  We propose a novel rank-adaptive higher-order orthogonal iteration (HOOI)
algorithm to compute the truncated Tucker decomposition of higher-order tensors
with a given error tolerance, and prove that the method is locally optimal and
monotonically convergent. A series of numerical experiments related to both
synthetic and real-world tensors are carried out to show that the proposed
rank-adaptive HOOI algorithm is advantageous in terms of both accuracy and
efficiency. Some further analysis on the HOOI algorithm and the classical
alternating least squares method are presented to further understand why rank
adaptivity can be introduced into the HOOI algorithm and how it works.
