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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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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17 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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13 views

Implementing PCA using Incremental approach [migrated]

I am trying to implement the algorithm proposed in the paper in Section (III) here in R. It uses incremental eigendecomposition and incremental SVD for calculating IPCA. Instead of working on images ...
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26 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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30 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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99 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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22 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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13 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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25 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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41 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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30 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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56 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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86 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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20 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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89 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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277 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
40 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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360 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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227 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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864 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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9 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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51 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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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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80 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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68 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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80 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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18 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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54 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 ...
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132 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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156 views

Incremental SVD Recommendation System

I have the following Octave/Matlab code to compute an SVD-like matrix-decomposition: ...
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1answer
28 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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559 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
2k 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 ...
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37 views

The bias of users and items in SVD++

I'm reading Yehuda Koren's paper: "Factorization Meets the Neighborhood: a Multifaceted Collaborative Filtering Model" SIGKDD 2008. I notice that in the traditional neighborhood method, say the ...
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35 views

Choosing between different methods when the first one raises error message for linear regression

I have a linear regression problem $$Ax=b$$ My initial approach that helped to solve some of my questions was using SVD and obtaining the chi-square and some other values that I am interested but it ...
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55 views

Estimating a vector from a rank-one symmetric matrix plus scaled identity

I have a problem regarding estimating a $M\times 1$ vector from a given $M\times M$ symmetric matrix. The known matrix is a scaled identity matrix with a rank-one update. I have some idea how to ...
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1answer
121 views

PCA vs SVD - understanding difference and preference of SVD over PCA [duplicate]

I understand that PCA and SVD are similar - PCA removes the mean and SVD doesn't? I think I have an understanding of PCA - you would use it to reduce dimensions of data and separate it out into ...
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70 views

should one always perform svd before doing KNN?

I am trying to perform a Collaborative filtering for recommendation of products to customers in fashion industry.I am using the usual KNN approach to bring similarities among products. I have seen ...
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1answer
79 views

Updating SVD in Recommender Systems for change in ratings

I have read that there are projection based methods to accomodate for new user's ratings or for the ratings for a new item in SVD. However, I want to know how to update my feature space for change in ...
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236 views

How to compare the outputs of two algorithms computing SVD?

For example we have 2 algorithms from R: SVD and irlba and I want to compare them int terms of speed,memory and precision. But I don't understand how to compare output of algorithms, they must be ...
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154 views

Understanding Singular Value Decomposition in the context of LSI

My question is generally on Singular Value Decomposition (SVD), and particularly on Latent Semantic Indexing (LSI). Say, I have $ A_{word \times document} $ that contains frequencies of 5 words for ...
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42 views

After applying SVD, how do I find which features from my original dataset were most significant?

I'm using MathNet.Numerics library in C# to find the SVD but the Sigma matrix gives no indication of which values correspond to which features. It simply lists them in the most significant order. ...
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27 views

What is the range of values that can be expected in the result of Principal Component Analysis (PCA)?

I want to normalize all of my preprocessing techniques between 0 and 1 so I want to know what the PCA range of values is so that I can apply a proper normalization to it. I applied PCA by using the ...
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2answers
264 views

How to use SVD for dimensionality reduction to reduce the number of columns (features) of the data matrix? [duplicate]

My original data has many more columns (features) than rows (users). I am trying to reduce the features of my SVD (I need all of the rows). I found one method of doing so in a book called "Machine ...
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1answer
62 views

Are SVD (Singular Value Decomposition) values always positive? Is there a relation between the maximum SVD value and the original data?

Assuming it's the standard SVD (no variation of it) with $A = USV^T$, would the $A$ matrix always have positive values (0 to $\infty$)? I noticed that the $U$ and $V^T$ matrices had some negative ...
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197 views

Different results for Singular Value Decomposition (SVD) using different tools

I am currently implementing Latent Semantic Analysis in Java using the EJML library for the preliminary Singular Value Decomposition (SVD). I am testing my code against the original term frequency ...
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24 views

Can LSA find correlations between multiple words?

I need to find correlations between multiple terms (say, 3 or 4) in a single-term search index. I'm trying to figure out if LSA fits to the problem. Am I right that LSA is no more than a term-to-term ...
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62 views

PCA eigenvalues meaning

When projecting the data set on the Eigen vectors of the co-variance matrix , the eigenvalues represent how much each example varies away from the mean of the data set in the projected direction , ...
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1answer
59 views

Question about PCA data recovery equation

In PCA , consider a 4 x 3 data matrix ( 4 examples each with 3 features ). After getting the 3 eigenvectors (a/b/c) and projecting data on the first 2 vectors, the equation looks like this : [ first ...