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, ^[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. ^{[6] [7] [8]} Robust and L1-norm-based variants of HOSVD have also been proposed. ^{[9] [10] [11] [12]}

Definition

For the purpose of this article, the abstract tensor ${\displaystyle {\mathcal {A}}}$ is assumed to be given in coordinates with respect to some basis as a M-way array, also denoted by ${\displaystyle {\mathcal {A}}\in \mathbb {C} ^{I_{1}\times I_{2}\cdots \times \cdots I_{m}\cdots \times I_{M}}}$, where M is the number of modes and the order of the tensor. ${\displaystyle \mathbb {C} }$ is the complex numbers and it includes both the real numbers ${\displaystyle \mathbb {R} }$ and the pure imaginary numbers.

Let ${\displaystyle {\mathcal {A}}_{[m]}\in \mathbb {C} ^{I_{m}\times (I_{1}I_{2}\cdots I_{m-1}I_{m+1}\cdots I_{M})}}$ denote the standard mode-m flattening of ${\displaystyle {\mathcal {A}}}$, so that the left index of ${\displaystyle {\mathcal {A}}_{[m]}}$ corresponds to the ${\displaystyle m}$'th index ${\displaystyle {\mathcal {A}}}$ and the right index of ${\displaystyle {\mathcal {A}}_{[m]}}$ corresponds to all other indices of ${\displaystyle {\mathcal {A}}}$ combined. Let ${\displaystyle {\bf {U}}_{m}\in \mathbb {C} ^{I_{m}\times I_{m}}}$be a unitary matrix containing a basis of the left singular vectors of the ${\displaystyle {\mathcal {A}}_{[m]}}$ such that the jth column ${\displaystyle \mathbf {u} _{j}}$ of ${\displaystyle {\bf {U}}_{m}}$ corresponds to the jth largest singular value of ${\displaystyle {\mathcal {A}}_{[m]}}$. Observe that the mode/factor matrix${\displaystyle {\bf {U}}_{m}}$ does not depend on the particular on the specific definition of the mode m flattening. By the properties of the multilinear multiplication, we have$${\displaystyle {\begin{array}{rcl}{\mathcal {A}}&=&{\mathcal {A}}\times ({\bf {I}},{\bf {I}},\ldots ,{\bf {I}})\\&=&{\mathcal {A}}\times ({\bf {U}}_{1}{\bf {U}}_{1}^{H},{\bf {U}}_{2}{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}{\bf {U}}_{M}^{H})\\&=&\left({\mathcal {A}}\times ({\bf {U}}_{1}^{H},{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}^{H})\right)\times ({\bf {U}}_{1},{\bf {U}}_{2},\ldots ,{\bf {U}}_{M}),\end{array}}}$$where ${\displaystyle \cdot ^{H}}$ denotes the conjugate transpose. The second equality is because the ${\displaystyle {\bf {U}}_{m}}$'s are unitary matrices. Define now the core tensor$${\displaystyle {\mathcal {S}}:={\mathcal {A}}\times ({\bf {U}}_{1}^{H},{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}^{H}).}$$Then, the HOSVD ^[5] of ${\displaystyle {\mathcal {A}}}$ is the decomposition$${\displaystyle {\mathcal {A}}={\mathcal {S}}\times ({\bf {U}}_{1},{\bf {U}}_{2},\ldots ,{\bf {U}}_{M}).}$$ The above construction shows that every tensor has a HOSVD.

Compact HOSVD

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.

Assume that ${\displaystyle {\bf {U}}_{m}\in \mathbb {C} ^{I_{m}\times R_{m}}}$ 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 ${\displaystyle {\mathcal {A}}_{[m]}}$ of ${\displaystyle {\mathcal {A}}}$. Let the columns of ${\displaystyle {\bf {U}}_{m}}$ be sorted such that the ${\displaystyle r_{m}}$ th column ${\displaystyle {\bf {u}}_{r_{m}}}$ of ${\displaystyle {\bf {U}}_{m}}$ corresponds to the ${\displaystyle r_{m}}$th largest nonzero singular value of ${\displaystyle {\mathcal {A}}_{[m]}}$. Since the columns of ${\displaystyle {\bf {U}}_{m}}$ form a basis for the image of ${\displaystyle {\mathcal {A}}_{[m]}}$, we have$${\displaystyle {\mathcal {A}}_{[m]}={\bf {U}}_{m}{\bf {U}}_{m}^{H}{\mathcal {A}}_{[m]}={\bigl (}{\mathcal {A}}\times _{m}({\bf {U}}_{m}{\bf {U}}_{m}^{H}){\bigr )}_{[m]},}$$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 ${\displaystyle m=1,2,\ldots ,m,\ldots ,M}$, we find as before that$${\displaystyle {\begin{array}{rcl}{\mathcal {A}}&=&{\mathcal {A}}\times ({\bf {U}}_{1}{\bf {U}}_{1}^{H},{\bf {U}}_{2}{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}{\bf {U}}_{M}^{H})\\&=&\left({\mathcal {A}}\times ({\bf {U}}_{1}^{H},{\bf {U}}_{2}^{H},\ldots ,{\bf {U}}_{M}^{H})\right)\times ({\bf {U}}_{1},{\bf {U}}_{2},\ldots ,{\bf {U}}_{M})\\&=&{\mathcal {S}}\times ({\bf {U}}_{1},{\bf {U}}_{2},\ldots ,{\bf {U}}_{M}),\end{array}}}$$where the core tensor ${\displaystyle {\mathcal {S}}}$ is now of size ${\displaystyle R_{1}\times R_{2}\times \cdots \times R_{M}}$.

Multilinear rank

The multilinear rank ^[1] of ${\displaystyle {\mathcal {A}}}$ is denoted with rank-${\displaystyle (R_{1},R_{2},\ldots ,R_{M})}$. The multilinear rank is a tuple in ${\displaystyle \mathbb {N} ^{M}}$ where ${\displaystyle R_{m}:=\mathrm {rank} ({\mathcal {A}}_{[m]})}$. Not all tuples in ${\displaystyle \mathbb {N} ^{M}}$ are multilinear ranks. ^[13] The multilinear ranks are bounded by ${\displaystyle 1\leq R_{m}\leq I_{m}}$ and it satisfy the constraint ${\textstyle R_{m}\leq \prod _{i\neq m}R_{i}}$ must hold. ^[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.

Interpretation

The following geometric interpretation is valid for both the full and compact HOSVD. Let ${\displaystyle (R_{1},R_{2},\ldots ,R_{M})}$ be the multilinear rank of the tensor ${\displaystyle {\mathcal {A}}}$. Since ${\displaystyle {\mathcal {S}}\in {\mathbb {C} }^{R_{1}\times R_{2}\times \cdots \times R_{M}}}$ is a multidimensional array, we can expand it as follows$${\displaystyle {\mathcal {S}}=\sum _{r_{1}=1}^{R_{1}}\sum _{r_{2}=1}^{R_{2}}\cdots \sum _{r_{M}=1}^{R_{M}}s_{r_{1},r_{2},\ldots ,r_{M}}\mathbf {e} _{r_{1}}\otimes \mathbf {e} _{r_{2}}\otimes \cdots \otimes \mathbf {e} _{r_{M}},}$$where ${\displaystyle \mathbf {e} _{r_{m}}}$ is the ${\displaystyle r_{m}}$th standard basis vector of ${\displaystyle {\mathbb {C} }^{I_{m}}}$. By definition of the multilinear multiplication, it holds that$${\displaystyle {\mathcal {A}}=\sum _{r_{1}=1}^{R_{1}}\sum _{r_{2}=1}^{R_{2}}\cdots \sum _{r_{M}=1}^{R_{M}}s_{r_{1},r_{2},\ldots ,r_{M}}\mathbf {u} _{r_{1}}\otimes \mathbf {u} _{r_{2}}\otimes \cdots \otimes \mathbf {u} _{r_{M}},}$$where the ${\displaystyle \mathbf {u} _{r_{m}}}$ are the columns of ${\displaystyle {\bf {U}}_{m}\in {\mathbb {C} }^{I_{m}\times R_{m}}}$. It is easy to verify that ${\displaystyle B=\{\mathbf {u} _{r_{1}}\otimes \mathbf {u} _{r_{2}}\otimes \cdots \otimes \mathbf {u} _{r_{M}}\}_{r_{1},r_{2},\ldots ,r_{M}}}$ is an orthonormal set of tensors. This means that the HOSVD can be interpreted as a way to express the tensor ${\displaystyle {\mathcal {A}}}$ with respect to a specifically chosen orthonormal basis ${\displaystyle B}$ with the coefficients given as the multidimensional array ${\displaystyle {\mathcal {S}}}$.

Computation

Let ${\displaystyle {\mathcal {A}}\in {\mathbb {C} }^{I_{1}\times I_{2}\times \cdots \times I_{M}}}$ be a tensor with a rank-${\displaystyle (R_{1},R_{2},\ldots ,R_{M})}$, where ${\displaystyle \mathbb {C} }$ contains the reals ${\displaystyle \mathbb {R} }$ as a subset.

Classic computation

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. It is a parallel computation that employs the matrix SVD to compute the orthonormal mode matrices.

M-mode SVD

Sources: ^{[6] [8]}

• For ${\displaystyle m=1,\ldots ,M}$, do the following:

1. Construct the mode-m flattening ${\displaystyle {\mathcal {A}}_{[m]}}$;
2. Compute the (compact) singular value decomposition ${\displaystyle {\mathcal {A}}_{[m]}={\bf {U}}_{m}{\bf {\Sigma }}_{m}{\bf {V}}_{m}^{T}}$, and store the left singular vectors ${\displaystyle {\bf {U}}\in \mathbb {C} ^{I_{m}\times R_{m}}}$;

• Compute the core tensor ${\displaystyle {\mathcal {S}}}$ via the multilinear multiplication ${\displaystyle {\mathcal {S}}={\mathcal {A}}\times _{1}{\bf {U}}_{1}^{H}\times _{2}{\bf {U}}_{2}^{H}\ldots \times _{m}{\bf {U}}_{m}^{H}\ldots \times _{M}{\bf {U}}_{M}^{H}}$

Interlacing computation

A strategy that is significantly faster when some or all ${\displaystyle R_{m}\ll I_{m}}$ consists of interlacing the computation of the core tensor and the factor matrices, as follows: ^{[14] [15] [16]}

• Set ${\displaystyle {\mathcal {A}}^{0}={\mathcal {A}}}$;
• For ${\displaystyle m=1,2\ldots ,M}$ perform the following:
  1. Construct the standard mode-m flattening ${\displaystyle {\mathcal {A}}_{[m]}^{m-1}}$;
  2. Compute the (compact) singular value decomposition ${\displaystyle {\mathcal {A}}_{[m]}^{m-1}=U_{m}\Sigma _{m}V_{m}^{T}}$, and store the left singular vectors ${\displaystyle U_{m}\in F^{I_{m}\times R_{m}}}$;
  3. Set ${\displaystyle {\mathcal {A}}^{m}=U_{m}^{H}\times _{m}{\mathcal {A}}^{m-1}}$, or, equivalently, ${\displaystyle {\mathcal {A}}_{[m]}^{m}=\Sigma _{m}V_{m}^{T}}$.

In-place computation

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.

Approximation

In applications, such as those mentioned below, a common problem consists of approximating a given tensor ${\displaystyle {\mathcal {A}}\in \mathbb {C} ^{I_{1}\times I_{2}\times \cdots \times I_{m}\cdots \times I_{M}}}$ by one with a reduced multilinear rank. Formally, if the multilinear rank of ${\displaystyle {\mathcal {A}}}$ is denoted by ${\displaystyle \mathrm {rank-} (R_{1},R_{2},\ldots ,R_{m},\ldots ,R_{M})}$, then computing the optimal ${\displaystyle {\mathcal {\bar {A}}}}$ that approximates ${\displaystyle {\mathcal {A}}}$ for a given reduced ${\displaystyle \mathrm {rank-} ({\bar {R}}_{1},{\bar {R}}_{2},\ldots ,{\bar {R}}_{m},\ldots ,{\bar {R}}_{M})}$ is a nonlinear non-convex ${\displaystyle \ell _{2}}$-optimization problem $${\displaystyle \min _{{\mathcal {\bar {A}}}\in \mathbb {C} ^{I_{1}\times I_{2}\times \cdots \times I_{M}}}{\frac {1}{2}}\|{\mathcal {A}}-{\mathcal {\bar {A}}}\|_{F}^{2}\quad {\text{s.t.}}\quad \mathrm {rank-} ({\bar {R}}_{1},{\bar {R}}_{2},\ldots ,{\bar {R}}_{M}),}$$where ${\displaystyle ({\bar {R}}_{1},{\bar {R}}_{2},\ldots ,{\bar {R}}_{M})\in \mathbb {N} ^{M}}$ is the reduced multilinear rank with ${\displaystyle 1\leq {\bar {R}}_{m}<R_{m}\leq I_{m}}$, and the norm ${\displaystyle \|.\|_{F}}$ is the Frobenius norm.

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 classicallytruncated HOSVD is obtained by replacing step 2 in the classic computation by

• Compute a rank-${\displaystyle {\bar {R}}_{m}}$ truncated SVD ${\displaystyle {\mathcal {A}}_{[m]}\approx U_{m}\Sigma _{m}V_{m}^{T}}$, and store the top ${\displaystyle {\bar {R}}_{m}}$ left singular vectors ${\displaystyle U_{m}\in F^{I_{m}\times {\bar {R}}_{m}}}$;

while a sequentially truncated HOSVD (or successively truncated HOSVD) is obtained by replacing step 2 in the interlaced computation by

• Compute a rank-${\displaystyle {\bar {R}}_{m}}$ truncated SVD ${\displaystyle {\mathcal {A}}_{[m]}^{m-1}\approx U_{m}\Sigma _{m}V_{m}^{T}}$, and store the top ${\displaystyle {\bar {R}}_{m}}$ left singular vectors ${\displaystyle U_{m}\in F^{I_{m}\times {\bar {R}}_{m}}}$. Unfortunately, truncation does not result in an optimal solution for the best low multilinear rank optimization problem,. ^{[5] [6] [14] [16]} However, both the classically and interleaved truncated HOSVD result in a quasi-optimal solution: ^{[14] [16] [15] [17]} if ${\displaystyle {\mathcal {\bar {A}}}_{t}}$ denotes the classically or sequentially truncated HOSVD and ${\displaystyle {\mathcal {\bar {A}}}^{*}}$ denotes the optimal solution to the best low multilinear rank approximation problem, then$${\displaystyle \|{\mathcal {A}}-{\mathcal {\bar {A}}}_{t}\|_{F}\leq {\sqrt {M}}\|{\mathcal {A}}-{\mathcal {\bar {A}}}^{*}\|_{F};}$$in 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.

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. ^{[22] [23] [24]} These applications also inspired a higher-order GSVD (HO GSVD) ^[25] and a tensor GSVD. ^[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. ^[32]

Robust L1-norm variant

L1-Tucker is the L1-norm-based, robust variant of Tucker decomposition. ^{[10] [11]} L1-HOSVD is the analogous of HOSVD for the solution to L1-Tucker. ^{[10] [12]}

References

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