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

TL;DR

This paper establishes the tightness of the approximation guarantees for HOSVD, ST-HOSVD, and HOOI algorithms by constructing specific tensors that reach these bounds, confirming no improvements are possible.

Contribution

It proves the classic approximation bounds for HOSVD, ST-HOSVD, and HOOI are tight by constructing worst-case tensors, showing these guarantees cannot be improved.

Findings

HOSVD approximation ratio is tight at N/(1+ε)

ST-HOSVD and HOOI bounds are also tight

Constructed tensors demonstrate worst-case scenarios

Abstract

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 $\varepsilon > 0$.