Product similarity: Hidden community detection tasks can be solved by independence measures relying on a similarity on the product space. Let km: Xm Xm R be kernels (m = [2]) with finite-dimensional feature maps, in other words there exist feature maps pm : Xm Rdm (m = [2]) such that km(xm,xm) = ($m(xm), 0m(x'm))Rm, for xm, x'm Xm, m = [2]. Prove that the function kk2: (X12)(X1X2) R defined as (kk2)((x1, x2), (x1, x2)) = k(x1, x1)k2(x2, x2) is a kernel by constructing a feature map to it.