A Tight Lower Bound for the Approximation Guarantee of Higher-Order Singular Value Decomposition

We prove that the classic approximation guarantee for the higher-order singular value decomposition (HOSVD) is tight by constructing a tensor for which HOSVD achieves an approximation ratio of $N/(1+\varepsilon)$, for any $\varepsilon > 0$. This matches the upper bound of De Lathauwer et al. (2000a) and shows that the approximation ratio of HOSVD cannot be improved. Using a more advanced construction, we also prove that the approximation guarantees for the ST-HOSVD algorithm of Vannieuwenhoven et al. (2012) and higher-order orthogonal iteration (HOOI) of De Lathauwer et al. (2000b) are tight by showing that they can achieve their worst-case approximation ratio of $N / (1 + \varepsilon)$, for any $\varepsilon > 0$.