Singular value decomposition (SVD) of a matrix $\mathbf{A}$ 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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Implementation of PCA using SVD without creating covariance matrix

So I'm currently taking a Machine Learning course and have correctly submitted my implementation of PCA. I used SVD. Here it is Octave. ...
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Problem with syntetic data generating for Probabilistic PCA and Factor Analysis (FA) comparison - methodology

I am trying to understand a short example related to dimension reduction from python scikit-learn.org official documentation for long time and unfortunately I am not successful. I don't have problems ...
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86 views

The proof of shrinking coefficients using ridge regression through “spectral decomposition”

I have understood how ridge regression shrinks coefficients towards zero geometrically. Moreover I know how to prove that in the special "Orthonormal Case," but I am confused how that works in the ...
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14 views

Creation of Labels for Label Spreading in skicit-learn

I am using Label Propagation in skicit-learn for finding labels for the unknown ones My Input is 'data_list' containing 400 000 sentences in sanskrit language like : ['tatra yad tad mahABAga SaMkara ...
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14 views

SVD - collaborative based filtering - Prediction matrix

On the movielens dataset, I used SVD to find U, s, and V matrices. Then performed the dimensional reduction by elimination of everything corresponding to lower valued eigen values( upto a threshhold). ...
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39 views

How to perform SVD to impute missing values, a concrete example

This might be a very stupid question, but I have read the great comments regarding how to deal with missing values before applying svd, but I would like to know how it is going to work, if I apply it ...
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1answer
38 views

Is the matrix coeff from MATLAB's pca the same as the left singular vectors of the centered data?

Consider the SVD of a centered data matrix: $$ X_{centered} = U \Sigma V^T$$ where a column of $X_{centered}$ is: $$ X_{centered} = x^{(i)} - \frac{1}{N} \sum^N_{n=1} x^{(n)} $$ is the matrix $ U ...
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1answer
80 views

Why does the reconstruction error of truncated SVD equal the sum of squared singular values?

I saw this formula in a textbook: squared Frobenius norm of the original matrix $\mathbf X$ minus its truncated SVD $\mathbf X_k$ (which can be seen as the approximation error) equals the sum of ...
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24 views

Principal Component Analysis PCA Terms and relationships: eigenvalues, eigenvectors, loadings, score matrix, and SVD [duplicate]

I've read many websites, blogs, pdfs on this top but struggle to put the picture together in simple math terms, that explains how some of the terms relate to each other / are computed. Let's assume ...
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1answer
26 views

How do I choose the initial features vectors for a Stochastic Gradient Descent trained SVD++ algorithm?

I'm reading the SVD++ Netflix Recommender Systems paper because I want to be able to properly assess this approach to building a recommender system. How should I choose the initial values of $q_i$ ...
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1answer
44 views

Rotating a new matrix into the same basis as another using SVD

I have collected some data and stored them in an $N\times P$ matrix $A$. Using SVD, we can rotate $A=UDV^T$ into a new basis, also discarding some dimensions: $A\approx U\tilde{D}V^T$, where $\tilde{D}...
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29 views

How you can you take the `min` of what looks like a single value calculation in SVD++?

I'm reading through this paper: http://www.cs.rochester.edu/twiki/pub/Main/HarpSeminar/Factorization_Meets_the_Neighborhood-_a_Multifaceted_Collaborative_Filtering_Model.pdf And I'm looking at the ...
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1answer
37 views

Can singular value decomposition be applied to a matrix of $n\times 1$ size?

Can singular value decomposition be applied to a matrix of $n \times 1$ size (a vector)? Usually I see that matrix is of size of $n \times m$. Any example?
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31 views

Unsupervised Aspect based opinion mining [closed]

I have been working on a project to create a model for Unsupervised Aspect based opinion mining on online reviews ( broadly domain independent). i want to automatically extract features of the product,...
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42 views

Is there a supervised/semi supervised version of pca for dimensionality reduction?

PCA can give me the proper result if "Large variances have important dynamics" holds true for the data. In other words if I want to know along which components the variance of my data is maximized, ...
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42 views

Orthogonal matching pursuit - am I using it wrong?

I am trying out this method as a regularized regression, as an alternative to lasso and elastic net. I have 40k data points and 40 features. Lasso selects 5 features, and orthogonal matching pursuit ...
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73 views

Is there any situation where PCA performs better than SVD?

It's for a text clustering application. There are around 25 documents and 50k features (from TF-IDF), so I was expecting SVD to be a better choice. I am using sklearn's PCA and TruncatedSVD functions ...
2
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1answer
56 views

Do the principal components change if we apply PCA more than once (recursively) on data?

Consider a set $X=(X_1; \dots; X_n)$ of $n$ data points such that $X_i \in \mathbb{R}^d$ is a column vector. Let $Y = \text{pca_proj}(X)$ denote the projection of points in $X$ according to the PCA ...
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44 views

Can SVD be used to perform factor analyis?

What is the relationship between SVD and factor analysis? How can use singular values and other matrices from SVD to perform factor analysis or cluster document-term matrix without using other ...
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1answer
27 views

Clustering based on SVD

I have a document-term matrix and I performed SVD on it. How can I cluster terms based on the singular values? Is there any relationship between SVD and factor analysis?
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80 views

A technique like truncated SVD that uses non-orthogonal components?

Question: Is there a technique like truncated SVD that instead of orthogonal components relies on non-orthogonal components? Specifically, just like a truncated SVD computes simultaneously the ...
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10 views

How can I add one continue variable without computing its interaction using factorization machines?

I have an input data which consist of 1) User, 2) Item, and 3) a continuous variable. It is a regression problem. So the input can be as below: ...
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choose singular vectors in co-clustering

I have a m*n Term-Document matrix A,Bipartite Graph matrix $$ G= \begin{matrix} 0&A\\ A^T&0\\ \end{matrix} $$ DegreeMatrix $$ D= \begin{matrix} D1&0\\ 0&D2\\ \end{matrix} $$ in the ...
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Different order and signs of eigenvectors when doing PCA via eig() or svd() functions in Matlab

Assume we have a matrix X = randn(5,3). I am doing two things: 1) [S D1 V1] = svd(X); 2) ...
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Amount of data needed and hypothesis for SVD

I was looking into the definition of SVD and trying to understand which are the conditions needed to be met in order to be able to use it. Is there any hypothesis concerning the distribution of the ...
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41 views

Train & predict probabilities using LDA having multiple collinearities

I am trying to fit an LDA model and predict conditional probabilities of class membership with it. I believe I understand the basic method to do this using the covariance matrix and class means, but ...
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44 views

PCA/SVD with datetime fields

I have a dataset or flight data with several columns including delay time, and date of departure. There are several other parameters, and I would like to run some sort of PCA with SVD to see how ...
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1answer
188 views

How does centering make a difference in PCA (for SVD and eigen decomposition)?

What difference does centering (or de-meaning) your data make for PCA? I've heard that it makes the maths easier or that it prevents the first PC from being dominated by the variables' means, but I ...
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1answer
144 views

Standard library for Funk SVD or other gradient descent SVD/eigenvalue

I want to get the first few eigenvectors of real symmetric matrices with missing values. Since it has missing values, I won't be able to use the common linear programming techniques, but stochastic ...
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1answer
132 views

Moving window PCA vs local PCA vs kernel PCA vs rolling PCA

All all these terms mean the same thing? Are there other terms for MWPCA? Are there any decent online references to theory and applications? Which term is most popular (if they are equivalent)? I'm ...
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53 views

Confirming an understanding of SVD

I'm trying to sift through the concepts of low-rank approximations, matrix factorizations and SVD. There's a lot of info out there and the rabbit hole is deep so I just want to make sure my high level ...
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35 views

LSA projections of documents and terms

I am trying to understand how Latent Semantic Analysis works, reading demonstrations based on singular value decomposition. Let's denote $X$ a $D \times W$ document-term matrix. The $D$ rows of $X$ ...
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719 views

Theory behind partial least squares regression

Can anyone recommend a good exposition of the theory behind partial least squares regression (available online) for someone who understands SVD and PCA? I have looked at many sources online and have ...
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61 views

What are the pros and cons of applying pointwise mutual information on a word cooccurrence matrix before SVD?

One way to generate word embeddings is as follows: Get a corpora, e.g. "I enjoy flying. I like NLP. I like deep learning." Build the word cooccurrence matrix from it: Perform SVD on $X$, and ...
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801 views

What is the intuition behind SVD?

I have read about singular value decomposition (SVD). In almost all textbooks it is mentioned that it factorizes the matrix into three matrices with given specification. But what is the intuition ...
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624 views

Updating SVD decomposition after adding one new row to the matrix

Suppose that I have a dense matrix $ \textbf{A}$ of $m \times n$ size, with SVD decomposition $$\mathbf{A}=\mathbf{USV}^\top.$$ In R I can calculate the SVD as ...
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66 views

PCA with non-zero mean Gaussian error

Question Re-written: I am trying to apply PCA to a data matrix which has non-zero mean Gaussian error. My data has variables out of which only some are independent and rest are dependent (they are ...
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23 views

Interpretation of PARAFAC/TUCKER components for factor mode?

I am working on my thesis at the moment and I'm stuck... I would need to use a multimode-PCA method like Parafac/Tucker-N. My task is to find any deviating structure from the data which is in 3-mode. ...
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63 views

Propagation of uncertainty through a linear system of equations - rectangular matrix, pseudo-inverse

I refer you to this post as I have very similar problem: Propagation of uncertainty through a linear system of equations Can the same technique, proposed by Glen_b, be used to find uncertainty in ...
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12 views

Get the components from singular value decomposition (SVD) in PCA [duplicate]

I am performing a principal component analysis (PCA) in MATLAB. I have a data matrix $X$ ...
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44 views

SVD of gaussian noise

On page 3 of this paper, it is stated that for white gaussian noise signal with variance equal to sigma^2, the sum of the squared singular values is equal to the variance: Now, I suppose the author ...
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1answer
77 views

Singular value decomposition on RGB images

Is singular value decomposition (SVD) only done for grayscale images? All examples in the literature seem to focus on grayscale images. I was wondering if SVD also makes sense if applied to each of ...
3
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1answer
112 views

Low-rank SVD reconstruction and linear projection of the data

Short version: I wonder if the singular value decomposition (SVD) low-rank approximation $X = U_q\Sigma_q V^T_q$ is a projection onto $V_q$ or $V_q^T$? My understanding of the answer: The basic ...
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1answer
158 views

Percentage of variation in each column explained by each SVD mode

I performed singular value decomposition (SVD) on a data matrix. The mean of each column is zero. One of the scores one can measure is the percentage of the total variation that is explained by each ...
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18 views

How ksvd algorithm is considered generalized kmean?

I am trying to understand more details of this paper "KSVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation" proposed a new algorithm called Ksvd and claims it's a ...
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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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91 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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58 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 $A_n = U_n\Sigma_nV_n^T$?
3
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94 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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37 views

R - SVD and PC full example [closed]

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