# 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 to compute the left singular eigenvector matrix (U) from the output of prcomp() for PCA in R?

I am examining the output of the prcomp() function in R for PCA in light of the singular value decomposition equation: $X = U \cdot \Sigma \cdot V^{T}$, where: $X$: ...
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### What kind of a matrix factorization is appropriate for my problem?

I have the following problem: I measure mass concentrations of a pollutant 'p' in n stations (same pollutant in all stations). And I have many observations. Theoretically, I believe that if I assume ...
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### Kernel matrix decomposition

I had a look at the sklearn.kernel_approxiamtion.Nystroem implementation, which is also described in this post: Nystroem Method for Kernel Approximation Here, a ...
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### How to predict for test set when training a recommender by decomposing the utility matrix X=UV?

This probably sounds stupid but I don't get the workflow of building a recommending system by the utility matrix: X[i,j] = how much the ith user likes the jth object. For practical issues I refer to ...
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### SVD for a complex data matrix — what is the meaning of the columns of $V$?

I've read this wonderful explanation of SVD, where the writer mentions that the columns of $V$ are the principal directions (Summary, #1). Is this also true when the data matrix $X$ is complex? If I'm ...
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### Convert the following expression w.r.t to the whole dataset instead of element of the dataset?

I am in the process of expressing the w in LSSVM with data points and constants. After I resolve the KKT conditions for the LSSVM I got $$w = \sum ^N _{i=1} \alpha_i x_i$$ Is it possible to convert ...
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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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### What is gravity in the context of “folding” in recommender systems?

What is "gravity" in the context of recommender systems? More specifically, how is it supposed to help with the "folding" problem where irrelevant queries may be returned if we don't provide ...
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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 ...
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### Cold starts in factorization - WALS projections

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 ...
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### SGD for matrix factorization: Negative sampling and gravity

I read here the following downside of using SGD for Matrix factorization: Harder to handle the unobserved entries (need to use negative sampling or gravity). What do they mean by negative sampling ...
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### Does SVD provide the best low rank approximation for any matrix regardless of shape?

Wikipedia states (link below) that by the Eckart-Young-Mirsky theorem, the SVD provides the best low rank matrix approximation (on the basis of Frobenius norm of the error matrix) for any matrix A in ...
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### How does multicollinearity affect the eigenvalues of a matrix?

I have been looking into ridge regression as a method to address multicollinearity in data. I am aware that multicollinearity can cause high variance in coefficient estimates. I have seen equations ...
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### What is finite precision arithmetic and how does it affect SVD when computed by computers?

Was reading the paper "DETECTING AND ASSESSING THE PROBLEMS CAUSED BY MULTICOLLINEARITY:A USE OF THE SINGULAR-VALUE DECOMPOSITION" by David Belsley and Virginia Klema. After performing SVD, while ...
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### Collaborative filtering movie recommender: how to account for missing ratings implying information about user preference?

I'm trying to learn about recommender systems with a fairly standard data set: I have a matrix with thousands of users, thousands of movies, and the ratings that users give to each movie. Obviously, ...
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### Infinitesimal generator of Markov chain using numpy

I am computing the infinitesimal generator of a continuous-time Markov chain from the transition probabilities p. I am following the methodology described here, ...
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### How can this L(2,1) problem be reduced to the orthogonal procrustes problem?

NOTE: Don't take this too serious -- the question is actually due to my misreading $\|y_i - Wx_i\|^2$ as $\|y_i - Wx_i\|_2$, see the answer. Smith et al. in Offline bilingual word vectors, orthogonal ...
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### Relation between eigenvalues of original and transformed matrices

Let the matrix $X$ be some data arranged in rows. Consider the following eigenvalue decomposition $X^\top X = Q \Theta Q^\top=\sum_{i=1}^n \theta_iq_iq_i^\top$ where $q_i$ are the eigenvectors and \$\...