# 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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### Decomposing Distance Matrix D for approximating Original Matrix A

Let's say we have a matrix $A \in R^{n \times d}$ where n is the number of elements and d is the dimension size. And we calculate the pairwise distances between each elements; say cosine for instance ...
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### Find the principal component and the proportion of the total population variance explained by each when the variance covariance matrix is given

I can understand the part where we have to find the principal component from the variane covariance matrix- find eigen values, make eigen vector and normalise. The principal component would be ...
1 vote
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### Matrix dimensions in Linear Algebra vs Time series Analysis

I am confused or may misunderstand the dimensions of a Matrix when I was reading about time series analysis. From what I understand in linear Algebra, if we have a Matrix $A \in \mathbf{R}^{m*n}$, ...
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### Funk SVD for binary data - product like or dislike

Assume the following situation: you have a user-item sparse matrix. However, instead of the usual 1 to 5 rating scale, items can only receive a positive (1) or negative (-1) feedback. Thus, the matrix ...
• 133
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### Can a covariance matrix be decomposed into a single vector of meaningful distances?

I am interested in creating a single vector to describe a covariance matrix such that the most dissimilar variables are the most extreme values and the other variables have some numeric distance from ...
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### CP Tensor Decomposition and Correlating Sample Magnitudes with Variables of Interest

I am learning about tensor decomposition, specifically CP, and am trying to understand if I can use it for my research. To give a bit more detail, I have brain imaging data from 10 participants, with ...
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1 vote
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### Understanding the decomposed matrices in Singular Value Decomposition [duplicate]

I am trying to get some intuition regarding the U and V matrices in SVD ($M=UEV^T$). I think these are orthonormal basis vectors, but I am struggling to get an intuition if they represent anything ...
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1 vote
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### Why is there a discrepancy between the eigenvalues of the covariance matrix (PCA) and the eigenvalues of the kernel matrix (kernel PCA)?

I've done PCA on my data matrix $\mathbf{X}$ which gives me i.a. the eigenvalues $\lambda$ and eigenvectors $v$ of the data covariance matrix $C=\mathbf{X}^T \mathbf{X}$. I'm now extending my ...
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### Relationship between Cholesky decomposition and matrix inversion?

I've been reviewing Gaussian Processes and, from what I can tell, there's some debate whether the "covariance matrix" (returned by the kernel), which needs to be inverted, should be done so ...
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1 vote
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### Alternating least squares --- what are the limitations?

I am taking an introductory course to Machine Learning and we learned alternating least squares for recommender systems. I learned that this method has some advantages --- easy to parallelize, and ...
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### Convex Optimization Algorithm for Additive Matrix Decomposition

Given an m $\times$ n matrix $Y = S + L + E$, where $S$ is a sparse matrix, $L$ is a low rank matrix and $E$ is a noise matrix, I want to recover $S$, $L$. One of the techniques given in Agarwal et al ...
1 vote
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### Covariance Matrix Decomposition - Data Decorrelation

So I recently found out about Mahalanobis distance. Given a r.v $x$ in N-dimensional space, an associated metric is defined by $$M(x) = \sqrt{(x-\mu)^T S^{-1}(x-\mu)}$$ where $\mu$ and $S$ are mean ...
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1 vote
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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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1 vote
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