# 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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### is it fair to use a subset of eigenvalues to evaluate the multidimensional variance

I want to find a single metric to assess how spread (or how much variance) a multidimensional dataset (a large number of features) is. I learned that the determinant (or pseudo-determinant) of the ...
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### Should I do documents transformation at once or a pair at a time for auto grading with cosine similarity?

I'm developing auto grading essay that compares the similarity between the answer key and student answer with cosine similarity. This one is written in php. Let's say in a course there are 30 - 100 ...
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### In principal component regression, how to show that $X \beta_\mathrm{PCR}=U\: \mathrm{diag}\left\{1,\ldots, 1, 0, \ldots 0\right\}U^Ty$ if $X=USV^T$

Regarding the third equation in this answer, I'm struggling to figure out how $X \beta_\mathrm{PCR} = U\: \mathrm{diag}\left\{1,\ldots, 1, 0, \ldots 0\right\} U^\top y$. Given $X = USV^T$ where $X$ ...
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### Time-Continuity of PCA/SVD [duplicate]

I have a daily set of observations (Jacard similarity matrix) that I want to embed in a much lower dimensional space. Thus I run PCA or SVD for each day and plot the projection on the top 2 dimensions ...
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### Procrustes Problem for rank-deficient input

It is well known that the solution to the orthogonal Procrustes problem $$\textrm{arg min}_{\Omega \in \text{SO}(n)} ||Y - \Omega X||^2_2,$$ can be expressed in terms of an SVD of the covariance ...
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### Why do we multiply the centered data with eigen vector instead of taking inverse of the eigen vector while performing PCA

The question may sound stupid but I really don't understand the logic behind this. Whenever we do a PCA, we take a covariance matrix on the centered data and do eigen decomposition. In order to ...
1 vote
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### Scaling before PCA & SVD - Iris Dataset [duplicate]

I've seen differing approaches from various resources online, on the necessity of scaling data before doing PCA/SVD. PCA Resource 1: CMU slides Center the means, does not divide by standard devitaion ...
1 vote
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### How do ‘arpack’, ‘propack’ and ‘lobpcg’ compare to each other?

I want to implement singular value decomposition on a sparse matrix using the Scipy implementation. The documentation shows three solvers: ‘arpack’, ‘propack’ and ‘lobpcg’. How do these solvers ...
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### SVD on the item-user interaction matrix

Factor-based recommenders (typically) apply SVD on the user-item interaction matrix (u-i matrix). If we apply SVD on the transpose of the u-i matrix (i-u matrix), will we obtain the same embedding ...
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### How can I interpret singular value decomposition analysis?

I am trying to understand singular value decomposition analysis. I compared two gridded atmospheric data. The Mode 1 has 79.5% squared covariance fraction. Modes 2 and 3 have 3% and 2%, respectively. ...
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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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### Determining ML Approach for calculating SVD using neural networks

I am currently working on a project where I need to perform SVD (Singular Value Decomposition) computation on a noisy data using neural networks. It doesn't have to be exact SVD, certain degree of ...
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### Singular value decomposition on a polynomial

I'm messing around with the SVD to find a best fit solution. The way I understand (never taking a stat class, only linear alg.) is that that the SVD captures the data variation by its projection onto ...
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### Does it make sense to use SVD to do a sort of "lossy compression"?

So - I know if you perform SVD to a matrix $X$, you can then use Echkart Young theorem to get the best rank $r$ approximation $\overline{X}$ to $X$ possible. Since the resultant ${\overline{X}}$ will ...
1 vote
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### Computational advantage for soft-impute method over other methods

I am reading in the soft-imputing paper for low-rank-based matrix completion. They suggested another solution for $$\hat{Z} = \text{argmin}_Z\lVert X - Z \rVert_F^2 + \lambda \lVert Z \rVert_*$$ ...
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Suppose I have some noisy dataset $\mathbf{X} \in \mathbb{R}^{N \times p}$ that I want to perform PCA on. Obtaining the (trimmed) SVD $\mathbf{X} = \mathbf{UDV}$ infers the q-dimensional principal ...