Year of Publication
Arts and Sciences
William S. Rayens
One of the differentiating features of PARAFAC decompositions is that, under certain conditions, unique solutions are possible. The search for uniqueness conditions for the PARAFAC Decomposition has a limited past, spanning only three decades. The complex structure of the problem and the need for tensor algebras or other similarly abstract characterizations provided a roadblock to the development of uniqueness conditions. Theoretically, the PARAFAC decomposition surpasses its bilinear counterparts in that it is possible to obtain solutions that do not suffer from the rotational problem. However, not all PARAFAC solutions will be constrained sufficiently so that the resulting decomposition is unique. The work of Kruskal, 1977, provides the most in depth investigation into the conditions for uniqueness, so much so that many have assumed, without formal proof, that his sufficient conditions were also necessary. Aided by the introduction of Khatri-Rao products to represent the PARAFAC decomposition, ten Berge and Sidiropoulos (2002) used the column spaces of Khatri-Rao products to provide the first evidence for countering the claim of necessity, identifying PARAFAC decompositions that were unique when Kruskals condition was not met. Moreover, ten Berge and Sidiropoulos conjectured that, with additional k-rank restrictions, a class of decompositions could be formed where Kruskals condition would be necessary and sufficient. Unfortunately, the column space argument of ten Berge and Sidiropoulos was limited in its application and failed to provide an explanation of why uniqueness occurred. On the other hand, the use of orthogonal complement spaces provided an alternative approach to evaluate uniqueness that would provide a much richer return than the use of column spaces for the investigation of uniqueness. The Orthogonal Complement Space Approach (OCSA), adopted here, would provide: (1) the answers to lingering questions about the occurrence of uniqueness, (2) evidence that necessity would require more than a restriction on k-rank, and (3) an approach that could be extended to cases beyond those investigated by ten Berge and Sidiropoulos.
Bush, Heather Michele Clyburn, "KHATRI-RAO PRODUCTS AND CONDITIONS FOR THE UNIQUENESS OF PARAFAC SOLUTIONS FOR IxJxK ARRAYS" (2006). University of Kentucky Doctoral Dissertations. 462.