Well-posedness of convex maximization problems on Stiefel manifolds and orthogonal tensor product approximations

Problems of best tensor product approximation of low orthogonal rank can be formulated as maximization problems on Stiefel manifolds. The functionals that appear are convex and weakly sequentially continuous. It is shown that such problems are always well-posed, even in the case of non-compact Stiefel manifolds. As a consequence, problems of finding a best orthogonal, strong orthogonal or complete orthogonal low-rank tensor product approximation and problems of best Tucker format approximation to any given tensor are always well-posed, even in spaces of infinite dimension. (The best rank-one approximation is a special case of all of them.) In addition, the well-posedness of a canonical low-rank approximation with bounded coefficients can be shown. The proofs are non-constructive and the problem of computation is not addressed here.