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.
Some aspects can be traced as far back as F. L. Hitchcock in 1928,[1] but it was L. R. Tucker who developed for third-order tensors the general Tucker decomposition in the 1960s,[2][3][4] further advocated by L. De Lathauwer et al.[5] 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.
is assumed to be given in coordinates with respect to some basis as a M-way array, also denoted by
be a unitary matrix containing a basis of the left singular vectors of the
corresponds to the jth largest singular value of
does not depend on the particular on the specific definition of the mode m flattening.
As in the case of the compact singular value decomposition of a matrix, where the rows and columns corresponding to vanishing singular values are dropped, it is also possible to consider a compact HOSVD, which is very useful in applications.
is a matrix with unitary columns containing a basis of the left singular vectors corresponding to the nonzero singular values of the standard factor-m flattening
where the first equality is due to the properties of orthogonal projections (in the Hermitian inner product) and the last equality is due to the properties of multilinear multiplication.
As flattenings are bijective maps and the above formula is valid for all
[13] The compact HOSVD is a rank-revealing decomposition in the sense that the dimensions of its core tensor correspond with the components of the multilinear rank of the tensor.
The following geometric interpretation is valid for both the full and compact HOSVD.
This means that the HOSVD can be interpreted as a way to express the tensor
with respect to a specifically chosen orthonormal basis
The strategy for computing the Multilinear SVD and the M-mode SVD was introduced in the 1960s by L. R. Tucker,[3] further advocated by L. De Lathauwer et al.,[5] and by Vasilescu and Terzopulous.
[8][6] The term HOSVD was coined by Lieven De Lathauwer, but the algorithm typically referred to in the literature as HOSVD was introduced by Vasilescu and Terzopoulos[6][8] with the name M-mode SVD.
consists of interlacing the computation of the core tensor and the factor matrices, as follows:[14][15][16] The HOSVD can be computed in-place via the Fused In-place Sequentially Truncated Higher Order Singular Value Decomposition (FIST-HOSVD) [16] algorithm by overwriting the original tensor by the HOSVD core tensor, significantly reducing the memory consumption of computing HOSVD.
In applications, such as those mentioned below, a common problem consists of approximating a given tensor
A simple idea for trying to solve this optimization problem is to truncate the (compact) SVD in step 2 of either the classic or the interlaced computation.
A classically truncated HOSVD is obtained by replacing step 2 in the classic computation by while a sequentially truncated HOSVD (or successively truncated HOSVD) is obtained by replacing step 2 in the interlaced computation by The HOSVD is most commonly applied to the extraction of relevant information from multi-way arrays.
Starting in the early 2000s, Vasilescu addressed causal questions by reframing the data analysis, recognition and synthesis problems as multilinear tensor problems.
The power of the tensor framework was showcased by decomposing and representing an image in terms of its causal factors of data formation, in the context of Human Motion Signatures for gait recognition,[18] face recognition—TensorFaces[19][20] and computer graphics—TensorTextures.
[21] The HOSVD has been successfully applied to signal processing and big data, e.g., in genomic signal processing.
[26] A combination of HOSVD and SVD also has been applied for real-time event detection from complex data streams (multivariate data with space and time dimensions) in disease surveillance.
[27] It is also used in tensor product model transformation-based controller design.
[28][29] The concept of HOSVD was carried over to functions by Baranyi and Yam via the TP model transformation.
[28][29] This extension led to the definition of the HOSVD-based canonical form of tensor product functions and Linear Parameter Varying system models[30] and to convex hull manipulation based control optimization theory, see TP model transformation in control theories.
HOSVD was proposed to be applied to multi-view data analysis[31] and was successfully applied to in silico drug discovery from gene expression.