Performance index for tensor polyadic decompositions

It is proposed to isolate the computation of the scaling matrix in CP tensor decompositions. This has two implications. First, the conditioning of the problem shows up explicitly, and could be controlled via a constraint on the so-called coherences. Second, a performance measure concerning only the factor matrices can be exactly calculated, and does not present the optimistic bias of the minimal error generally utilized in the literature. In fact, for tensors of order d, it suffices to solve a degree-2 polynomial system in d variables. We subsequently give an explicit solution when d = 3.