  The higher-order orthogonality iteration (HOOI) has been popularly used for
finding a best low-multilinear-rank approximation of a tensor. However, its
iterate sequence convergence is still an open question. In this paper, we first
analyze a greedy HOOI, which updates each factor matrix by selecting from the
best candidates one that is closest to the current iterate. Assuming the
existence of a block-nondegenerate limit point, we establish its global
convergence through the so-called Kurdyka-Lojasiewicz (KL) property. In
addition, we show that if the starting point is sufficiently close to any
block-nondegenerate globally optimal solution, the greedy HOOI produces a
sequence convergent to a globally optimal solution. Relating the iterate
sequence by the original HOOI to that by the greedy HOOI, we then show that the
same convergence results hold for the original HOOI and thus positively address
the open question.
