The Carroll and Chang conjecture of equal Indscal components when Candecomp/Parafac gives perfect fit

The Candecomp/Parafac algorithm approximates a set of matrices X1,…,XI by products of the form ACiB′, with Ci diagonal, i=1,…,I. Carroll and Chang have conjectured that, when the matrices are symmetric, the resulting A and B will be column wise proportional. For cases of perfect fit, Ten Berge et al. have shown that the conjecture holds true in a variety of cases, but may fail when there is no unique solution. In such cases, obtaining proportionality by changing (part of) the solution seems possible. The present paper extends and further clarifies their results. In particular, where Ten Berge et al. solved all I×2×2 cases, now all I×3×3 cases, and also the I×4×4 cases for I=2,8, and 9 are clarified. In a number of cases, A and B necessarily have column wise proportionality when Candecomp/Parafac is run to convergence. In other cases, proportionality can be obtained by using specific methods. No cases were found that seem to resist proportionality.