Symmetry transformations for square sliced three-way arrays, with applications to their typical rank

AbstractThe typical 3-tensorial rank has been much studied over algebraically closed fields, but very little has been achieved in the way of results pertaining to the real field. The present paper examines the typical 3-tensorial rank over the real field, when the slices of the array involved are square matrices. The typical rank of 3×3×3 arrays is shown to be five. The typical rank of p×q×q arrays is shown to be larger than q+1 unless there are only two slices (p=2), or there are three slices of order 2×2 (p=3 and q=2). The key result is that when the rank is q+1, there usually exists a rank-preserving transformation of the array to one with symmetric slices

Degeneracy in Candecomp/Parafac and Indscal Explained For Several Three-Sliced Arrays With A Two-Valued Typical Rank

The Candecomp/Parafac (CP) method decomposes a three-way array into a prespecified number R of rank-1 arrays, by minimizing the sum of squares of the residual array. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate sequences of solutions, in which several rank-1 arrays become highly correlated in all three modes and some elements of the rank-1 arrays become arbitrarily large. We consider the real-valued CP decomposition of all known three-sliced arrays, i.e., of size p×q×3, with a two-valued typical rank. These are the 5×3×3 and 8×4×3 arrays, and the 3×3×4 and 3×3×5 arrays with symmetric 3×3 slices. In the latter two cases, CP is equivalent to the Indscal model. For a typical rank of {m,m+1}, we consider the CP decomposition with R=m of an array of rank m+1. We show that (in most cases) the CP objective function does not have a minimum but an infimum. Moreover, any sequence of feasible CP solutions in which the objective value approaches the infimum will become degenerate. We use the tools developed in Stegeman (2006), who considers p×p×2 arrays, and present a framework of analysis which is of use to the future study of CP degeneracy related to a two-valued typical rank. Moreover, our examples show that CP uniqueness is not necessary for degenerate solutions to occur

On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal

Reduction of asymmetry by rank-one matrices

Gower has shown how to partition the sum of squares of an asymmetric matrix into independent parts associated with the symmetric and the skew-symmetric parts of the matrix; and has pointed out that asymmetry can be removed by subtracting certain unit-rank matrices, which improve the symmetry in equally contributing pairs. In the present paper, the definition of asymmetry implied in Gower's approach is analyzed and contrasted with the standard definition of asymmetry, entailing a different method of removing asymmetry. In particular, it is shown that, when the standard definition is adopted, only half the number of rank-one matrices is needed to remove asymmetry as is needed in Gower's approach. Some implications for graphical data analysis are discussed