The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
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The well-known INDSCAL model for simultaneous metric multidimensional scaling (MDS) of three-way data analyzes doubly centered matrices of squared dissimilarities. An alternative approach, called for short DINDSCAL, is proposed for analyzing directly the input matrices of squared dissimilarities. An important consequence is that missing values can be easily handled. The DINDSCAL problem is solved by means of the projected gradient approach. First, the problem is transformed into a gradient dynamical system on a product matrix manifold (of Stiefel sub-manifold of zero-sum matrices and non-negative diagonal matrices). The constructed dynamical system can be numerically integrated which gives a globally convergent algorithm for solving the DINDSCAL. The DINDSCAL problem and its solution are illustrated by well-known data routinely used in metric MDS and INDSCAL. Alternatively, the problem can also be solved by iterative algorithm based on the conjugate (projected) gradient method, which MATLAB implementation is enclosed as an appendix.