Higher-order singular value decomposition
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A higher-order singular value decomposition (HOSVD) is a singular value decomposition (SVD). It may be referred to as M-mode SVD, or Multilinear SVD. The M-mode SVD and the Tucker Decomposition terminology was coined in the 1980s by P. Kroonenberg, but it was later referred to as Multilinear SVD/HOSVD by L. De Lathauwer. The term HOSVD is a misnomer since in multilinear algebra, there does not exist an SVD method for multi-way arrays (also known as M-way arrays, or informally data tensors) with all the properties of a matrix SVD. A matrix SVD computes both a
- (a) a rank-R decomposition and
- (b) the orthonormal row/column matrices.
These two capabilities are embodied by two different decompositions for multi-way arrays.
Property (a) may be computed by employing the CANDECOMP/PARAFAC (CP) decomposition. CP is a linear decomposition that represents a tensor as the sum N of rank-1 tensors, for apriori user specified R. This should not be confused with the rank-R decomposition which is the minimum number of R rank-1 tensors that decomposes a tensor exactly. A rank-1 tensor is the outer product of m vectors, where m is the number of the tensor modes.
Property (b) is extended to higher order tensors by a class of methods known variably as Tucker3, M-mode SVD, "Multilinear SVD" or "Higher Order SVD". (This article will use the term "Tucker decomposition".) These methods compute the orthonormal spaces associated with the different modes (axes) of a tensor. The Tucker decomposition is used in multilinear principal component analysis.
Historically, much of the interest in multilinear SVDs was driven by the need to analyze empirical data, especially in psychometrics and chemometrics. Abstract and general mathematical theorems are rare (though see Kruskal with regard to the CP decomposition); instead, the methods are often designed for analyzing specific data types. The 2008 review article by Kolda and Bader provides a compact summary of the history of these decompositions, and many references for further reading.
Tucker decomposition / M-mode SVD / Multilinear SVD / Higher Order SVD
The Tucker Decomposition is the Multilinear SVD.
In 1966, L. Tucker proposed a decomposition method for three-way arrays (referred to as a 3-mode "tensors") as a multidimensional extension of factor analysis. This decomposition was further developed in the 1980s by P. Kroonenberg, who coined the terms Tucker3, Tucker3ALS (an alternating least squares dimensionality reduction algorithm), 3-Mode SVD, and 3-Mode PCA. In the intervening years, several authors developed the decomposition for N-way arrays. Most recently, this work was treated in an elegant fashion and introduced to the SIAM community by L. De Lathauwer et al. who referred to the decomposition as a multilinear SVD and HOSVD.
Let the SVD of a real matrix be , then it can be written in an element-wise form as
and give, in a certain sense optimal, orthonormal basis for the column and row space, is diagonal with decreasing elements. The higher-order singular value decomposition (HOSVD) can be defined by the multidimensional generalization of this concept:
where the matrices and the core tensor should satisfy certain requirements (similar ones to the matrix SVD), namely
- Each is an orthogonal matrix.
- Two subtensors of the core tensor are orthogonal i.e., if .
- The subtensors in the core tensor are ordered according to their Frobenius norm, i.e. for n = 1, ..., N.
- Given a tensor , construct the mode-k flattening . That is, the matrix that corresponds to .
- Compute the singular value decomposition , and store the left singular vectors .
- The core tensor is then the projection of onto the tensor basis formed by the factor matrices , i.e.,
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  (CVPR 2001, ICPR 2002), face recognition - TensorFaces,   (ECCV 2002, CVPR 2003, etc.) and computer graphics—TensorTextures (Siggraph 2004).
The Multilinear SVD has been successfully applied to signal processing and big data, e.g., in genomic signal processing. These applications also inspired a higher-order GSVD (HO GSVD) and a tensor GSVD.
In machine learning, the CP-decomposition is the central ingredient in learning probabilistic latent variables models via the technique of moment-matching. For example, consider the multi-view model which is a probabilistic latent variable model. In this model, the generation of samples are posited as follows: there exists a hidden random variable that is not observed directly, given which, there are several conditionally independent random variables known as the different "views" of the hidden variable. For simplicity, assume there are three symmetrical views of a -state categorical hidden variable . Then the empirical third moment of this latent variable model can be written as: .
In applications such as topic modeling, this can be interpreted as the co-occurrence of words in a document. Then the eigenvalues of this empirical moment tensor can be interpreted as the probability of choosing a specific topic and each column of the factor matrix corresponds to probabilities of words in the vocabulary in the corresponding topic.
The concept of Multilinear SVD 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.
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