Applications of lp-norms and their smooth approximations for gradient based learning vector quantization

Learning vector quantization applying non-standard metrics became quite popular for classification performance improvement compared to standard approaches using the Euclidean distance. Kernel metrics and quadratic forms belong to the most promising approaches. In this paper we consider Minkowski distances (lp-norms). In particular, l1-norms are known to be robust against noise in data, such that, if this structural knowledge is available in advance about the data, this norm should be utilized. However, application in gradient based learning algorithms based on distance evaluations need to calculate the respective derivatives. Because lp-distance formulas contain the absolute approximations thereof are required. We consider in this paper several approaches for smooth consistent approximations for numerical evaluations and demonstrate the applicability for exemplary real world applications.