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Yeast Infection No More Review Summary<br><br>
In [[linear algebra]], the '''generalized singular value decomposition''' ('''GSVD''') is the name of two different techniques based on the [[singular value decomposition]]. The two versions differ because one version decomposes two (or more) matrices (much like [[Principal component analysis#Higher order|higher order PCA]]) and the other version uses a set of constraints imposed on the left and right singular vectors.


==Higher order version==
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The '''generalized singular value decomposition''' ('''GSVD''') is a [[matrix decomposition]] more general than the [[singular value decomposition]]. It is used to study the [[condition number|conditioning]] and [[regularization (mathematics)|regularization]] of linear systems with respect to quadratic [[semi-norm]]s.
 
Let <math>\mathbb{F} = \mathbb{R}</math>, or <math>\mathbb{F} = \mathbb{C}</math>.
Given matrices <math>A \in \mathbb{F}^{m \times n}</math> and <math>B \in \mathbb{F}^{p \times n}</math>, their GSVD is given by
:<math>A=U\Sigma_1 [ X, 0] Q^*</math>
and
:<math>B=V\Sigma_2 [ X, 0] Q^*</math>
 
where <math>U \in \mathbb{F}^{m \times m},V \in \mathbb{F}^{p \times p}</math>, and <math>Q \in \mathbb{F}^{n \times n}</math> are [[unitary matrix|unitary matrices]], and <math>X \in \mathbb{F}^{r \times r} </math> is non-singular, where <math>r = rank([A^*,B^*])</math>. Also,
<math>\Sigma_1 \in \mathbb{F}^{m \times r}</math> is non-negative diagonal, and <math>\Sigma_2 \in \mathbb{F}^{p \times r}</math> is non-negative block-diagonal, with diagonal blocks; <math>\Sigma_2 </math> is not always diagonal. It holds that <math>\Sigma_1^T \Sigma_1 = \lceil\alpha_1^2, \dots, \alpha_r^2\rfloor</math> and <math>\Sigma_2^T \Sigma_2 = \lceil\beta_1^2, \dots, \beta_r^2\rfloor</math>, and that <math>\Sigma_1^T \Sigma_1 + \Sigma_2^T \Sigma_2 = I_r</math>. This implies <math> 0 \le \alpha_i,\beta_i\le 1</math>.
The ratios <math>\sigma_i=\alpha_i/\beta_i</math> are called the ''generalized singular values'' of <math>A</math> and <math>B</math>. If <math>B</math> is square and invertible, then the generalized singular values ''are'' the singular values, and <math>U</math> and <math>V</math> are the matrices of singular vectors, of the matrix <math> AB^{-1}</math>. Further, if <math>B = I</math>, then the GSVD reduces to the singular value decomposition, explaining the name.
 
==Weighted version==
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>Jolliffe I.T. [http://www.springer.com/west/home/new+%26+forthcoming+titles+%28default%29?SGWID=4-40356-22-2285433-0 Principal Component Analysis], Series: [http://www.springer.com/west/home/statistics/statistical+theory+and+methods?SGWID=4-10129-69-173621571-0 Springer Series in Statistics], 2nd ed., Springer, NY, 2002, XXIX, 487 p. 28 illus. ISBN 978-0-387-95442-4
</ref><ref>{{Cite book
| author = Greenacre, Michael
| publisher=Academic Press |location= London
| year = 1983
| title = Theory and Applications of Correspondence Analysis
| isbn = 0-12-299050-1
}}</ref><ref>{{Cite journal| author = Abdi. H., & Williams, L.J. |year = 2010 | title = Principal component analysis. | journal = Wiley Interdisciplinary Reviews: Computational Statistics, | volume = 2 | pages = 433–459}}</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 diagonal matrices of weights. This form of the ''GSVD'' is the core of certain techniques, such as [[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>Abdi, H. (2007). Singular Value Decomposition (SVD) and Generalized Singular Value Decomposition (GSVD). In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage. pp. 907-912.</ref>
 
== Applications ==
The GSVD has been successfully applied to signal processing and big data, e.g., in genomic signal processing.<ref>{{Cite journal
| author = O. Alter, P. O. Brown and D. Botstein
| title = Generalized Singular Value Decomposition for Comparative Analysis of Genome-Scale Expression Datasets of Two Different Organisms
| journal = PNAS
| volume = 100
| issue = 6
| pages = 3351-3356
| date = March 2003
| doi = 10.1073/pnas.0530258100
| url = http://dx.doi.org/10.1073/pnas.0530258100
}}</ref><ref>{{Cite journal
| author = C. H. Lee,* B. O. Alpert,* P. Sankaranarayanan and O. Alter
| 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
| doi =  10.1371/journal.pone.0030098
| url = http://dx.doi.org/10.1371/journal.pone.0030098
| id = [http://www.alterlab.org/research/highlights/pone.0030098_Highlight.pdf Highlight]
}}</ref>
These applications also inspired a higher-order GSVD (HO GSVD).<ref>{{Cite journal
| author = S. P. Ponnapalli, M. A. Saunders, C. F. Van Loan and O. Alter
| 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
| doi = 10.1371/journal.pone.0028072
  | url = http://dx.doi.org/10.1371/journal.pone.0028072
| id = [http://www.alterlab.org/research/highlights/pone.0028072_Highlight.pdf Highlight]
}}</ref>
 
==See also==
 
* C. C. Paige, and M. A. Saunders: Towards a Generalized Singular Value Decomposition, SIAM J. Numer. Anal., Volume 18, Number 3, June 1981.
* [[Gene Golub]], and [[Charles Van Loan]], Matrix Computations, Third Edition, Johns Hopkins University Press, Baltimore, 1996, ISBN 0-8018-5414-8
* Hansen, Per Christian, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Monographs on Mathematical Modeling and Computation 4. ISBN 0-89871-403-6
* [[LAPACK]] manual [http://www.netlib.org/lapack/lug/node36.html]
 
==References==
{{Reflist}}
 
[[Category:Linear algebra]]
[[Category:Singular value decomposition]]

Latest revision as of 14:26, 29 October 2014

Yeast Infection No More Review Summary

The system 'Yeast Infections No More' shows you how to take out male, genital and mouth candidiasis and never having to use prescription drugs for your condition. Infection No Longer can easily help you to track down the reason for your consistent recurring candida albicans and get rid of it for better should you have been obtaining trouble using a chronic candida albicans and you just desire a get rid of that actually works. This is a product or service which has been put into practice and used with positive results to lower candida symptoms in a long time useful, helping to make this reserve by Linda Allen among the most very effective available in the market today.

But Does Yeast Infection No More Really Work?

Sure, Yeast Infection No More does job. Men and women from around the globe used the content within this program to reduce lasting, persistent yeast infections effectively, and so can you. With tips on reducing intestinal issues, alleviating migraines, and handling skin disorders due to candida albicans, you will see this in depth guide book offers more than you expect. Yeast infection is amongst the most commonly encountered of bacteria that affect the healthiness of both men and women, but there is no need to merely tolerate the problem,. That's the truth.

There are several issues that yeast infections could potentially cause, including depression and frame of mind swings, migraines and cystic acne respiratory system microbe infections, menstrual pain and discomfort, and perhaps muscle tissue discomforts and discomfort. This is a tiny trial on the problems that any infections can cause. Yeast Infection No More can reduce the Candida causing your soreness, and relieve the many indications of the problem in just fourteen days. You may start finding the impact of your program throughout a long time, even so.

What Have Some Other Clients Mentioned?

There are many those who suffer from recurring, recurrent candida albicans each and every year, and most of these men and women have begun looking for alternative cures for their yeast infection. These fortunate enough folks have identified Yeast Infection No More, and also have still left their responses in regards to the system that you should examine via so you can provide an educated choice concerning the product. This is a tiny trial in the purchaser opinions offered, and this is what they need to say about Yeast Infection No More:

These customers during these testimonials have experienced respite from their warning signs of yeast infection inside hours of making use of the program. You may implement the Yeast Infection No More program in your own home, minus the expense and hassle of employing anti-biotics or non-prescription medications for your own ailment and determine success speedily. There are plenty of symptoms from creating a infection, but with Yeast Infection No More, you will not have to handle any of them any further.

Information

Any time you pick the Yeast Infection No More program, you will definately get:

- A 150 page normal infection remedies guidebook

- A number of top quality totally free additional bonuses

- Cost-free life-time revisions in the program

- No cost Just one-On-1 Advice With Linda Allen For 90 days

- A choice in between the Basic Release or Deluxe Release

- An 8 full week completely money-back guarantee

Additionally there is a Yeast Infection No More Luxurious Version for $57 which include every little thing provided during the Fundamental model as well as the new Complete Candida Candida Cookbook containing in excess of 200 tested recipes with anti--yeast infection regular food list and dinner plans.

The purchase price for Yeast Infection No More is actually $39.97 as well as Deluxe Model is $57. Also, the materials are generally in electronic digital PDF style, in order to download and install them immediately and both read through them on your computer or print them out.

Can You Advocate Getting Yeast Infection No More?

Candida Forget About can provide you the help you require in seeking out reduction and removal of continual, continuing infections, should you be experiencing difficulty with yeast yeast infection symptoms in men infections. You may decide to speak to your physician to make certain the difficulties you might be obtaining, no matter if genital, oral or complexion established, have been in connection with Candida albicans. Other conditions do exist that directly present the exact same signs or symptoms as a infections, however are essentially due to unique bacterias or fungi. Once you are certainly and have determined that the issue is yeast, you may simply and efficiently use Yeast Infection No More to stop your problems for decent.

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