We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of O(k^{-1/(2^{d+1}-2)}), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of O(1/k^{0.5}).

Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data recon-struction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode ma-tricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known.

In this paper, we present an efficient QR algorithm for solving the linear response eigenvalue problem H x= x, where H is -symmetric with respect to 0= diag (I n, I n). Based on newly introduced -orthogonal transformations, the QR algorithm preserves the -symmetric structure of H throughout the whole process, and thus guarantees the computed eigenvalues to appear pairwise (, ) as they should. With the help of a newly established implicit -orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the QR algorithm. Numerical experiments are given to show the effectiveness of the algorithm.

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