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. in their Multilinear SVD work that employs the power method, or advocated by Vasilescu and Terzopoulos that developed M-mode SVD a parallel algorithm that employs the matrix SVD. The term higher order singular value decomposition (HOSVD) was coined be DeLathauwer, but the algorithm referred to commonly in the literature as the HOSVD and attributed to either Tucker or DeLathauwer was developed by Vasilescu and Terzopoulos. Robust and L1-norm-based variants of HOSVD have also been proposed. For the purpose of this article, the abstract tensor is assumed to be given in coordinates with respect to some basis as a M-way array, also denoted by , where M is the number of modes and the order of the tensor. is the complex numbers and it includes both the real numbers and the pure imaginary numbers. Let be a unitary matrix containing a basis of the left singular vectors of the standard mode-m flattening of such that the jth column of corresponds to the jth largest singular value of . Observe that the mode/factor matrix does not depend on the particular on the specific definition of the mode m flattening. By the properties of the multilinear multiplication, we havewhere denotes the conjugate transpose. The second equality is because the 's are unitary matrices. Define now the core tensorThen, the HOSVD of is the decomposition The above construction shows that every tensor has a HOSVD. As in the case of the compact singular value decomposition of a matrix, it is also possible to consider a compact HOSVD, which is very useful in applications.