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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prove for Eckart-young in Frobenius norm

On page 74, linear algebra and learning from data. P74 the prove for Eckart-young in the frobenius norm. I couldnot understand why G = 0 in the proof, anybody can help me? Thank you!
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Assessing the size of a cone by the singular values of $M$

Suppose I work with vectors from a high dimensional space with $100<N<1000$, e.g. word-embeddings. Say I have, already selected $R$ vectors, with $R\simeq10$, which form a matrix $M \in \mathbb{...
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Applying SVD on dataset with 4 columns

I have a dataset with following format and 200000 rows: X Y Z A 5608 142 740 1 4533 142 741 2 5620 143 740 0 4732 142 744 1 5500 143 742 1 5514 142 741 2 I am ...
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Can I use the matrix $U$ instead of the matrix $V$ in Principal Component Analysis?

I'm taking Andrew NG's Machine Learning Course and got to the part of Principal Component Analysis. Andrew's implementation of PCA aroused 2 questions for me. 1. Let's say that we have the data ...
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SVD -> PCA -> t-SNE; Does it make sense?

I have a data set of size (4600, 10000). I did L2 normalization at first, then I did the following two steps to visualize it in a lower dimension: ...
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Is each row of latent factors obtained from matrix decomposition (SVD) dependent on the other rows of the higher dimensional matrix?

I implemented a recommendation system using user-user interaction data, learning missing ratings through alternating least squares and matrix factorization, which as I understand it, adjusts and ...
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Why is there a reconstruction loss in PCA with orthonormal eigenvectors?

I've already read How to reverse PCA and reconstruct original variables from several principal components? and I understand conceptually and visually why there has to be a reconstruction loss. ...
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How to make a scree plot out of SVD data to validate PCA

After doing a singular value decomposition (SVD) of a data set, I'm left with three matrices: 1. An orthogonal Left Singular Vector (U) 2. diagonal matrix with elements in descending order (S) 3. ...
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How to measure changes in condition indices over time

I am trying to understand how adding data, one observation at a time, affects the condition indices of a model. A similar question is how adding individual observations affects the principal ...
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Intuition About Principal Component Directions

I am trying to really get a deep understanding of PCA. From my understanding, a principal component is defined as $$\mathbf{z}_k = \phi_{1,k} \mathbf{x}_1 + \ldots + \phi_{p,k} \mathbf{x}_p = \mathbf{...
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dimensionality reduction using SVD for forecasting with machine learning

I'm using a LSTM model to forecast time series data. My dataset has far too many variables and I would like to perform dimensionality reduction. My LSTM model works on a rolling window of 500. I ...
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Different number of Eigen/Singular values from PCA and SVD

My understanding is that a SVD done on a raw data matrix M and a PCA done on its covariance matrix C should return the same eigen/singular values. I have a 2736 x 356 data matrix and am using the ...
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Why is more Covariance Matrix different in these two computations? [duplicate]

I am trying to break down a matrix into its principal components step-by-step using SVD in order to better understand the process. What I don't understand is why I'm getting different forms of the ...
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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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Why does kmeans after SVD result in ideal clusters

I am clustering tweets which are related to eye fashion and they are extracted using keywords like mascara, eyeliner, eyeshadow, etc from twitter. I constructed a Tf-idf matrix (tweets x words) ...
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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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Since SVDs are derived from existing variables, should you push the existing variables AND derived SVDs into a model?

wouldn't multicollinearity occur between the variables since SVDs are derived from the existing variables in the first place?
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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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Why do PCA and PCoA give the same components but different explained variances?

I'm quite familiar with Principal Component Analysisis, as I use it to study genetic structure. Lately, I was revisiting some of the functions I was using in R (...
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Calculating PCA coefficients using SVD, PCA (sklearn) and Covariance Matrix

I am trying to understand PCA implemented in different methods on python. I am failing to get equal PCA coefficients in each of the methods. By PCA coefficients I mean data projected in the principle ...
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Singular values for a latent-factor model

Suppose we build a latent-factor model using alternating least squares (ALS) or stochastic gradient descent (SGD). Can we calculate weights for each latent factor, in a similar way to how the singular-...
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How SVD factorisation -based recomendation algos deal with new user interaction

Classic SVD and SVD++ alogritms generate predictions based on a current known ratings only for known users and known items. But I need to make prediction for some new user on the old items. In the ...
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SVD matrixes do not coincide with Eigen decomposition for covariance matrix [duplicate]

I am comparing the output from the singular value decomposition with the eigendecomposition of the covariance matrix (symmetric matrix). I am expecting that the Eigenvector and a non-diagonal matrix ...
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preparing free text column for regression

I have a column X which contains occupation/profession as an independent variable as free text, which is very much correlated with a continuous dependent variable. What techniques do you usually use ...
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What is the meaning of these principal components?

I have a matrix of data. I computed the principal components of my matrix using SVD (code shown below): subtract mean...then $$[U,S,V] = SVD({\rm matrix})$$ for $V$, which is the principal ...
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SVD PCA reconstruction of data [duplicate]

I have some data about the $\{noise,~ size,~ speed,~ length,~ width\}$ of cars. I have performed SVD, and I want to reconstruct my data using only the first 2 principal components. I subtracted mean ...
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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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Why is computing ridge regression with a Cholesky decomposition much quicker than using SVD?

By my understanding, for a matrix with n samples and p features: Ridge regression using Cholesky decomposition takes O(p^3) time Ridge regression using SVD takes O(p^3) time Computing SVD when only ...
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Why is it much quicker to compute ridge regression than regular linear regression?

By my understanding, for a matrix with n samples and p features: Ridge regression using cholesky takes O(p^3) time Ordinary linear regression takes O(p^3) time Singular value decomposition if u, v ...
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Using numpy SVD to calculate factor loadings [duplicate]

I'm doing PCA (Principal Component Analysis) in Python using the numpys Singular Value Decomposition. Effectively extracting the principal components like so: ...
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Meaning of matrix U in LSA?

LSA uses SVD algorithm i.e. the terms-documents matrix $=U \Sigma V^T$ What is the exact meaning of matrix U here: I know that it is a rotation but rotation of what? Is it a concepts space? If yes ...
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Can anyone help me with step by step procedure in MATLAB for missing data imputation using Singular Value Decomposition (SVD [duplicate]

I need step by step procedure for imputing data using SVD in Matlab. Please provide resources if any !
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Explanation of SVD in relation to specific research question (building vector on multiple 3D coordinates)

unfortunately I'm completely out of my comfort zone when it comes to the math involved in my question, appologies in advance. Background: I am evaluating the presicion of electrode placement in the ...
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Reference point in projection axis of SVD (singular value decomposition)

I am watching a YouTube video on SVD, and attempting to recreate some of its examples to better understand the internal machinery of the algorithm. In one of the slides, the instructor mentions that ...
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Additional Property of Singular Value Decomposition

I am new to SVD so forgive me if the question is trivial. Following is my question. If I have two sets of linear equations, Y1 = T1.X Y2 = T2.X where T1 and T2 are mxn rectangular matrices. Now let'...
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Idea behind change of basis and how it relates to projecting your points onto principal components

I would like to clarify if my understanding is correct. In the traditional X-Y coordinate system, our choice of basis vectors are $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$ and when you I have a point $...
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SVD for simultaneous row and column reduction of a squared matrix

I have an n x n similarity matrix which I'd like to reduce to a smaller square matrix. I am aware of this answer: How to use SVD for dimensionality reduction to reduce the number of columns (features)...
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Decomposing SVD of this dataset by hand (perspective)

I have the following data matrix $\left[\begin{array}{ccccc} 1 & 1 & 1 & 0 & 0\\ -3 & -3 & -3 & 0 & 0\\ 2 & 2 & 2 & 0 & 0\\ 0 & 0 & 0 & -1 &...
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Is low rank finite-iteration manifold identification possible?

In sparse optimization, I am trying to solve the problem $$ \min_{x\in \mathbb R^{n}} \quad f(x) + \|x\|_1 $$ and at optimality, $x^*$ may be sparse. If I define the sparse manifold as $\mathcal M = ...
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Eigenvalue decomposition/SVD and the filtering perspective

I have been studying the SVD algorithm recently and I can understand how it might be used for compression but I am trying to figure out if there is a perspective of SVD where it can be seen as a low ...
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Why truncated SVD can denoise images

There are a lot of empirical results about that truncated SVD (TSVD) can help denoise the noises of images, but I wonder what is the theoretical support behind that? We know that TSVD is the best low-...
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Does Column ordering matter in QR decomposition?

I am trying to understand if the ordering of columns matters in QR decompsoition. In general it seems that column ordering won't matter. I guess for SVD or any matrix factorization the way columns ...
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Eckart-Young-Mirsky theorem: rank $≤k$ or rank $=k$

The Eckart-Young-Mirsky theorem is sometimes stated with rank $\le k$ and sometimes with rank $= k$. Why? More specifically, given a matrix $X \in \mathbb{R}^{n \times d}$, and a natural number $k \...
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Extracting latent vectors from autoencoder similar to SVD

I have read that there is an equivalency between a linear autoencoder and performing SVD. SVD can be used in collaborative filtering, for example, factorization of a user-movies matrix $\mathbf{M}$ ...
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354 views

PCA in psych package with more columns than rows

Why is it impossible to do a PCA in R using principal from psych package without warnings with a matrix, which has more columns ...
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PCA for three-dimensional linear fit on time-resolved trajectory

I study the behavior of organisms that are able of self-locomotion and that show directed motion toward one another. This directed motion occurs through the detection of chemical trails released by ...
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Derivation of Procrustes transformation

I'm curious about the derivation for the Procrustes transformation. I'm following ESLII: see figure 14.25 and problem 14.8 (Procrustes distance with scaling). Given matrices $\mathbf{X}_1$ and $\...
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matrix factorization with non-negative constraint only on one of the factors

I have a 2D spectral data time series with a wavelength dimension and a time dimension, and I'd like to decompose it to the time evolution ($SV^T$ for SVD and $H$ for NNMF) of several spectral ...
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Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

X-Posted on math.stackexchange, apologies, though I thought this was equally relevant to both communities. I'm wondering if there exists any higher-order SVD for dimensionality reduction. Note that ...
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How SVD is used for dimensionality reduction? [duplicate]

Given a $M \times N$ data matrix $ D = (x_1, x_2, \cdots, x_M)^{T}$. Applying singular value decomposition to $D$ yields $$D = USV^{T} = (u_1, u_2, \cdots, u_M) \begin{pmatrix} s_1 &&0\\...