Linear combination of weighted t-cost and chamfering weighted distances

摘要

Previously, we have studied linear combinations of a few pairs of norms, and reported their effectiveness in providing better approximation of Euclidean norms. In particular, we showed good approximation property of a combination of a pair of norms, namely CWD_(eu) and WtD_(isr) by experimentally computing their approximate maximum relative errors (MRE) with respect to the Euclidean norm. In this work, we have considered a pairing of any two members from the families of chamfering weighted distances (CWD) and weighted t-cost distances (WtD), respectively, and derive theoretical values of MREs with respect to the Euclidean norm by exploiting geometry of its hypersphere. Towards this we have computed the vertices of the hypersphere. Subsequently, in addition to our previously reported combination of CWD_(eu) and WtD_(isr), we have also considered a few other combinations and showed their good approximation properties by computing theoretical MREs, as well as by validating those values experimentally. Further, by minimizing the theoretical expressions of MRE locally in the coefficient space of a linear combination, we obtain good approximators of Euclidean norm in any arbitrary dimension.