Do Fractional Norms and Quasinorms Help to Overcome the Curse of Dimensionality?

The curse of dimensionality causes well-known and widely discussed problems for machine learning methods. There is a hypothesis that usage of Manhattan distance and even fractional quasinorms lp (for p less than 1) can help to overcome the curse of dimensionality in classification problems. In this study, we systematically test this hypothesis for 37 binary classification problems on 25 databases. We confirm that fractional quasinorms have greater relative contrast or coefficient of variation than Euclidean norm l2, but we demonstrate also that the distance concentration shows qualitatively the same behaviour for all tested norms and quasinorms and the difference between them decays while dimension tends to infinity. Estimation of classification quality for kNN based on different norms and quasinorms shows that the greater relative contrast does not mean the better classifier performance and the worst performance for different databases was shown by the different norms (quasinorms). A systematic comparison shows that the difference in performance of kNN based on lp for p=2, 1, and 0.5 is statistically insignificant.