2023/04/28 by Terézia Fulová, Fulová, Terézia, Mária Trnovská +1 · 1 citation
Computer Science · Decision Sciences · Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Multi-Criteria Decision Making #Optimization and Control (math.OC) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2304.14961
openalex publication_date 2023/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Procrustes problems are matrix approximation problems searching for a~transformation of the given dataset to fit another dataset. They find applications in numerous areas, such as factor and multivariate analysis, computer vision, multidimensional scaling or finance. The known methods for solving Procrustes problems have been designed to handle specific sub-classes, where the set of feasible solutions has a special structure (e.g. a Stiefel manifold), and the objective function is defined using a specific matrix norm (typically the Frobenius norm). We show that a wide class of Procrustes problems can be formulated and solved as a (rank-constrained) semi-definite program. This includes balanced and unbalanced (weighted) Procrustes problems, possibly to a partially specified target, but also oblique, projection or two-sided Procrustes problems. The proposed approach can handle additional linear, quadratic, or semi-definite constraints and the objective function defined using the Frobenius norm but also standard operator norms. The results are demonstrated on a set of numerical experiments and also on real applications.