# Questions tagged [matrix-decomposition]

Matrix decomposition refers to the process of factorizing a matrix into a product of smaller matrices. By decomposing a large matrix, one can efficiently perform many matrix algorithms.

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### Eckart–Young–Mirsky theorem for $n \gg m$

It has been proven that the best reconstruction error in the $k$ rank matrix estimation problem in terms of Frobenius or $L2$ norm is given by the $k$-truncated SVD as shown here. I've read in ...
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### Why PCA is invariant under rotation

Lets say that we have a matrix of variables (the columns are variables and rows are the observations) called X whenre X = [x1, x2, ...., xp] where ...
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### Generate dataset that adhere to a given correlation matrix [duplicate]

I'm newbie in statistics and this question might be naive so kindly excuse me in advance. I have a correlation matrix of 40 features and I want to generate a dataset of hundreds of observations that ...
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### Decomposition analysis for data between zero and one

I want to analyze latent components of data that has values between zero and one (including zero and one). In detail, the data structure is n x m and I'm looking to find the r underlying components. ...
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### How to find a decomposition of multivariate X along which y varies the most?

I'm looking for an existing algorithm which carries out the task shown in the title. My use-case in other words: I have a set of continuous independent variables (X) and a continuous dependent ...
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### What is the correlation between random variables after being multiplied by the same lower triangle matrix decomposed from a covariance matrix?

Assume $C_{n \times n}$ is a positive, symmetric and semi-definite covariance matrix, we know that the LU decomposition exists, i.e., $C_{n \times n}=L_{n \times n}U_{n \times n}$. Now $n$ ...
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### Constrained Matrix Decomposition

I am working on a structural vector autoregression that requires imposing constraints on a matrix factorization. In particular, I have an N-dimensional positive definite matrix $\Sigma$ that I need to ...
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### Why does the ridge penalty shrink the singular values? [duplicate]

I am trying to understand the following analysis of ridge regression. I am new to SVD but I think I have a sufficient grasp on most of the content. There are two things I am struggling with. The ...
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### How to decompose a random walk (array) into its Markov Chain transition matrix?

The algorithm, PageRank, receives a Markov Chain transition matrix (page links from one to another.) Either by random walk, or more efficiently, eigenvectors, the stationary distribution of the Markov ...
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### Is the first independent component of independent component analysis always important?

I was looking at a neuroscience paper that used ICA to reduce dimensionality of calcium signaling profiles in 20 randomly selected neurons of a zebrafish brain. I presume that in Figure 2, ICA was ...
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### Why does Non-Negative Matrix Factorization reconstructs exactly the same matrix?

I'm trying recently to get into recommender systems and almost all tutorials I find mention collaborative filtering done with matrix factorization. I found this tutorial that describes how to build ...
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### Relation between low-rank approximation, nuclear norm of a matrix and Singular Value Decomposition

I'm reading the following paper https://arxiv.org/pdf/2005.10203.pdf which proposes improvements on robustness of large graphs to defend against adversarial attacks that are nothing but slight ...
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### Latent factors are the same in both decomposed matrices?

This question is in the context of recommendation systems. We can use matrix factorization techniques to decompose a user-product explicit/implicit matrix(R) into two matrices(U, P). Let's say R is a ...
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### Why absolute value of eigenvalues are used in PCA or LDA?

In PCA and LDA techniques, eigenvectors with the $k$ largest eigenvalues give principal components. However, when selecting these eigenvalues, are they to be sorted by the absolute value (regardless ...
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### Drawing samples from matrix normal

I have to generate $n \times m$ sample ($A$) from a matrix normal distribution, given two covariance matrices: $n \times n$ row covariance matrix (matrix $B$) (defines the covariance between the rows ...
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### Are principal component analysis (PCA) and empirical orthogonal function (EOF) methods the same?

As far as I've seen, EOF is just PCA but instead of thinking about the data matrix X as (number of samples, number of features), you consider it as (number of time points, number of different spatial ...
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### Matrix Factorization and Linear Regression

Which matrix factorization algorithm is used in LinearRegression() function of scikit-learn?
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### Fastest way to find Leading singular value and vector (power iteration, rsvd etc)

I want to know the fastest way to find out the leading singular value and vector of a large rectangular matrix. I have seen 2 suggestions and have questions on both of them : Power Method : For this ...
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### How to explain the numerical discrepancy between FactoMineR::PCA() and the svd() in their output of the U matrix?

I am comparing the output of two functions in R to do Principal Component Analysis (PCA), the FactoMineR::PCA() and the ...
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### Human intuition behind SVD in case of recommendation system

This does not answer my question. I struggled very hard to understand the SVD from a linear-algebra point of view. But in some cases I failed to connect the dots. So, I started to see all the ...
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### SVD : Why right singular matrix is written as transpose

The SVD is always written as, A = U Σ V_Transpose The question is, Why is the right singular matrix written as V_Transpose? I mean lets say, W = V_Transpose and then write SVD as A = U Σ W SVD Image ...
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### Penalized Canonical Correlation in R with PMA Module

I am trying to use sparse canonical correlation analysis as implemented in the R PMA package. I'm finding that the correlations output by the package seem slightly inconsistent with the ones you would ...
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### Symbolic Singular Value Decomposition? U,S,V as function of the elements of M [closed]

Suppose we want to compute the SVD of $\mathbf{M} = \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix}$ (...
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### Interpreting PCA results of first two components

I don't like the looks of my PCA graph here. PCA coordinates should be uncorrelated, yet the variance between the coordinates of the second component increases as the first component decreases. What's ...
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I have a dataset where I have a sparse utility matrix (user-product) with binary input: 1 if the user $i$ bought the product $j$, and 0 if it hasn't. However it has a different meaning on the test set,...