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The approximation of tensors has important applications in variousdisciplines, but it remains an extremely challenging task. It is well knownthat tensors of higher order can fail to have best low-rank approximations, butwith an important exception that best rank-one approximations always exists.The most popular approach to low-rank approximation is the alternating leastsquares (ALS) method. The convergence of the alternating least squaresalgorithm for the rank-one approximation problem is analysed in this paper. Inour analysis we are focusing on the global convergence and the rate ofconvergence of the ALS algorithm. It is shown that the ALS method can convergesublinearly, Q-linearly, and even Q-superlinearly. Our theoretical results areillustrated on explicit examples.

Convergence of Alternating Least Squares Optimisation for Rank-One Approximation to High Order Tensors