Higher-order singular value decomposition explained

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

l{A} is assumed to be given in coordinates with respect to some basis as a M-way array, also denoted by l{A}\in C ^{I _{1} x I _{2} … x … I _{m} … x I _{M}} , where M is the number of modes and the order of the tensor. C is the complex numbers and it includes both the real numbers R and the pure imaginary numbers.

Let

l{A} _{[m]} \in C ^{I _{m} x (I _{1} I _{2} … I _{m-1} I _{m+1} … I _{M)}} denote the standard mode-m flattening of l{A} , so that the left index of l{A} _{[m]} corresponds to the m 'th index l{A} and the right index of l{A} _{[m]} corresponds to all other indices of l{A} combined. Let {\bf U} _{m} \in C ^{I _{m} x I _{m}} be a unitary matrix containing a basis of the left singular vectors of the l{A} _{[m]} such that the jth column u _{j} of {\bf U} _{m} corresponds to the jth largest singular value of l{A} _{[m]} . Observe that the mode/factor matrix {\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 havewhere ⋅ ^{H} denotes the conjugate transpose. The second equality is because the {\bf U} _{m} 's are unitary matrices. Define now the core tensor$$\backslash mathcal\; :=\; \backslash mathcal\; \backslash times\; (\_1^H,\; \_2^H,\; \backslash ldots,\; \_M^H).$$Then, the HOSVD of l{A} is the decomposition$$\backslash mathcal\; =\; \backslash mathcal\backslash times\; (\_1,\; \_2,\; \backslash ldots,\; \_M).$$ 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

{\bf U} _{m} \in C ^{I _{m} x 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 l{A} _{[m]} of l{A} . Let the columns of {\bf U} _{m} be sorted such that the r _{m} th column {\bf u} _{r _{m}} of {\bf U} _{m} corresponds to the r _{m} th largest nonzero singular value of l{A} _{[m]} . Since the columns of {\bf U} _{m} form a basis for the image of l{A} _{[m]} , we have$$\backslash mathcal\_\; =\; \_m\; \_m^H\; \backslash mathcal\_\; =\; \backslash bigl(\backslash mathcal\; \backslash times\_m\; (\_m\; \_m^H)\; \backslash bigr)\_,$$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 m =1 , 2 ,\ldots,m,\ldots,M , we find as before thatwhere the core tensor l{S} is now of size R _{1} x R _{2} x … x R _{M} .

Multilinear rank

The multilinear rank of

l{A} is denoted with rank-(R _{1,} R _{2,} \ldots, R _{M)} . The multilinear rank is a tuple in N ^{M} where R _{m} : = rank ( l{A} _{[m]} ) . Not all tuples in N ^{M} are multilinear ranks.^[13] The multilinear ranks are bounded by 1 \le R _{m} \le I _{m} and it satisfy the constraint $R\_m\; \backslash le\; \backslash prod\_\; R\_i$ must hold.

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

(R _{1,} R _{2,} \ldots, R _{M)} be the multilinear rank of the tensor l{A} . Since l{S} \in { C} ^{R _{1} x R _{2} x … x R _{M}} is a multidimensional array, we can expand it as follows$$\backslash mathcal\; =\; \backslash sum\_^\; \backslash sum\_^\; \backslash cdots\; \backslash sum\_^\; s\_\; \backslash mathbf\_\; \backslash otimes\; \backslash mathbf\_\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash mathbf\_,$$where e _{r _{m}} is the r _{m} th standard basis vector of { C} ^{I _{m}} . By definition of the multilinear multiplication, it holds that$$\backslash mathcal\; =\; \backslash sum\_^\; \backslash sum\_^\; \backslash cdots\; \backslash sum\_^\; s\_\; \backslash mathbf\_\; \backslash otimes\; \backslash mathbf\_\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash mathbf\_,$$where the u _{r _{m}} are the columns of {\bf U} _{m} \in { C} ^{I _{m} x R _{m}} . It is easy to verify that B = \{ u _{r _{1}} ⊗ u _{r _{2}} ⊗ … ⊗ 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 l{A} with respect to a specifically chosen orthonormal basis B with the coefficients given as the multidimensional array l{S} .

Computation

Let

l{A} \in { C} ^{I _{1} x I _{2} x … x I _{M}} be a tensor with a rank-(R _{1,} R _{2,} \ldots, R _{M)} , where C contains the reals 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, further advocated by L. De Lathauwer et al., and by Vasilescu and Terzopulous. 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] 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

m =1 ,\ldots,M , do the following:

1. Construct the mode-m flattening

l{A} _{[m]} ;l{A} _{[m]} = {\bf U} _{m} {\bf \Sigma} _{m} {\bf V} ^{T} _{m} , and store the left singular vectors {\bf U} \in C ^{I _{m} x R _{m}} ;

• Compute the core tensor

l{S} via the multilinear multiplication l{S} = l{A} x _{1} {\bf U} ^{H} _{1} x _{2} {\bf U} ^{H} _{2} \ldots x _{m} {\bf U} ^{H} _{m} \ldots x _{M} {\bf U} ^{H} _{M}

Interlacing computation

A strategy that is significantly faster when some or all

R _{m} \ll I _{m} consists of interlacing the computation of the core tensor and the factor matrices, as follows:^{[14] [15] [16]}

• Set

l{A} ^{0} = l{A} ;

• For

m = 1 , 2 \ldots, M perform the following:

• 1. Construct the standard mode-m flattening

l{A} ^{m -1} _{[m]} ;

• 1. Compute the (compact) singular value decomposition

l{A} ^{m -1} _{[m]} = U _{m} \Sigma _{m} V ^{T} _{m} , and store the left singular vectors U _{m} \in F ^{I _{m} x R _{m}} ;

• 1. Set

l{A} ^{m} = U ^{H} _{m} x _{m} l{A} ^{m-1} , or, equivalently, l{A} ^{m} _{[m]} = \Sigma _{m} V ^{T} _{m} .

In-place computation

The HOSVD can be computed in-place via the Fused In-place Sequentially Truncated Higher Order Singular Value Decomposition (FIST-HOSVD) 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

l{A} \in C ^{I _{1} x I _{2} x … x I _{m} … x I _{M}} by one with a reduced multilinear rank. Formally, if the multilinear rank of l{A} is denoted by rank- (R _{1,R} _{2,\ldots,R} _{m,\ldots,R} _{M)} , then computing the optimal l{\bar A} that approximates l{A} for a given reduced rank- (\bar R _{1,\bar} R _{2,\ldots,\bar} R _{m,\ldots,\bar} R _{M)} is a nonlinear non-convex \ell _{2} -optimization problem$$\backslash min\_\; \backslash frac\; \backslash |\; \backslash mathcal\; -\; \backslash mathcal\; \backslash |\_F^2\backslash quad\backslash text\backslash quad\; \backslash mathrm(\backslash bar\; R\_1,\; \backslash bar\; R\_2,\; \backslash ldots,\; \backslash bar\; R\_M),$$where (\bar R _{1,} \bar R _{2,} \ldots, \bar R _{M)} \in N ^{M} is the reduced multilinear rank with 1 \le \bar R _{m} < R _{m} \le I _{m} , and the norm \|.\| _{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 classically truncated HOSVD is obtained by replacing step 2 in the classic computation by

• Compute a rank-

\bar R _{m} truncated SVD l{A} _{[m]} ≈ U _{m} \Sigma _{m} V ^{T} _{m} , and store the top \bar R _{m} left singular vectors U _{m} \in F ^{I _{m} x \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-

\bar R _{m} truncated SVD l{A} ^{m -1} _{[m]} ≈ U _{m} \Sigma _{m} V ^{T} _{m} , and store the top \bar R _{m} left singular vectors U _{m} \in F ^{I _{m} x \bar R _{m}} . Unfortunately, truncation does not result in an optimal solution for the best low multilinear rank optimization problem,.^[6] However, both the classically and interleaved truncated HOSVD result in a quasi-optimal solution:^[17] if l{\bar A} _{t} denotes the classically or sequentially truncated HOSVD and l{\bar A} ^{*} denotes the optimal solution to the best low multilinear rank approximation problem, then$$\backslash |\; \backslash mathcal\; -\; \backslash mathcal\_t\; \backslash |\_F\; \backslash le\; \backslash sqrt\; \backslash |\; \backslash mathcal\; -\; \backslash mathcal^*\; \backslash |\_F;$$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. 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

1. Hitchcock. Frank L. 1928-04-01. Multiple Invariants and Generalized Rank of a M-Way Array or Tensor. Journal of Mathematics and Physics. en. 7. 1–4. 39–79. 10.1002/sapm19287139. 1467-9590.
2. Tucker. Ledyard R.. 1966-09-01. Some mathematical notes on three-mode factor analysis. Psychometrika. en. 31. 3. 279–311. 10.1007/bf02289464. 5221127. 44301099. 0033-3123.
3. Tucker. L. R.. 1963. Implications of factor analysis of three-way matrices for measurement of change. In C. W. Harris (Ed.), Problems in Measuring Change. Madison, Wis.: Univ. Wis. Press.. 122–137.
4. Tucker. L. R.. 1964. The extension of factor analysis to three-dimensional matrices. In N. Frederiksen and H. Gulliksen (Eds.), Contributions to Mathematical Psychology. New York: Holt, Rinehart and Winston. 109–127.
5. De Lathauwer. L.. De Moor. B.. Vandewalle. J.. 2000-01-01. A Multilinear Singular Value Decomposition. SIAM Journal on Matrix Analysis and Applications. 21. 4. 1253–1278. 10.1137/s0895479896305696. 0895-4798. 10.1.1.102.9135.
6. M. A. O. Vasilescu, D. Terzopoulos (2002) with the name M-mode SVD. The M-mode SVD is suitable for parallel computation and employs the matrix SVD "Multilinear Analysis of Image Ensembles: TensorFaces", Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002
7. M. A. O. Vasilescu, D. Terzopoulos (2003) "Multilinear Subspace Analysis of Image Ensembles", "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003"
8. M. A. O. Vasilescu, D. Terzopoulos (2005) "Multilinear Independent Component Analysis", "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’05), San Diego, CA, June 2005, vol.1, 547–553."
9. Godfarb. Donald. Zhiwei. Qin. Robust low-rank tensor recovery: Models and algorithms. SIAM Journal on Matrix Analysis and Applications. 35. 1. 225–253. 2014. 10.1137/130905010. 1311.6182. 1051205.
10. Chachlakis. Dimitris G.. Prater-Bennette. Ashley. Markopoulos. Panos P.. L1-norm Tucker Tensor Decomposition. IEEE Access. 22 November 2019. 7. 178454–178465. 10.1109/ACCESS.2019.2955134. free. 1904.06455.
11. Markopoulos. Panos P.. Chachlakis. Dimitris G.. Papalexakis. Evangelos. The Exact Solution to Rank-1 L1-Norm TUCKER2 Decomposition. IEEE Signal Processing Letters. 25. 4. April 2018. 511–515. 10.1109/LSP.2018.2790901. 1710.11306. 2018ISPL...25..511M. 3693326.
12. Book: Markopoulos. Panos P.. Chachlakis. Dimitris G.. Prater-Bennette. Ashley. 2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP) . L1-Norm Higher-Order Singular-Value Decomposition . 21 February 2019. 1353–1357. 10.1109/GlobalSIP.2018.8646385. 978-1-7281-1295-4. 67874182.
13. Carlini. Enrico. Kleppe. Johannes. Ranks derived from multilinear maps. Journal of Pure and Applied Algebra. 215. 8. 1999–2004. 10.1016/j.jpaa.2010.11.010. 2011. free.
14. Vannieuwenhoven. N.. Vandebril. R.. Meerbergen. K.. 2012-01-01. A New Truncation Strategy for the Higher-Order Singular Value Decomposition. SIAM Journal on Scientific Computing. 34. 2. A1027–A1052. 10.1137/110836067. 2012SJSC...34A1027V . 15318433 . 1064-8275.
15. Book: Hackbusch, Wolfgang. Tensor Spaces and Numerical Tensor Calculus SpringerLink. 42. en-gb. 10.1007/978-3-642-28027-6. Springer Series in Computational Mathematics. 2012. 978-3-642-28026-9. 117253621 .
16. Cobb. Benjamin. Kolla. Hemanth. Phipps. Eric. Çatalyürek. Ümit V.. 2022. FIST-HOSVD: Fused in-Place Sequentially Truncated Higher Order Singular Value Decomposition. Platform for Advanced Scientific Computing(PASC) . en. 9781450394109. 10.1145/3539781.3539798. free.
17. Grasedyck. L.. 2010-01-01. Hierarchical Singular Value Decomposition of Tensors. SIAM Journal on Matrix Analysis and Applications. 31. 4. 2029–2054. 10.1137/090764189. 0895-4798. 10.1.1.660.8333.
18. M. A. O. Vasilescu (2002) "Human Motion Signatures: Analysis, Synthesis, Recognition," Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460.
19. M.A.O. Vasilescu, D. Terzopoulos (2003) "Multilinear Subspace Analysis for Image Ensembles, M. A. O. Vasilescu, D. Terzopoulos, Proc. Computer Vision and Pattern Recognition Conf. (CVPR '03), Vol.2, Madison, WI, June, 2003, 93–99.
20. M.A.O. Vasilescu, D. Terzopoulos (2002) "Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002, in Computer Vision -- ECCV 2002, Lecture Notes in Computer Science, Vol. 2350, A. Heyden et al. (Eds.), Springer-Verlag, Berlin, 2002, 447–460.
21. M.A.O. Vasilescu, D. Terzopoulos (2004) "TensorTextures: Multilinear Image-Based Rendering", M. A. O. Vasilescu and D. Terzopoulos, Proc. ACM SIGGRAPH 2004 Conference Los Angeles, CA, August, 2004, in Computer Graphics Proceedings, Annual Conference Series, 2004, 336–342.
22. L. Omberg. G. H. Golub. O. Alter. November 2007. A Tensor Higher-Order Singular Value Decomposition for Integrative Analysis of DNA Microarray Data From Different Studies. PNAS. 104. 47. 18371–18376. 10.1073/pnas.0709146104. 2147680. 18003902. 2007PNAS..10418371O. free.
23. L. Omberg. J. R. Meyerson. K. Kobayashi. L. S. Drury. J. F. X. Diffley. O. Alter. October 2009. Global Effects of DNA Replication and DNA Replication Origin Activity on Eukaryotic Gene Expression. Molecular Systems Biology. 5. 312. 10.1038/msb.2009.70. 2779084. 19888207. Highlight.
24. C. Muralidhara. A. M. Gross. R. R. Gutell. O. Alter. April 2011. Tensor Decomposition Reveals Concurrent Evolutionary Convergences and Divergences and Correlations with Structural Motifs in Ribosomal RNA. PLOS ONE. 6. 4. e18768. 10.1371/journal.pone.0018768. 3094155. 21625625. Highlight. 2011PLoSO...618768M. free.
25. S. P. Ponnapalli. M. A. Saunders. C. F. Van Loan. O. Alter. December 2011. A Higher-Order Generalized Singular Value Decomposition for Comparison of Global mRNA Expression from Multiple Organisms. PLOS ONE. 6. 12. e28072. 10.1371/journal.pone.0028072. 3245232. 22216090. Highlight. 2011PLoSO...628072P. free.
26. P. Sankaranarayanan. T. E. Schomay. K. A. Aiello. O. Alter. April 2015. Tensor GSVD of Patient- and Platform-Matched Tumor and Normal DNA Copy-Number Profiles Uncovers Chromosome Arm-Wide Patterns of Tumor-Exclusive Platform-Consistent Alterations Encoding for Cell Transformation and Predicting Ovarian Cancer Survival. PLOS ONE. 10. 4. e0121396. 10.1371/journal.pone.0121396. 4398562. 25875127. AAAS EurekAlert! Press Release and NAE Podcast Feature. 2015PLoSO..1021396S. free.
27. Hadi Fanaee-T. João Gama. May 2015. EigenEvent: An algorithm for event detection from complex data streams in Syndromic surveillance. Intelligent Data Analysis. 19. 3. 597–616. 1406.3496. 2014arXiv1406.3496F. 10.3233/IDA-150734. 17966555.
28. P. Baranyi. April 2004. TP model transformation as a way to LMI based controller design. IEEE Transactions on Industrial Electronics. 51. 387 - 400. 10.1109/tie.2003.822037. 2. 7957799.
29. P. Baranyi. D. Tikk. Y. Yam. R. J. Patton. 2003. From Differential Equations to PDC Controller Design via Numerical Transformation. Computers in Industry. 51. 3. 281 - 297. 10.1016/s0166-3615(03)00058-7.
30. Definition of the HOSVD-based canonical form of polytopic dynamic models. P. Baranyi. L. Szeidl. P. Várlaki. Y. Yam. July 3–5, 2006. Budapest, Hungary. 660–665. 3rd International Conference on Mechatronics (ICM 2006).
31. Y-h. Taguchi. August 2017. Tensor decomposition-based unsupervised feature extraction applied to matrix products for multi-view data processing. PLOS ONE. 12. 8. e0183933. 10.1371/journal.pone.0183933. 5571984. 28841719. 2017PLoSO..1283933T. free.
32. Y-h. Taguchi. October 2017. Identification of candidate drugs using tensor-decomposition-based unsupervised feature extraction in integrated analysis of gene expression between diseases and DrugMatrix datasets. Scientific Reports. 7. 1. 13733. 10.1038/s41598-017-13003-0. 5653784. 29062063. 2017NatSR...713733T.