# 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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### How can I construct a desired variance-covariance matrix for simulating multivariate Gaussian distribution samples using MATLAB?

I want to simulate multivariate normal distribution samples to help understand PCA, biplot, etc. For example, I want to see how the correlation structure affects the appearance of 2-D biplot. Two ...
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### Compressed sensing and matrix recovery

I'm reading a paper estimation of (near) low-rank matrices with noise and high-dimensional scaling by Negahban and Wainwright where authors defied compressed sensing as a particular instance of the ...
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### SVD on demeaned matrix

I'm trying to understand the effect of de-meaning with SVD. Suppose I have matrix $WW^T = \sum_{i=1}^n w_iw_i^T$ where $W$ is $n \times m$ and $w_i$ are its columns. Running SVD on this yields ...
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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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### can we use a hybrid optimization schem for NMF

The NMF problem of the form $$X \simeq WH$$ is a constrained biconvex optimization problem, and is often solved by alternating updates schemes. For example, the multiplicative update rules use ...
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### variance explained for CP factorisation

What is the best way to compute the variance explained for one or more terms in a CP tensor factorisation? Is it even defined? In PCA this is made easy by the fact that the eigenvalues are orthogonal. ...
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### Recommending item category based on customer shopping behavior?

I'm trying to build some recommender system for online wine shop. What I'm thinking is recommending item category (ex: red_pinot, ...
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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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### Use Matrix Factorization to predict probability of a recommendation system?

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,...
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### Is there a version of NMF that normalizes the sum of scores of each sample?

I want to decompose a nonnegative data matrix $A \in \mathbb{R}^{n\times m}$ into nonnegative basis vectors $U \in \mathbb{R}^{n \times k}$ and a score matrix $V \in \mathbb{R}^{m \times k}$ such that ...
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### ML model for Signal Decomposition [closed]

So recently I got a task which can be summarized as follows: Suppose we have 3 functions f1, f2, f3 and a certain combination of the functions gives us ...
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### Decomposition of $x^T K x$ as $y^T y$

When $x$ is a vector of size Nx1, and $K$ is a very large symmetric sparse matrix of size NxN (say N=100K), is it possible to decompose $x^T K x$ as $y^T y$? As if I could get $y = K^{1/2} x$. Edit ...
I read here (Google's crash course on recommendations) the following: Given a new item $i_0$ not seen in training, if the system has a few interactions with users, then the system can easily compute ...