On the (non)existence of best low-rank approximations of generic IxJx2 arrays

Abstract

Several conjectures and partial proofs have been formulated on the (non)existence of a best low-rank approximation of real-valued IxJx2 arrays. We analyze this problem using the Generalized Schur Decomposition and prove (non)existence of a best rank-R approximation for generic IxJx2 arrays, for all values of I,J,R. Moreover, for cases where a best rank-R approximation exists on a set of positive volume only, we provide easy-to-check necessary and sufficient conditions for the existence of a best rank-R approximation.