Can unbounded distance measures mitigate the curse of dimensionality?

In this work, we revisit the curse of dimensionality, especially the concentration of the norm phenomenon which is the inability of distance functions to separate points well in high dimensions. We study the influence of the different properties of a distance measure, viz., triangle inequality, boundedness and translation invariance and on this phenomenon. Our studies indicate that unbounded distance measures whose expectations do not exist are to be preferred. We propose some new distance measures based on our studies and present many experimental results which seem to confirm our analysis. In particular, we study these distance measures w.r.t. indices like relative variance and relative contrast and further compare and contrast these measures in the setting of nearest neighbour/proximity searches and hierarchical clustering.