Least-squares orthogonalization using semidefinite programming

We consider the problem of constructing an optimal set of orthogonal vectors from a given set of vectors in a real Hilbert space. The vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. We show that the design of the optimal vectors, referred to as the least-squares (LS) orthogonal vectors, can be formulated as a semidefinite programming (SDP) problem. Using the many well-known algorithms for solving SDPs, which are guaranteed to converge to the global optimum, the LS vectors can be computed very efficiently in polynomial time within any desired accuracy. By exploiting the connection between our problem and a quantum detection problem we derive a closed form analytical expression for the LS orthogonal vectors, for vector sets with a broad class of symmetry properties. Specifically, we consider geometrically uniform (GU) sets with a possibly non-abelian generating group, and compound GU sets which consist of subsets that are GU.