SVD refers to the singular value decomposition of a matrix. Given a matrix $\mathbf{A}$, the SVD is given by $\mathbf{A} = \mathbf{USV}^\top$ where $\mathbf{U}$ and $\mathbf{V}$ are orthogonal matrices and $\mathbf{S}$ is a diagonal matrix.

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Meaning of SVD plot of $U$ and $V^T$

I am using SVD/PCA for text mining purposes. Having a $(|terms|,|documents|)$ normalized matrix $M$, by applying SVD, I should be able to reduce the dimensionality and just keep the most meaningful ...
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6 views

Suggestions on passing objects in r to an external executable [migrated]

I would like to do SVD on big matrix (90,000 x 60,000), but using svd is too slow for many iterations (as in permutation). fast.svd and irlba render similar runtime. Redsvd (C++ executable) is ...
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41 views

SVD from Matrix formulation to objective function

I'm writing the question to try to complete the circle after reading the 2 other questions on Cross Validated and the link on the third bullet point: What is the objective function of PCA? What ...
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18 views

Sequential Singular Value Decomposition

Is there any way (or any instructive paper) to implement singular value decomposition sequentially? i.e. compute $U_{n}$ using $U_{n-1}$ for a matrix $A_n$ such that $A_n = U_n\Sigma_nV_n^T$?
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45 views

Definition of orthogonal matrix

I am reading the book Elements of Statistical Learning and trying to understand singular value decomposition (SVD). In particular, what is an orthogonal matrix as it relates to SVD? According to ...
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15 views

R - SVD and PC full example

I would like to understand if I'm using the packages correctly: A complete example would be: I create a random matrix ...
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37 views

How to go from sparse matrix to linear regression model (using SVD)?

I am trying to replicate the Kosinski, Stillwell, & Graepel (2013) study about predicting private traits and attributes from Facebook like data for study purposes. First I have admit, however, ...
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27 views

Low rank approximation of binary valued matrix

How does one get the low rank approximation of binary matrix? Is the low rank approximation also a binary matrix? Note - Here binary matrix just means that any entry of the matrix can either be 0 or ...
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81 views

What fast algorithms exist for computing truncated SVD?

Possibly off topic here, but there exist several (one, two) related questions already. Poking around in the literature (or a google search for Truncated SVD Algorithms) turns up a lot of papers that ...
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22 views

Can PCA explained variance be computed from the components (or from SVD matrices)?

I'm looking to use Spark to calculate PCAs. However I need to get the explained variance for each component and the PCAModel class doesn't appear to provide that. Is there a way to calculated the ...
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21 views

Too small baseline predictors

I'm trying to implement a recommender system, based on SVD-algorithm. I have a matrix with binary rates, i.e. 0 and 1. This matrix is very sparse. I'm using a formula for learning process: ...
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1answer
277 views

SVD interpretation of Climate data

I was going through my homework. I have a question to interpret the plot which i obtain from first and second left singular vectors after computing SVD. What can i exactly interpret, is it specific to ...
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1answer
25 views

Projecting a new entry onto the two largest eigenvectors of a PCA model

I have a bit of problems understanding how PCA and SVD works, as most materials focus on calculating the factors rather than the classification of new entries. In order to provide some context of my ...
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38 views

Using SVD or PCA for reducing dimensionality [duplicate]

I have always heard that I can reduce dimensionality of a matrix using SVD. So, I'd like to ask something hypothetically. Suppose that the following matrix A has a high dimensionality and I want to ...
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57 views

Sparse features and dimension reduction

Let sparse feature be a feature which values are subsets of some set. For example, the set of countries from which user logged to server is a sparse feature, because for each user we've got the set of ...
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2answers
54 views

Singular value and eigen-decomposition of a square symmetric matrix should be identical, but differ in sign

As far as I know, singular value decomposition (SVD) and eigendecomposition give the same result for symmetric square matrices. But when I check the results in R, that's not what I see. Please see ...
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1answer
59 views

PCA on a rank-deficient matrix using SVD of the covariance matrix

I have a high-dimensional data matrix X where sample size is smaller than the variable size. I want to use PCA as a dimensionality reduction method, but I cannot ...
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1answer
83 views

Latent Semantic Indexing and Data Centering

In PCA it's common to center the data, i.e. preprocess the data matrix such that the columns have zero mean. PCA can be done via SVD, but in this case the data matrix also has to be mean-centered. If ...
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20 views

SVD of a large sparse matrix in R [duplicate]

I have a huge matrix with roughly around half a billion entries. However, the matrix is hugely sparse. If I store it as a matrix in R, it will be around 6 gb and if I store it in spare matrix format, ...
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88 views

Which SVD matrix do we use for cosine similarity

In Latent Semantic Analysis, we get 3 matrices from the singular value decomposition (SVD), but I am confused - which matrix do we use for cosine similarity?
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53 views

What is the relationship between SVD and UV decomposition?

I am aware that both are Matrix decomposition techniques. SVD decomposes the Matrix as $\mathbf C = \mathbf U_1 \mathbf L \mathbf V_1^\top$. UV decomposes the Matrix as $\mathbf C = \mathbf ...
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122 views

Treating missing data in voting pattern analysis

I'm trying to analyze voting patterns of Ukraine's parliament deputies. I scraped all the data on their voting during last session. Each data entry has following information: Deputy name, date, bill ...
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47 views

Single value decomposition applied to Lee Carter in R

I have a problem regarding the svd in R. It is applied to matrix m, 101 rows by 69 columns, which is the demeaned log mortality rates. When I apply svd(m) (I left nv and nu out), I get d which is a ...
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40 views

Getting Rotation Matrix from SVD

I'd like to use SVD to get the rotation of ellipsoid data. When I create ellipsoid data with a known rotation, I can see that there is some kind of relationship between the $V'$ and the original ...
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45 views

how to solve weighted low rank approximation problem with diagonal weight matrix

Without weights the low rank approximation problem can be solved in terms of svd of the original matrix.But is there any way to solve the problem following problem $$\min_C{\sum_{i=1}^n \sum_{j=1}^d ...
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60 views

Fastest algorithm for rank-one approximation of a matrix [closed]

I need to compute the best (in the sense of sum of squares) rank-one approximation to a matrix, i.e. a one-component SVD, $$ \mathbf{X} \approx \sigma\mathbf{u}\mathbf{v}^T $$ such that $$ ...
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40 views

minimization of weighted frobenius norm for pca

So my problem is i like to derive pca solution as the maximum likelihood estimate for the true data.So basically i am assuming that my measured data has two component one is low rank component and ...
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1answer
78 views

Singular Value decomposition positive components

I am using Singular Value Decomposition (SVD) applied to Singular Spectrum Analysis (SSA) of a timeseries. ...
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89 views

Rotation in PCA and Factor Analysis [duplicate]

I want to know what elements are (varimax-)rotated when I rotate after PCA and after Factor Analysis. Let’s assume a standardized data vector $X$ of dimension $N \times q$. In PCA, I have the ...
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23 views

PCA using prcomp and princomp in R [duplicate]

I have been trying to do Principal Component Analysis (PCA) via R. The data set is available at https://www.dropbox.com/s/s3jstl8pu1e1xcp/Cars.csv?dl=0 I tried to do PCA via 2 different methods - ...
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2answers
98 views

Is dimensionality reduction almost always useful for classification?

Is singular value decomposition almost always useful in practice for enhancing the predicative power of a trained classification model? E.x. A dataset for classification has 20,000 features. Run SVD ...
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1answer
611 views

PCA and Correspondence analysis in their relation to Biplot

Biplot is often used to display results of principal component analysis (and of related techniques). It is a dual or overlay scatterplot showing component loadings and component scores simultaneously. ...
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1answer
60 views

Clustering singular vectors [closed]

I have a $n\times m$ matrix $A$ that I have done SVD on and picked off $k$ out of $n$ highest singular values and vectors. That is, I have decomposition for the $k$ rank approximation of $A$: $A^{k} ...
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655 views

Positioning the arrows on a PCA biplot

I am looking to implement a biplot for principal component analysis (PCA) in JavaScript. My question is, how do I determine the coordinates of the arrows from the $U,V,D$ output of the singular vector ...
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257 views

Does a correlation matrix of two variables always have the same eigenvectors?

I wanted to conduct a total least squares regression on two variables. My statistical programme does not provide TLS, but TLS luckily equals Principal Component Analysis, as far as I know. Since all ...
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3answers
906 views

Weird correlations in the SVD results of random data; do they have a mathematical explanation or is it a LAPACK bug?

I observe a very weird behaviour in the SVD outcome of random data, which I can reproduce in both Matlab and R. It looks like some numerical issue in the LAPACK library; is it? I draw $n=1000$ ...
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14 views

Interpolating singular values

So I have the singular values associated with a data matrix and I would like to interpolate them and then find the maximum curvature of the interpolation in order to decide how many singular values to ...
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1answer
67 views

Matrix Factorization in Recommender Systems: Multiple solutions?

I have implemented a recommender system for predicting user ratings based on the matrix factorization approach. $$ r_{ui}=μ+b_u+b_i+q_i^T p_u $$ Where q and p are found by mimization of the squared ...
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1answer
3k views

Relationship between SVD and PCA. How to use SVD to perform PCA?

Principal component analysis (PCA) is usually explained via an eigen-decomposition of the covariance matrix. However, it can also be performed via singular value decomposition (SVD) of the data matrix ...
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2answers
116 views

Maximally reducing the rank of a matrix by removing some rows or columns

I have a $N \times M$ matrix, and the rank of matrix, $r$, is near $\min(M,N)$. I want to minimize the rank by removing some of the rows or columns to get $r \ll \min(M,N)$. The goal is to achieve ...
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109 views

How to verify implementation of SVD in Javascript

I have implemented the SVD algortihm for my Node.js project for collaborative filtering of a sparse dataset based on this paper by GroupLens. For calculating the SVD, I am using the package node-svd ...
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127 views

Fisher information matrix with negative eigenvalues, fix through singular value decomposition?

I have the following Fisher information matrix (just an example): ...
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63 views

What does it mean to compute eigenvectors of a covariance matrix if the data were not centered first? [duplicate]

Say $\mathbf{X} \in \mathbb{R}^{n \times p}$ and $\boldsymbol{\Sigma} = \frac{1}{n}\mathbf{X}'\mathbf{X}$. The eigenvector decomposition of $\boldsymbol{\Sigma}$ gives $\boldsymbol{\Sigma} = ...
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38 views

Singular Value Decomposition (data reduction) on non-numerical data

I have a large amount of data where each datapoint contains string valued attributes, for example: ...
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82 views

Using SVD on features before SVM classification, when p >> N

So I am going through Hastie's Elements of Statistical computing, and in section 18.3.5 which deals with computational shortcuts when the number of dimensions $p$ is much larger that the number of ...
4
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1answer
162 views

Why are the singular values of a standardized data matrix not equal to the eigenvalues of its correlation matrix? [duplicate]

Conceptually, aren't the eigenvalues of a correlation matrix and the singular values of the associated scaled data matrix supposed to be the same? The below illustration is saying that it isn't so. ...
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196 views

Incremental SVD Recommendation System

I have the following Octave/Matlab code to compute an SVD-like matrix-decomposition: ...
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34 views

svd adds value before an elastic net model?

I learned that SVD eliminates redundancies. If you use an elastic net model, is it still greedy as stepwise models in general? or the fact that it has penalization factors reduces the greedy ...
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745 views

Is there any advantage of SVD over PCA?

I know how to calculate PCA and SVD mathematically, and I know that both can be applied to Linear Least Squares regression. The main advantage of SVD mathematically seems to be that it can be applied ...
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1answer
3k views

K-means on cosine similarities vs. Euclidean distance (LSA)

I am using latent semantic analysis to represent a corpus of documents in lower dimensional space. I want to cluster these documents into two groups using k-means. Several years ago, I did this using ...