# Questions tagged [svd]

Singular value decomposition (SVD) of a matrix $\mathbf{A}$ is given by $\mathbf{A} = \mathbf{USV}^\top$ where $\mathbf{U}$ and $\mathbf{V}$ are orthogonal matrices and $\mathbf{S}$ is a diagonal matrix.

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### Adding New Regularization Terms to Closed Form LSE

I am trying to align the input space X to the output space Y by using least squares method in a closed form solution. To do that I use svd for finding the rotation matrix (W). And I find a solution ...
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### What is the difference between two forms of matrix coherence?

I am stuck on the definition of coherence of a matrix. Let $x_1, \dots, x_p \in \mathbb{R}^p$ be the columns of the matrix $X$, which are assumed to be normalized such that $x'_i x_i = 1$. Wikipedia ...
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### How to use SVD to perform PCA for complex data matrix?

Most researches I have learnt are using SVD to perform PCA for real data mtrix. If the data is complex, will the corresponding solution be different? How to use SVD to perform PCA for complex data ...
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### What is the principal axis means in PCA? [duplicate]

I want to know the main trend of data by PCA method. There was a question that explained what the PCA method works. In this page. And there was also a question that told how to implement this method. ...
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### Why is the first canonical direction equal to the left singular vector i.e. why is $w_1 = a = u_1$ in CCA (Canonical Correlation Analysis)?

I want to understand why the canonical direction $a$ is equal to the left singular value of $M = \Sigma^{-1/2}_X \Sigma_{X, Y} \Sigma^{-1/2}_Y$ and not $a = \sigma_1 u_1$. My calculation tell me that ...
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### Singular value decomposition for a matrix with missing entries

Suppose $A$ is a matrix consisting of real numbers and nan's. What are some of the robust formulations and the associated algorithms for estimating its singular ...
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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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### Relation between first left-singular vector and row-wise mean

I am working on some imaging data: let $X$ be a matrix of size $t\times v$ ($v$ is the number of pixels of my images, $t$ a time dimension), centered and scaled to unit variance. I am interested in ...
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### What is so special about the least norm solution in case of an undetermined system of equations

In particular this is the go to approach in case of solving a least squares problem that lacks a unique solution, how does being the closest point to the origin among all the solutions make it any ...
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### Computing Pairwise Distances Through PCA or SVD

What should I do to reduce an mxn (m=17, n=650,000) matrix, where m are samples and n are features of these samples, into a matrix of pairwise distances (which I will then use to generate a dendrogram)...