Higher-order singular value decomposition

In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one type of generalization of the matrix singular value decomposition. It has applications in computer vision, computer graphics, machine learning, scientific computing, and signal processing. Some aspects can be traced as far back as F. L. Hitchcock in 1928, but it was L. R. Tucker who developed for third-order tensors the general Tucker decomposition in the 1960s, further advocated by L. De Lathauwer et al. , or advocated by Vasilescu and Terzopoulos.

Although the term HOSVD was coined by De Lathauwer, the algorithm most commonly referred to as the Tucker or Higher-Order Singular Value Decomposition (HOSVD) in the literature was originally introduced by Vasilescu and Terzopoulos under the name M-mode SVD..

The M-mode SVD is a simple and elegant algorithm suitable for parallel computation. The algorithm developed by Tucker/Kroonenberg and De Lathauwer et al. are sequential algorithms that employ gradient descent or the power method, respectively. The M-mode SVD parallel formulation is computationally distinct from prior approaches.

This misattribution has had lasting impact on the scholarly record, obscuring the original source of a widely adopted algorithm, and complicating efforts to trace its development, reproduce results, or properly credit foundational contributions in multilinear algebra and tensor methods.

Robust and L1-norm-based variants of this decomposition framework have since been proposed.