In multilinear algebra, the higher-order singular value decomposition of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one generalization of the matrix singular value decomposition. The HOSVD has applications in computer graphics, machine learning, scientific computing, and signal processing. Some key ingredients of the HOSVD 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, including the HOSVD. The HOSVD as decomposition in its own right was further advocated by L. De Lathauwer et al. in 2000. Robust and L1-norm-based variants of HOSVD have also been proposed. As the HOSVD was studied in many scientific fields, it is sometimes historically referred to as multilinear singular value decomposition, m-mode SVD, or cube SVD, and sometimes it is incorrectly identified with a Tucker decomposition.
Definition
For the purpose of this article, the abstract tensor is assumed to be given in coordinates with respect to some basis as a multidimensional array, also denoted by, in, where d is the order of the tensor and is either or. Let be a unitary matrix containing a basis of the left singular vectors of the standard factor-k flattening of such that the jth column of corresponds to the jth largest singular value of. Observe that the factor matrix does not depend on the particular freedom of choice in the definition of the standard factor-k 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 decompositionThe above construction shows that every tensor has a HOSVD.
Compact 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. Assume that is a matrix with unitary columns containing a basis of the left singular vectors corresponding to the nonzero singular values of the standard factor-k flattening of. Let the columns of be sorted such that the jth column of corresponds to the jth largest nonzero singular value of. Since the columns of form a basis for the image of, we havewhere the first equality is due to the properties of orthogonal projections 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, we find as before thatwhere the core tensor is now of size.
Multilinear rank
The tuple where is called the multilinear rank of. By definition, every tensor has a unique multilinear rank, and its components are bounded by. Not all tuples in are multilinear ranks. In particular, it is known that must hold. The compact HOSVD is a rank-revealing factorization in the sense that the dimensions of its core tensor correspond with the components of the multilinear rank of the tensor.
Interpretation
The following geometric interpretation is valid for both the full and compact HOSVD. Let be the multilinear rank of the tensor. Since is a multidimensional array, we can expand it as followswhere is the jth standard basis vector of. By definition of the multilinear multiplication, it holds thatwhere the are the columns of. It is easy to verify that is an orthonormal set of tensors. This means that the HOSVD can be interpreted as a way to express the tensor with respect to a specifically chosen orthonormal basis with the coefficients given as the multidimensional array.
Computation
Let, where is either or, be a tensor of multilinear rank.
Classic computation
The classic strategy for computing a compact HOSVD was introduced in 1966 by L. R. Tucker and further advocated by L. De Lathauwer et al.; it is based on the definition of the decomposition. The next three steps are to be performed:
For, do the following:
Construct the standard factor-k flattening ;
Compute the singular value decomposition, and store the left singular vectors ;
Compute the core tensor via the multilinear multiplication
Interlaced computation
A strategy that is significantly faster when some or all consists of interlacing the computation of the core tensor and the factor matrices, as follows:
Set ;
For perform the following:
# Construct the standard factor-k flattening ;
# Compute the singular value decomposition, and store the left singular vectors ;
# Set, or, equivalently,.
Approximation
In applications, such as those mentioned below, a common problem consists of approximating a given tensor by one of low multilinear rank. Formally, if denotes the multilinear rank of, then the nonlinear non-convex -optimization problem iswhere with, is a target multilinear rank that is assumed to be given, and where the norm is the Frobenius norm. Based on the foregoing algorithms for computing a compact HOSVD, a natural idea for trying to solve this optimization problem is to truncate the SVD in step 2 of either the classic or the interlaced computation. A classicallytruncated HOSVD is obtained by replacing step 2 in the classic computation by
Compute a rank- truncated SVD, and store the top left singular vectors ;
while a sequentially truncated HOSVD is obtained by replacing step 2 in the interlaced computation by
Compute a rank- truncated SVD, and store the top left singular vectors ;
Unfortunately, neither of these strategies results in an optimal solution of the best low multilinear rank optimization problem, contrary to the matrix case where the Eckart-Young theorem holds. However, both the classically and sequentially truncated HOSVD result in a quasi-optimal solution: if denotes the classically or sequentially truncated HOSVD and denotes the optimal solution to the best low multilinear rank approximation problem, thenin practice this means that if there exists an optimal solution with a small error, then a truncated HOSVD will for many intended purposes also yield a sufficiently good solution.
Applications
The HOSVD is most commonly applied to the extraction of relevant information from multi-way arrays. Circa 2001, Vasilescu reframed the data analysis, recognition and synthesis problems as multilinear tensor problems based on the insight that most observed data are the result of several causal factors of data formation, and are well suited for multi-modal data tensor analysis. The power of the tensor framework was showcased in a visually and mathematically compelling manner by decomposing and representing an image in terms of its causal factors of data formation, in the context of Human Motion Signatures, face recognition—TensorFaces and computer graphics—TensorTextures. The HOSVD has been successfully applied to signal processing and big data, e.g., in genomic signal processing. These applications also inspired a higher-order GSVD and a tensor GSVD. A combination of HOSVD and SVD also has been applied for real-time event detection from complex data streams in disease surveillance. It is also used in tensor product model transformation-based controller design. In multilinear subspace learning, it was applied to modeling tensor objects for gait recognition. The concept of HOSVD was carried over to functions by Baranyi and Yam via the TP model transformation. This extension led to the definition of the HOSVD-based canonical form of tensor product functions and Linear Parameter Varying system models 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 and was successfully applied to in silico drug discovery from gene expression.
