inline1829 to the neuron i only.


The equation 3.12 will now give the final corrected winning strengths tex2html_wrap_inline1663 . The last corrections to the magnitudes of the winning strengths are not globally uniform, because the ratios tex2html_wrap_inline1829 may vary in different directions. The corrections are locally uniform, however. For neurons with similar weight vectors the ratios are also similar, and therefore the last correction preserves the local relations between winning strengths.

If equations 3.11 and 3.12 were iterated, the solution would approach the optimal solution of equation 3.8. We don't want to do this, however, since the simple linear reconstruction mapping does not promote sparsity in any way. We have obtained a sparse coding by using equation 3.7 and we don't want to lose it. Even a single application of equation 3.12 suffices to bring the outputs to a satisfactory agreement with equation 3.8.

next up previous contents
Next: Finding the set of Up: Derivation of the new Previous: Combining the competitions

Harri Lappalainen
Thu May 9 14:06:29 DST 1996