Successive projection method for solving the unbalanced Procrustes problem (Q867767)

The Procrustes problem (PP) considered here is to minimize the Frobenius norm \(f(Q)=\| AQ-B\| _F\) over all matrices \(Q\) satisfying \(Q^TQ=I\) when \(A\) and \(B\) are given matrices. When \(Q\) is square, the PP is called balanced, when \(Q\) is \(n\times k\) with \(k<n\), the PP is unbalanced. An unbalanced PP with optimal or approximate solution \(Q\), can be expanded to a balanced one by taking the orthogonal complement \(H\) of \(Q\) and minimizing the norm of \(A G - [B,AH]\) with \(G\) orthogonal. Besides a survey of alternatives methods, a successive projection method is proposed which updates the columns of \(Q\) by solving PPs with \(k=1\) for each column separately. Such a PP with \(k=1\) is equivalent to a linear least squares problem with a quadratic equality constraint. After each cycle of the \(k\) columns, a global correction is obtained e.g., by solving an expanded balanced PP. It is shown that the method converges to a matrix \(Q\) which satisfies the necessary (but not the sufficient) conditions for an optimal solution of the PP.