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In this paper, we propose a general algorithmic framework to solve a class ofoptimization problems on the product of complex Stiefel manifolds based on thematrix polar decomposition. We establish the weak convergence, globalconvergence and linear convergence rate of this general algorithmic approachusing the \L{}ojasiewicz gradient inequality and the Morse-Bott property. Thisgeneral algorithm and its convergence results are applied to the simultaneousapproximate tensor diagonalization and simultaneous approximate tensorcompression, which include as special cases the low rank orthogonalapproximation, best rank-1 approximation and low multilinear rank approximationfor higher order complex tensors. We also present a symmetric variant of thisgeneral algorithm to solve a symmetric variant of this class of optimizationmodels, which essentially optimizes over a single Stiefel manifold. Weestablish its weak convergence, global convergence and linear convergence ratein a similar way. This symmetric variant and its convergence results areapplied to the simultaneous approximate symmetric tensor diagonalization, whichincludes as special cases the low rank symmetric orthogonal approximation andbest symmetric rank-1 approximation for higher order complex symmetric tensors.It turns out that well-known algorithms such as LROAT, S-LROAT, HOPM, S-HOPMare all special cases of this general algorithmic framework and its symmetricvariant, and our convergence results subsume the results found in theliterature designed for those special cases. All the algorithms and convergenceresults in this paper also apply to the real case.

Polar decomposition based algorithms on the product of Stiefel manifolds with applications in tensor approximation