1.5 In the class, we use the Euclidean distance as the distance metric for the nearest neighbor classification. Given data xRD, the square of Euclidean distance between xi and xj is defined as following: E(xi,xj)=xixj22=d=1D(xidxjd)2 In some applications such as information retrieval and neural language processing, the cosine distance function is widely applied. It is defined as: C(xi,xj)=1xi2xj2xiTxj=1xi2xj2d=1D(xidxjd), where the norm of x is defined as x2=d=1Dxd2. 2 Now you are asked to prove that for any xi and xj normalized to the unit norm, i.e. xi2=xj2= 1, changing the distance measure from the square of Euclidean distance to the cosine distance function will NOT affect the nearest neighbor classification results. Specifically, for any xi,xj and x0, show that, if C(xi,xj)C(xi,xo), then E(xi,xj)E(xi,xo), where xi2=xj2=xo2=1. Hint: don't be intimated by the equations. Try to derive both equations (1) and (2) with the given condition, then compare the derived forms of (1) and (2). (8 points)