Editing Generalized singular value decomposition

{{short description|Name of two different techniques based on the singular value decomposition}}
{{Use dmy dates|date=October 2020}}
In [[linear algebra]], the '''generalized singular value decomposition''' ('''GSVD''') is the name of two different techniques based on the [[singular value decomposition|singular value decomposition (SVD)]]. The two versions differ because one version decomposes two matrices (somewhat like the [[Higher-order singular value decomposition|higher-order or tensor SVD]]) and the other version uses a set of constraints imposed on the left and right singular vectors of a single-matrix SVD.

==First version: two-matrix decomposition==
The '''generalized singular value decomposition''' ('''GSVD''') is a [[matrix decomposition]] on a pair of matrices which generalizes the [[singular value decomposition]]. It was introduced by Van Loan <ref name="VanLoan"/> in 1976 and later developed by Paige and [[Michael Saunders (academic)|Saunders]],<ref name = "Paige"/> which is the version described here. In contrast to the SVD, the GSVD decomposes simultaneously a pair of matrices with the same number of columns. The SVD and the GSVD, as well as some other possible generalizations of the SVD,<ref>{{Cite book | last = Hansen | first = Per Christian | name-list-style = vanc | publisher = SIAM Monographs on Mathematical Modeling and Computation | year = 1997 | title = Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion | isbn = 0-89871-403-6 }}</ref><ref>{{cite web | last1 = de Moor | first1 = Bart L. R. | last2 = Golub | first2 = Gene H. | name-list-style = vanc | year = 1989 | title = Generalized Singular Value Decompositions A Proposal for a Standard Nomenclauture | url = http://ftp.esat.kuleuven.be/pub/SISTA/ida/reports/89-10.pdf  }}</ref><ref>{{cite journal | last1 = de Moor | first1 = Bart L. R. | last2 = Zha| first2 = Hongyuan | name-list-style = vanc | year = 1991 | title = A tree of generalizations of the ordinary singular value decomposition|  journal = Linear Algebra and Its Applications | volume = 147 | pages = 469–500  | doi = 10.1016/0024-3795(91)90243-P | doi-access = free }}</ref> are extensively used in the study of the [[condition number|conditioning]] and [[regularization (mathematics)|regularization]] of linear systems with respect to quadratic [[semi-norm]]s. In the following, let <math>\mathbb{F} = \mathbb{R}</math>, or <math>\mathbb{F} = \mathbb{C}</math>. 

=== Definition === 

The '''generalized singular value decomposition''' of matrices <math>A_1 \in \mathbb{F}^{m_1 \times n}</math> and <math>A_2 \in \mathbb{F}^{m_2 \times n}</math> is<math display="block">
\begin{align}
A_1 & = U_1\Sigma_1 [ W^* D, 0_D] Q^*, \\
A_2 & = U_2\Sigma_2 [ W^* D, 0_D] Q^*,
\end{align}
</math>where
* <math>U_1 \in \mathbb{F}^{m_1 \times m_1}</math> is [[unitary matrix|unitary]], 
* <math>U_2 \in \mathbb{F}^{m_2 \times m_2}</math> is unitary, 
* <math>Q \in \mathbb{F}^{n \times n}</math> is unitary,
*<math>
W \in \mathbb{F}^{k \times k}
</math>is unitary,
*<math>
D \in \mathbb{R}^{k \times k}
</math> is real diagonal with positive diagonal, and contains the non-zero singular values of <math>C = \begin{bmatrix} A_1 \\ A_2 \end{bmatrix}</math> in decreasing order,
* <math>0_D = 0 \in \mathbb{R}^{k \times (n - k)} </math>,
* <math>\Sigma_1 = \lceil I_A, S_1, 0_A \rfloor \in \mathbb{R}^{m_1 \times k}</math> is real non-negative [[Block matrix|block-diagonal]], where <math>S_1 = \lceil \alpha_{r + 1}, \dots, \alpha_{r + s} \rfloor</math> with <math> 1 > \alpha_{r + 1} \ge \cdots \ge \alpha_{r + s} > 0</math>, <math>I_A = I_r</math>, and <math>0_A = 0 \in \mathbb{R}^{(m_1 - r - s) \times (k - r - s)} </math>,
* <math>\Sigma_2 = \lceil 0_B, S_2, I_B \rfloor \in \mathbb{R}^{m_2 \times k}</math> is real non-negative block-diagonal, where <math>S_2 = \lceil \beta_{r + 1}, \dots, \beta_{r + s} \rfloor </math> with <math> 0 < \beta_{r + 1} \le \cdots \le \beta_{r + s} < 1</math>, <math>I_B = I_{k - r - s}</math>, and <math>0_B = 0 \in \mathbb{R}^{(m_2 - k + r) \times r} </math>,
* <math>\Sigma_1^* \Sigma_1 = \lceil\alpha_1^2, \dots, \alpha_k^2\rfloor</math>,
* <math>\Sigma_2^* \Sigma_2 = \lceil\beta_1^2, \dots, \beta_k^2\rfloor</math>,
* <math>\Sigma_1^* \Sigma_1 + \Sigma_2^* \Sigma_2 = I_k</math>,
* <math>k = \textrm{rank}(C)</math>.

We denote <math>\alpha_1 = \cdots = \alpha_r = 1</math>, <math>\alpha_{r + s + 1} = \cdots = \alpha_k = 0</math>, <math>\beta_1 = \cdots = \beta_r = 0</math>, and <math>\beta_{r + s + 1} = \cdots = \beta_k = 1</math>. While <math>\Sigma_1</math> is diagonal, <math>\Sigma_2 </math> is not always diagonal, because of the leading rectangular [[zero matrix]]; instead <math>\Sigma_2</math> is "bottom-right-diagonal". 

=== Variations ===

There are many variations of the GSVD. These variations are related to the fact that it is always possible to multiply <math>Q^*</math> from the left by <math>E E^* = I</math> where <math>E \in \mathbb{F}^{n \times n}</math> is an arbitrary unitary matrix. We denote 

* <math>X = ([W^* D, 0_D] Q^*)^*</math> 
* <math>
X^* = [0, R] \hat{Q}^*

</math>, where <math>
R \in \mathbb{F}^{k \times k}

</math> is upper-triangular and invertible, and <math>
\hat{Q} \in \mathbb{F}^{n \times n}

</math> is unitary. Such matrices exist by [[QR decomposition|RQ-decomposition]].
* <math>Y = W^* D</math>. Then <math>
Y
</math> is invertible. 

Here are some variations of the GSVD: 

* [[MATLAB]] (gsvd):<math display="block">
\begin{aligned}
A_1 & = U_1 \Sigma_1 X^*, \\
A_2 & = U_2 \Sigma_2 X^*.
\end{aligned}

</math> 
* [[LAPACK]] (LA_GGSVD):<math display="block">
\begin{aligned}
A_1 & = U_1 \Sigma_1 [0, R] \hat{Q}^*, \\
A_2 & = U_2 \Sigma_2 [0, R] \hat{Q}^*.
\end{aligned}

</math> 
* Simplified:<math display="block">
\begin{align}
A_1 & = U_1\Sigma_1 [ Y, 0_D] Q^*, \\
A_2 & = U_2\Sigma_2 [ Y, 0_D] Q^*.
\end{align}
</math>

=== Generalized singular values ===

A ''generalized singular value'' of <math>A_1</math> and <math>A_2</math> is a pair <math>(a, b) \in \mathbb{R}^2</math> such that 

<math display="block">
\begin{align}
\lim_{\delta \to 0} \det(b^2 A_1^* A_1 - a^2 A_2^* A_2 + \delta I_n) / \det(\delta I_{n - k}) & = 0, \\
a^2 + b^2 & = 1, \\
a, b & \geq 0.
\end{align}
</math>We have 

*<math> A_i A_j^* = U_i \Sigma_i Y Y^* \Sigma_j^* U_j^*</math>
*<math> A_i^* A_j = Q \begin{bmatrix} Y^* \Sigma_i^* \Sigma_j Y & 0 \\ 0 & 0 \end{bmatrix} Q^* = Q_1 Y^* \Sigma_i^* \Sigma_j Y Q_1^* </math>


By these properties we can show that the generalized singular values are exactly the pairs <math>(\alpha_i, \beta_i)</math>. We have<math display="block">
\begin{aligned}
& \det(b^2 A_1^* A_1 - a^2 A_2^* A_2 + \delta I_n) \\
= & \det(b^2 A_1^* A_1 - a^2 A_2^* A_2 + \delta Q Q^*) \\
= & \det\left(Q \begin{bmatrix} Y^* (b^2 \Sigma_1^* \Sigma_1 - a^2 \Sigma_2^* \Sigma_2) Y + \delta I_k & 0 \\ 0 & \delta I_{n - k} \end{bmatrix} Q^*\right) \\
= & \det(\delta I_{n - k}) \det(Y^* (b^2 \Sigma_1^* \Sigma_1 - a^2 \Sigma_2^* \Sigma_2) Y + \delta I_k).
\end{aligned}
</math>Therefore
:<math>
\begin{aligned}
{} & \lim_{\delta \to 0} \det(b^2 A_1^* A_1 - a^2 A_2^* A_2 + \delta I_n) / \det(\delta I_{n - k}) \\
= & \lim_{\delta \to 0} \det(Y^* (b^2 \Sigma_1^* \Sigma_1 - a^2 \Sigma_2^* \Sigma_2) Y + \delta I_k) \\
= & \det(Y^* (b^2 \Sigma_1^* \Sigma_1 - a^2 \Sigma_2^* \Sigma_2) Y) \\
= & |\det(Y)|^2 \prod_{i = 1}^k (b^2 \alpha_i^2 - a^2 \beta_i^2).
\end{aligned}
</math>
This expression is zero exactly when <math>a = \alpha_i</math> and <math>b = \beta_i</math> for some <math>i</math>.

In,<ref name="Paige" /> the generalized singular values are claimed to be those which solve <math>\det(b^2 A_1^* A_1 - a^2 A_2^* A_2) = 0</math>. However, this claim only holds when <math>k = n</math>, since otherwise the determinant is zero for every pair <math>(a, b) \in \mathbb{R}^2</math>; this can be seen by substituting <math>\delta = 0</math> above.

=== Generalized inverse ===

Define <math>E^+ = E^{-1}</math> for any invertible matrix <math>E \in \mathbb{F}^{n \times n}</math> , <math>0^+ = 0^*</math> for any zero matrix <math>0 \in \mathbb{F}^{m \times n}</math>,  and <math>\left\lceil E_1, E_2 \right\rfloor^+ = \left\lceil E_1^+, E_2^+ \right\rfloor</math> for any block-diagonal matrix. Then define<math display="block">A_i^+ = Q \begin{bmatrix} Y^{-1} \\ 0 \end{bmatrix} \Sigma_i^+ U_i^*</math>It can be shown that <math>A_i^+</math> as defined here is a [[generalized inverse]] of <math>A_i</math>; in particular a <math>\{1, 2, 3\}</math>-inverse of <math>A_i</math>. Since it does not in general satisfy <math>(A_i^+ A_i)^* = A_i^+ A_i</math>, this is not the [[Moore–Penrose inverse]]; otherwise we could derive <math>(AB)^+ = B^+ A^+</math> for any choice of matrices, which only holds for [[Moore–Penrose inverse|certain class of matrices]]. 

Suppose <math> Q = \begin{bmatrix}Q_1 & Q_2\end{bmatrix} </math>, where <math>Q_1 \in \mathbb{F}^{n \times k}</math> and <math>Q_2 \in \mathbb{F}^{n \times (n - k)}</math>. This generalized inverse has the following properties:

* <math> \Sigma_1^+ = \lceil I_A, S_1^{-1}, 0_A^T \rfloor </math>
* <math> \Sigma_2^+ = \lceil 0^T_B, S_2^{-1}, I_B \rfloor </math>
* <math> \Sigma_1 \Sigma_1^+ = \lceil I, I, 0 \rfloor </math>
*<math> \Sigma_2 \Sigma_2^+ = \lceil 0, I, I \rfloor </math>
*<math> \Sigma_1 \Sigma_2^+ = \lceil 0, S_1 S_2^{-1}, 0 \rfloor </math>
* <math> \Sigma_1^+ \Sigma_2 = \lceil 0, S_1^{-1} S_2, 0 \rfloor </math>
* <math> A_i A_j^+ = U_i \Sigma_i \Sigma_j^+ U_j^*</math>
* <math> A_i^+ A_j = Q \begin{bmatrix} Y^{-1} \Sigma_i^+ \Sigma_j Y & 0 \\ 0 & 0 \end{bmatrix} Q^* = Q_1 Y^{-1} \Sigma_i^+ \Sigma_j Y Q_1^* </math>

=== Quotient SVD ===

A ''generalized singular ratio'' of <math>A_1</math> and <math>A_2</math> is <math>\sigma_i=\alpha_i \beta_i^+</math>. By the above properties, <math> A_1 A_2^+ = U_1 \Sigma_1 \Sigma_2^+ U_2^*</math>. Note that <math> \Sigma_1 \Sigma_2^+ = \lceil 0, S_1 S_2^{-1}, 0 \rfloor </math> is diagonal, and that, ignoring the leading zeros, contains the singular ratios in decreasing order. If <math>A_2</math> is invertible, then <math> \Sigma_1 \Sigma_2^+ </math> has no leading zeros, and the generalized singular ratios are the singular values, and <math>U_1</math> and <math>U_2</math> are the matrices of singular vectors, of the matrix <math>A_1 A_2^+ = A_1 A_2^{-1}</math>. In fact, computing the SVD of <math>A_1 A_2^{-1}</math> is one of the motivations for the GSVD, as "forming <math>AB^{-1}</math> and finding its SVD can lead to unnecessary and large numerical errors when <math>B</math> is ill-conditioned for solution of equations".<ref name = "Paige"/> Hence the sometimes used name "quotient SVD", although this is not the only reason for using GSVD. If <math>A_2</math> is not invertible, then <math> U_1 \Sigma_1 \Sigma_2^+ U_2^*</math>is still the SVD of <math> A_1 A_2^+</math> if we relax the requirement of having the singular values in decreasing order. Alternatively, a decreasing order SVD can be found by moving the leading zeros to the back: <math> U_1 \Sigma_1 \Sigma_2^+ U_2^* = (U_1 P_1) P_1^* \Sigma_1 \Sigma_2^+ P_2 (P_2^* U_2^*)</math>, where <math> P_1</math> and <math> P_2</math> are appropriate permutation matrices. Since rank equals the number of non-zero singular values, <math> \mathrm{rank}(A_1 A_2^+)=s</math>.

=== Construction ===
Let 

* <math>C = P \lceil D, 0 \rfloor Q^*</math> be the SVD of <math>C = \begin{bmatrix} A_1 \\ A_2 \end{bmatrix}</math>, where <math>P \in \mathbb{F}^{(m_1 + m_2) \times (m_1 \times m_2)}</math> is unitary, and <math>Q</math> and <math>D</math> are as described,
* <math>P = [P_1, P_2]</math>, where <math>P_1 \in \mathbb{F}^{(m_1 + m_2) \times k}</math> and <math>P_2 \in \mathbb{F}^{(m_1 + m_2) \times (n - k)}</math>,
* <math>P_1 = \begin{bmatrix} P_{11} \\ P_{21} \end{bmatrix}</math>, where <math>P_{11} \in \mathbb{F}^{m_1 \times k}</math> and <math>P_{21} \in \mathbb{F}^{m_2 \times k}</math>,
* <math>P_{11} = U_1 \Sigma_1 W^*</math> by the SVD of <math>P_{11}</math>, where <math>U_1</math>, <math>\Sigma_1</math> and <math>W</math> are as described,
* <math>P_{21} W = U_2 \Sigma_2</math> by a decomposition similar to a [[QR decomposition|QR-decomposition]], where <math>U_2</math> and <math>\Sigma_2</math> are as described.

Then<math display="block">\begin{aligned}
C & = P \lceil D, 0 \rfloor Q^* \\
{} & = [P_1 D, 0] Q^* \\
{} & = \begin{bmatrix} U_1 \Sigma_1 W^* D & 0 \\ U_2 \Sigma_2 W^* D & 0 \end{bmatrix} Q^* \\
{} & = \begin{bmatrix} U_1 \Sigma_1 [W^* D, 0] Q^* \\ U_2 \Sigma_2 [W^* D, 0] Q^* \end{bmatrix} .
\end{aligned}</math>We also have<math display="block">\begin{bmatrix} U_1^* & 0 \\ 0 & U_2^* \end{bmatrix} P_1 W = \begin{bmatrix} \Sigma_1 \\ \Sigma_2 \end{bmatrix}.</math>Therefore<math display="block">\Sigma_1^* \Sigma_1 + \Sigma_2^* \Sigma_2 = \begin{bmatrix} \Sigma_1 \\ \Sigma_2 \end{bmatrix}^* \begin{bmatrix} \Sigma_1 \\ \Sigma_2 \end{bmatrix} = W^* P_1^* \begin{bmatrix} U_1 & 0 \\ 0 & U_2 \end{bmatrix} \begin{bmatrix} U_1^* & 0 \\ 0 & U_2^* \end{bmatrix} P_1 W = I.</math>Since <math>P_1</math> has orthonormal columns, <math>||P_1||_2 \leq 1</math>. Therefore<math display="block">||\Sigma_1||_2 = ||U_1^* P_1 W||_2 = ||P_1||_2 \leq 1.</math>We also have for each <math>x \in \mathbb{R}^k</math> such that <math>||x||_2 = 1</math> that<math display="block">||P_{21} x||_2^2 \leq ||P_{11} x||_2^2 + ||P_{21} x||_2^2 = ||P_{1} x||_2^2 \leq 1.</math>Therefore <math>||P_{21}||_2 \leq 1</math>, and<math display="block">||\Sigma_2||_2 = || U_2^* P_{21} W ||_2 = ||P_{21}||_2 \leq 1.</math>

== Applications ==
[[File:Tensor Generalized Singular Value Decomposition following et int. Alter PLoS One 2015 and Alter NCI Physical Sciences in Oncology 2015.jpg|thumb|The tensor GSVD is one of the comparative spectral decompositions, multi-tensor generalizations of the SVD, invented to simultaneously identify the similar and dissimilar among, and create a single coherent model from any data types, of any number and dimensions.]]

The GSVD, formulated as a comparative spectral decomposition,<ref>{{cite journal | vauthors = Alter O, Brown PO, Botstein D | title = Generalized singular value decomposition for comparative analysis of genome-scale expression data sets of two different organisms | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 100 | issue = 6 | pages = 3351–6 | date = March 2003 | pmid = 12631705 | pmc = 152296 | doi = 10.1073/pnas.0530258100 | bibcode = 2003PNAS..100.3351A | doi-access = free }}</ref> has been successfully applied to signal processing and data science, e.g., in genomic signal processing.<ref>{{cite journal | vauthors = Lee CH, Alpert BO, Sankaranarayanan P, Alter O | title = GSVD comparison of patient-matched normal and tumor aCGH profiles reveals global copy-number alterations predicting glioblastoma multiforme survival | journal = PLOS ONE| volume = 7 | issue = 1 | pages = e30098 | date = January 2012 | pmid = 22291905 | pmc = 3264559 | doi = 10.1371/journal.pone.0030098 | bibcode = 2012PLoSO...730098L | doi-access = free }}</ref><ref>{{cite journal | vauthors = Aiello KA, Ponnapalli SP, Alter O | title = Mathematically universal and biologically consistent astrocytoma genotype encodes for transformation and predicts survival phenotype | journal = APL Bioengineering | volume = 2 | issue = 3 | pages = 031909 | date = September 2018 | pmid = 30397684 | pmc = 6215493 | doi = 10.1063/1.5037882 }}</ref><ref>{{cite journal | vauthors = Ponnapalli SP, Bradley MW, Devine K, Bowen J, Coppens SE, Leraas KM, Milash BA, Li F, Luo H, Qiu S, Wu K, Yang H, Wittwer CT, Palmer CA, Jensen RL, Gastier-Foster JM, Hanson HA, [[Jill S. Barnholtz-Sloan|Barnholtz-Sloan JS]], Alter O | title = Retrospective Clinical Trial Experimentally Validates Glioblastoma Genome-Wide Pattern of DNA Copy-Number Alterations Predictor of Survival | journal = APL Bioengineering | volume = 4 | issue = 2 | pages = 026106 | date = May 2020 | doi = 10.1063/1.5142559 | pmid = 32478280 | pmc = 7229984 | id = [https://www.eurekalert.org/pub_releases/2020-05/uouh-gpf051320.php Press Release] | doi-access = free }}</ref>

These applications inspired several additional comparative spectral decompositions, i.e., the higher-order GSVD (HO GSVD)<ref name="Ponnapalli2011"/> and the tensor GSVD.<ref name="Sankaranarayanan2015"/> <ref name = "Bradley2019"/>

It has equally found applications to estimate the spectral decompositions of linear operators when the eigenfunctions are parameterized with a [[linear model]], i.e. a [[reproducing kernel Hilbert space]].<ref>{{Cite arXiv|last1=Cabannes|first1=Vivien|last2=Pillaud-Vivien|first2=Loucas|last3=Bach|first3=Francis|last4=Rudi|first4=Alessandro|date=2021|title=Overcoming the curse of dimensionality with Laplacian regularization in semi-supervised learning|class=stat.ML|eprint=2009.04324}}</ref>

==Second version: weighted single-matrix decomposition==
The weighted version of the '''generalized singular value decomposition''' ('''GSVD''') is a constrained [[matrix decomposition]] with constraints imposed on the left and right singular vectors of the [[singular value decomposition]].<ref>{{cite book | vauthors = Jolliffe IT | title = Principal Component Analysis | url = https://archive.org/details/principalcompone00joll_0 | series = Springer Series in Statistics | edition = 2nd | publisher = Springer | location = NY | date = 2002 | isbn = 978-0-387-95442-4 | url-access = registration }}
</ref><ref>{{Cite book | last = Greenacre | first = Michael | name-list-style = vanc | publisher = Academic Press | location = London | year = 1983 | title = Theory and Applications of Correspondence Analysis | isbn = 978-0-12-299050-2 }}</ref><ref>{{Cite journal| vauthors = Abdi H, Williams LJ |year = 2010 | title = Principal component analysis. | journal = Wiley Interdisciplinary Reviews: Computational Statistics | volume = 2 |issue=4 | pages = 433–459 | doi=10.1002/wics.101|s2cid = 122379222 }}</ref> This form of the ''GSVD'' is an extension of the ''SVD'' as such. Given the ''SVD'' of an ''m×n'' real or complex matrix ''M''
:<math>M = U\Sigma V^* \,</math>
where
:<math>U^* W_u U = V^* W_v V = I.</math>
Where ''I'' is the [[identity matrix]] and where <math>U</math> and <math>V</math> are orthonormal given their constraints (<math>W_u</math> and <math>W_v</math>). Additionally, <math>W_u</math> and <math>W_v</math> are positive definite matrices (often diagonal matrices of weights). This form of the ''GSVD'' is the core of certain techniques, such as generalized [[principal component analysis]] and [[Correspondence analysis]].

The weighted form of the ''GSVD'' is called as such because, with the correct selection of weights, it ''generalizes'' many techniques (such as [[multidimensional scaling]] and [[linear discriminant analysis]]).<ref>{{cite book | vauthors = Abdi H | date = 2007 | chapter = Singular Value Decomposition (SVD) and Generalized Singular Value Decomposition (GSVD). | veditors = Salkind NJ | title = Encyclopedia of Measurement and Statistics. | url = https://archive.org/details/encyclopediameas00salk | url-access = limited | location = Thousand Oaks (CA) | publisher = Sage | pages = [https://archive.org/details/encyclopediameas00salk/page/n939 907]–912 }}</ref>

== References ==

{{Reflist|refs=

<ref name = "Bradley2019">{{cite journal | vauthors = Bradley MW, Aiello KA, Ponnapalli SP, Hanson HA, Alter O | title = GSVD- and tensor GSVD-uncovered patterns of DNA copy-number alterations predict adenocarcinomas survival in general and in response to platinum | journal = APL Bioengineering | volume = 3 | issue = 3 | pages = 036104 | date = September 2019 | pmid = 31463421 | pmc = 6701977 | doi = 10.1063/1.5099268 | id = [https://alterlab.org/publications/Bradley_et_al_APL_Bioeng_2019_Supplementary_Material.pdf Supplementary Material] }}</ref>

<ref name="Sankaranarayanan2015">{{cite journal | vauthors = Sankaranarayanan P, Schomay TE, Aiello KA, Alter O | title = 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 | journal = PLOS ONE| volume = 10 | issue = 4 | pages = e0121396 | date = April 2015 | pmid = 25875127 | pmc = 4398562 | doi = 10.1371/journal.pone.0121396  | bibcode = 2015PLoSO..1021396S | doi-access = free }}</ref>

<ref name="VanLoan"> {{cite journal | last= Van Loan | first = Charles F. | name-list-style = vanc | year = 1976 | title = Generalizing the Singular Value Decomposition | journal = SIAM J. Numer. Anal. | volume = 13 | issue = 1| pages = 76–83 | doi = 10.1137/0713009 | bibcode = 1976SJNA...13...76V }}</ref> 

<ref name = "Paige">{{cite journal | last1 = Paige | first1 = C. C. | last2 = Saunders | first2 = M. A. | name-list-style = vanc | year = 1981 | title = Towards a Generalized Singular Value Decomposition | journal = SIAM J. Numer. Anal. | volume = 18 | issue = 3| pages = 398–405| doi = 10.1137/0718026 | bibcode = 1981SJNA...18..398P }}</ref> 

<ref name="Ponnapalli2011">{{cite journal | vauthors = Ponnapalli SP, Saunders MA, Van Loan CF, Alter O | title = A higher-order generalized singular value decomposition for comparison of global mRNA expression from multiple organisms | journal = PLOS ONE| volume = 6 | issue = 12 | pages = e28072 | date = December 2011 | pmid = 22216090 | pmc = 3245232 | doi = 10.1371/journal.pone.0028072 | bibcode = 2011PLoSO...628072P | doi-access = free }}</ref>

}}

== Further reading ==
{{refbegin}}
* {{Cite book | last1 = Golub | first1 = Gene | last2 = Van Loan | first2 = Charles | name-list-style = vanc | publisher = Johns Hopkins University Press | location = Baltimore | year = 1996 | title = Matrix Computation | edition = Third | isbn = 0-8018-5414-8 }}
* [[LAPACK]] manual [http://www.netlib.org/lapack/lug/node36.html]
{{refend}}

[[Category:Linear algebra]]
[[Category:Singular value decomposition]]