# 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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### 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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### How to diagnose convergence in CF for low-rank matrix factorization trained with SGD?

I have a huge dataset of user-item ratings and would like to make predictions. To this end I'm using the low-rank matrix factorization with SGD for CF (collaborative filtering). I pre-compute the ...
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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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### Why modeling known entries improves Tensor completion problem?

In the following paper: "Scalable tensor factorizations with missing data" by Evrim Acar Authors mention on page 2 that unlike conventional "CP" method which models both known and unknown values, they ...
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### SVD versus RSVD

In the so-called incremental SVD used for collaborative filtering: http://www.machinelearning.org/proceedings/icml2007/papers/407.pdf http://www2.research.att.com/~volinsky/papers/ieeecomputer.pdf ...
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### Interpretation of Eigenvalue vs. Singular Value plot

I'm doing some preliminary analysis on the feature matrix for a certain dataset (rows are observations, columns are feature dimensions). I have computed the SVD and PCA decompositions for this matrix ...
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### Deriving Multiplicative Update Rules for Regularized NMF

After reading the following CrossValidated post, I cannot derived the correct multiplicative rules for regularized NMF from this paper. They obtain the coefficients $|I_u|$ and $|U_i|$ in the ...
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### Symmetric decomposition of a covariance matrix that is not given explicitly

I am implementing a Monte-Carlo method that requires decomposition of a $k \times k$ covariance matrix $\Sigma=A^TA$ where the dimensionality of $A$ should also be $k\times k$. No further constraints ...
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### QR decomposition used for estimating condition numbers

I have read that the QR decomposition is often used to estimate the condition number of a matrix but I don't understand... what is the benefit of using the QR decomposition for this? Is it purely ...
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### Factorizing a matrix of distributions [closed]

Let's say we have a matrix $X \in \mathbb{R}^{m \times n}$, then the (R-truncated) SVD allows to approximate: $X_{i,j} \approx \sum\limits_{r=1}^{R} \sigma_r \times U_{r,i} \times V_{r, j}$ Now I ...
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### Implementation of Proximal alternating linearized minimization

The updates of the gradients are somehow wrong. I have implemented the below given algorithm. I have done something wrong ...
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### Factorized matrix for recommendations, what then?

I have a dataset that looks like this: Image taken from this blog Let's assume that I have applied Matrix factorization and have learned the zero values for the items missing for every user. I now ...
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### Rank 1 SVD with constraint on U

I need to perform a particular rank 1 decomposition of a sparse matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$. In particular I am looking for the positive vector $\mathbf{u} \in \mathbb{R}^{+n}$ ...
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### Improvements on using factorization machines?

I am fairly new to factorization machines, I have read papers about it and seen examples of it online. My current goal is to solve a recommendation problem and I'm not sure if what I'm doing is ...
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### Which is more numerically stable for OLS: pinv vs QR

If I am doing standard OLS and want to calculate beta values (OLS estimators), which of the following is the more numerically stable method? And why? Assuming that the columns of $X$ are already mean-...
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### Generalised least squares using QR decomposition

I know that the calculation of parameter values of a standard OLS can be made more efficient using a QR decomposition; i.e. if $X=QR$ and we are using the model $Y=X\beta+\epsilon$; Then it is true ...
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### Orthogonal polynomials with respect to weighted inner product

Recently I have posted a question on SO, but maybe here is better place to ask. So, I have data and I want to fit polynomial of order $k$ orthogonal with respect to weighted inner product of functions:...
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