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.

Filter by
Sorted by
Tagged with
1
vote
0answers
82 views

Adding explicit user info to matrix factorization

In the paper Matrix Factorization Techniques for Recommender Systems, it is claimed that we can incorporate extra user information into our recommender model by doing something like this: $$ \hat{r}_{...
0
votes
0answers
21 views

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. ...
1
vote
1answer
17 views

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 ...
1
vote
0answers
31 views

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 ...
0
votes
0answers
24 views

PCA and SVD not effective at capturing variance in fewer dimensions. Any good alternatives?

I have four columns, which I have z-scaled and tried PCA and SVD on, hoping that I can obtain one dimension that explains the majority of the variance. However, with PCA there is approximately 25% ...
0
votes
0answers
13 views

SVD - vectors in matrix A

In SVD we have $A = U \Sigma V^T$. When applying it for ML, e.g. to calculate Moore-Penrose pseudoinverse for linear regression, I have seen that we take columns of $A$ as vectors. Typically in ML I ...
0
votes
0answers
8 views

How to fit a constrained harmonic fit in R to mean daily temperature data?

I am trying to reproduce the statistics procedures for creating daily normal temperatures according to this paper. I was provided code in IDL (a Fortran based scientific language), but it is so ...
0
votes
0answers
37 views

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 ...
1
vote
0answers
27 views

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 ...
0
votes
0answers
25 views

Correct use of S.V.D. regression on my data

All I(barley) know is electrical engineering. Only ever taken one basic stat class and a linear alg. class. So please use layman terms. I'm working on a wireless power transfer system. (Like those ...
1
vote
0answers
35 views

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 $$\hat{Z} = \text{argmin}_Z\lVert X - Z \rVert_F^2 + \lambda \lVert Z \rVert_*$$ ...
0
votes
0answers
18 views

PCA with sample-specific prior information about principal components

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 ...
0
votes
0answers
7 views

Physical significance of non-negative factors of a matrix?

I was trying to make a recommender system using matrix factorization techniques on rating data. I came across 2 algorithms - SVD and NMF. While the basic difference is very clear , I was wondering ...
0
votes
0answers
8 views

How can I regularize data pre-processing parameter estimates (center, scale, rotation matrix)?

Suppose I have an $N×1$ vector $Y_{in}$ of response values an $N×P$ matrix of predictors $X_{in}$ whose individual columns exhibit significant correlation. Let's further suppose that this matrix $X$ ...
0
votes
1answer
23 views

Variable Selection : Removing Linear Dependency by SVD using the Condition number and then eliminating the variable causing multicollinearity

I am trying to perform regression with over 5000 feature variables(X) and I would like to eliminate multicollinearity. Incremental VIF computation is expensive. Incremental PCA works but I might lose ...
0
votes
0answers
32 views

Optimisation of reconstruction error for PCA

I am trying to understand the math behind the minimal reconstruction error for PCA models in this book. Given a data matrix $Y$ which has dimensions $D\times n$ and assuming that it is centered. The ...
1
vote
0answers
21 views

Why are the directions of eigenvectors in SVD and Eigen-Decomposition for PCA opposite? [duplicate]

As you may know, scikit-learn library utilizes singular value decomposition (SVD) of data matrix X to produce eigenvectors for PCA. I decided to code PCA by using ...
0
votes
0answers
26 views

Principal Component analysis - SVD U matrix projection

I have a bit mathematical question I am interested in. Principal component analysis (PCA) has mathematically multiple solutions. One way is to use SVD. I have prepared an example bellow. I am curious ...
1
vote
0answers
23 views

Fastest way to find Leading singular value and vector (power iteration, rsvd etc)

I want to know the fastest way to find out the leading singular value and vector of a large rectangular matrix. I have seen 2 suggestions and have questions on both of them : Power Method : For this ...
2
votes
1answer
41 views

Largest singular values

Given the positive semi-definite, symmetric matrix $A = bb^T + \sigma^2I$ where b is a column vector is it possible to find the singular values and singular vectors of the matrix analytically? I know ...
2
votes
1answer
29 views

Using SVD to write the least squares fitted vector

Elements of statistics p.66 Please I know the least squares solution for $\hat\beta = (X^TX)^{-1}X^Ty$ but I don't know how they were able to get $X\hat\beta= X(X^TX)^{-1}X^Ty = UU^Ty$ These are the ...
1
vote
1answer
27 views

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: $X$: ...
2
votes
0answers
21 views

Singular value decomposition used for dimensionality reduction in brain signal topographic data

I am trying to replicate the localizer method described in this paper (page 4). I am stuck on a step which I don't completely understand, and I would like your input and interpretation to progress. ...
4
votes
0answers
72 views

Physical interpretation of $U$ and $V$ matrices in SVD

I have a question about the physical interpretation of $U$ and $V$ matrices in SVD. I collect measurements at multiple devices across time are collected into an $m$ × $T$ matrix $M$, where m is the ...
1
vote
0answers
21 views

Kernel matrix decomposition

I had a look at the sklearn.kernel_approxiamtion.Nystroem implementation, which is also described in this post: Nystroem Method for Kernel Approximation Here, a ...
0
votes
0answers
25 views

Equal variance along left and right singular vectors?

Please confirm or reject my line of reasoning: Given SVD of $X$: $X_{NxP}=U_{NxP}D_{PxP}V_{PxP}'$, Variance along ith column vector of $U$ is given by $||X'u_i||^2=u'_iXX'u_i=u'_id_i^2u_i=d_i^2$, ...
0
votes
0answers
29 views

How to predict for test set when training a recommender by decomposing the utility matrix X=UV?

This probably sounds stupid but I don't get the workflow of building a recommending system by the utility matrix: X[i,j] = how much the ith user likes the jth object. For practical issues I refer to ...
0
votes
0answers
15 views

How does the Zeiger-McEwin & Kung algorithm work for fitting a sum of exponentials?

I am trying to understand this paper fit sum of exponentials but am having a bit of difficulty. Let me go through what I have understood so far. One has a certain time-series data set and want's to ...
1
vote
1answer
46 views

Question about the Proof of PCA in “Learning from Data” by Shwartz and Ben-David, P. 280-281

Does anyone know how to justify the red and blue line in the attached proof of PCA? Red line: $B \in \mathbb{R}^{ d \times n}$, arrange $B = [B_{j,1} | B_{j,2} | \cdots | B_{j,n}]$, then $B^\top B = \...
1
vote
0answers
21 views

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 ...
0
votes
0answers
25 views

Principal components with and without centering [duplicate]

Suppose that I am given various samples of a vector random variable as the columns $v_1,v_2,\dots,v_n$ of a certain matrix $A$. Is there a relation between the SVD $A = USV^T$ the SVD $\hat{A} = \hat{...
1
vote
0answers
36 views

SVD for a complex data matrix — what is the meaning of the columns of $V$?

I've read this wonderful explanation of SVD, where the writer mentions that the columns of $V$ are the principal directions (Summary, #1). Is this also true when the data matrix $X$ is complex? If I'm ...
0
votes
0answers
34 views

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)...
0
votes
0answers
18 views

Which form of SVD is most commonly in stats/ML (e.g., PCA, least squares)?

SVM comes in several forms: e.g., the full form and the reduced forms https://en.wikipedia.org/wiki/Singular_value_decomposition#Reduced_SVDs. Is there one that is more commonly used in stats? If so, ...
3
votes
1answer
127 views

Human intuition behind SVD in case of recommendation system

This does not answer my question. I struggled very hard to understand the SVD from a linear-algebra point of view. But in some cases I failed to connect the dots. So, I started to see all the ...
0
votes
0answers
28 views

How vector projection works behind SVD?

I was reading a blog on mathematical intuition behind SVD. Here, author pointed out three information we get after vector decomposition. The directions of projection — the unit vectors (v₁ and v₂) ...
10
votes
3answers
603 views

SVD : Why right singular matrix is written as transpose

The SVD is always written as, A = U Σ V_Transpose The question is, Why is the right singular matrix written as V_Transpose? I mean lets say, W = V_Transpose and then write SVD as A = U Σ W SVD Image ...
1
vote
0answers
42 views

Relationship between the SVD and correlation matrices

I'm reading Data Driven Science and Engineering by Kutz and Brunton to understand more about the SVD. Consider $X = U\Sigma V$, $XX^*$, and $X^*X$ where $X \in \mathbb{R}^{m\times n} $ In particular, ...
0
votes
1answer
70 views

Penalized Canonical Correlation in R with PMA Module

I am trying to use sparse canonical correlation analysis as implemented in the R PMA package. I'm finding that the correlations output by the package seem slightly inconsistent with the ones you would ...
5
votes
1answer
363 views

How are eigenvalues/singular values related to variance (SVD/PCA)?

Let $X$ be a data matrix of size $n \times p$. Assume that $X$ is centered (column means subtracted). Then, the $p \times p$ covariance matrix is given by $$C = \frac{X^TX}{n-1}$$ Since $C$ is ...
1
vote
0answers
17 views

Symbolic Singular Value Decomposition? U,S,V as function of the elements of M [closed]

Suppose we want to compute the SVD of $\mathbf{M} = \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix}$ (...
0
votes
0answers
20 views

Variance of a principal component

Suppose that we have a SVD for our data matrix centered $X = UDV^T$. Then it is stated that the i-th principal component, $Xv_i$, has variance $\frac{d_i^2}{N}$. Consider these steps. $$ var(Xv_i) = ...
0
votes
0answers
20 views

Whitening on projection matrices

The projection matrix $P = I -xx^T\in \mathbf{R}^{d \times d}$ has a zero eigenvalue and eigenvalues equal to one with multiplicity $d-1$. Is it possible to apply whitening transform on $P$ taking ...
0
votes
0answers
13 views

How does partial least squares algorithm return more than one factor?

My understanding of PLS regression is that we find an eigen vector such that it maximises the covariance between X(matrix of independent variables) and Y(vector/matrix of dependent variable) i.e. find ...
1
vote
0answers
19 views

Use Matrix Factorization to predict probability of a recommendation system?

I have a dataset where I have a sparse utility matrix (user-product) with binary input: 1 if the user $i$ bought the product $j$, and 0 if it hasn't. However it has a different meaning on the test set,...
0
votes
0answers
52 views

In Probabilistic PCA, Where does the arbitrary orthogonal matrix(rotation matrix) come from?

I'm working on studying Probabilistic PCA based on the paper (Tipping & Bishop, 1999), I can follow the idea that the maximum likelihood function would reach the stationary point when the the ...
2
votes
1answer
106 views

Why is non-centered SVD accepted in LSA

In Latent Semantic Analysis (LSA) , we apply SVD to a term-document matrix $A$, then choose to ignore all but $k$ largest singular values. The term-document matrix is not centered, or normalised, ...
0
votes
0answers
28 views

Interpreting SVD on non-centered matrix

I have a very large, very sparse matrix $A \in \mathbb{N}^{n \times m}$ I'd like to perform SVD on. It is non-centered. When I center it to $A'$, I can't even fit it in memory (because $A'$ is in $\...
0
votes
0answers
13 views

Variability of K SVD components [duplicate]

Let's say I have a SVD of a matrix $A = U \Sigma V^T$, $A \in \mathbb{R}^{n \times m}$, and I'm using top-k components corresponding to $\sigma_1, ..\sigma_k$, the k largest values on the diagonal of $...
0
votes
0answers
18 views

Is there any relationship between SVD low rank approximation and DCT low frequency approximation?

Matrix can be approximated by low rank approximation using SVD by using major principle components. On the other hand, if we look at the frequency domain, matrix can also be approximated using DCT (...

1
2 3 4 5
8