On reconstruction error of Kohonen self-organizing mapping

In this paper, we analyze how the number of weights and the size of neighborhood in the SOM algorithm affect the reconstruction error. One-dimensional random variables with a uniform distribution and fixed sizes of neighborhood are considered. First, we give the linear equation that for any number of weights and any size of neighborhood can satisfy the equilibrium state. We find that the SOM algorithm converges to the unique minimal point of the reconstruction error if and only if the size of neighborhood is one. If the size is greater than one, the ∞-norm between an equilibrium state and the minimal point is proportional to the size of neighborhood and inversely proportional to the number of weights. Upper and lower bounds of the reconstruction error are given. The upper bound is proportional to a cube of the neighborhood size and inversely proportional to a square of the number of weights. The main part of the lower bound is proportional to a cube of the ratio of the size of neighborhood to the number of weights. These mean that in order to achieve a small reconstruction error, the size of neighborhood must be small. All the results are proved