Simplicity of core arrays in three-way principal component analysis and the typical rank of [formula omitted] arrays

Abstract

Interpreting the solution of a Principal Component Analysis of a three-way array is greatly simplified when the core array has a large number of zero elements. The possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand, and by mathematical analysis on the other. In the present paper, it is shown that a p×q×2 array, with p>q⩾2 , can almost surely be transformed to have all but 2 q elements zero. It is also shown that arrays of that form have three-way rank p at most. This has direct implications for the typical rank of p×q×2 arrays, also when p= q. When p⩾2q, the typical rank is 2 q; when q<p<2q it is p, and when p= q, the rank is typically (almost surely) p or p + 1. These typical rank results pertain to the decomposition of real valued three-way arrays in terms of real valued rank one arrays, and do not apply in the complex setting, where the typical rank of p×q×2 arrays is also min[ p, 2q] when p > q, but it is p when p= q.