On the uniqueness of correspondence analysis solutions

In correspondence analysis (CA), the rows and columns of a contingency table are optimally represented in a k-dimensional approximation, where it is common to set k=3 (which includes a so-called trivial dimension). Since CA is a dimension reduction technique, we might expect that the k-dimensional approximation is not unique, i.e. there exist several contingency tables with the same k-dimensional approximation. Interestingly, Van de Velden et al. [17] find in their computational experiments that 3-dimensional CA solutions are unique up to rotation, which leads to the question whether this is always the case. We show that k-dimensional CA solutions are not necessarily unique. That is, two distinct contingency tables may have the same k-dimensional approximation. We present necessary and sufficient conditions for the non-uniqueness of CA solutions, which hold for any value of k. Based on our sufficient conditions, we present a procedure to generate contingency tables with the same k-dimensional solution. Finally, we note that it is difficult to satisfy the necessary conditions, which suggests that CA solutions are most likely unique in practice.