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Many idealized problems in signal processing, machine learning and statisticscan be reduced to the problem of finding the symmetric canonical decompositionof an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawinginspiration from the matrix case, the successive rank-one approximations (SROA)scheme has been proposed and shown to yield this tensor decomposition exactly,and a plethora of numerical methods have thus been developed for the tensorrank-one approximation problem. In practice, however, the inevitable errors(say) from estimation, computation, and modeling necessitate that the inputtensor can only be assumed to be a nearly SOD tensor---i.e., a symmetric tensorslightly perturbed from the underlying SOD tensor. This article shows that evenin the presence of perturbation, SROA can still robustly recover the symmetriccanonical decomposition of the underlying tensor. It is shown that when theperturbation error is small enough, the approximation errors do not accumulatewith the iteration number. Numerical results are presented to support thetheoretical findings.

Successive Rank-One Approximations for Nearly Orthogonally Decomposable Symmetric Tensors