Questions tagged [svd]

Singular Value Decomposition (SVD) is a decomposition (factorization) of rectangular real or complex matrix into the product of a unitary rotation matrix, a diagonal scaling matrix, and a second unitary rotation matrix.

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My Complex Matrix SVD is Correct according to rule A = USV' but Wrong according to Matlab or any linear algebra library

I am working on Singular Value Decomposition for complex matrices. I implemented One Sided Jacobi algorithm. It gives exactly the same result as the svd function in Matlab for the real matrices. ...
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Generate polynomial basis through a sequence of SVD

I need help to understand how to use the result given by an algorithm for constructing an orthonormal polynomial basis over $L^{2}(X)$, where $X\subset\mathbb{R}^2$, with respect to the inner product $...
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Solving linear system and obtaining operator norm

I need to solve a linear system of the form $(\mathrm{Id} + \mathbf{J})\mathbf{x} = \mathbf{b}$ for $\mathbf{x}$ and I also need to compute the operator norm of $\mathbf{J}$ (i.e. the largest singular ...
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2 votes
2 answers
192 views

Calculate determinant of unitary matrices based on SVD implementation

I have a real square matrix $X$ which I need to perform a Singular Value Decomposition on. Now, performing the operation $$ X = USV^T $$ as $U$ and $V$ are orthogonal, we know that $\det(X)=\pm\det(S)$...
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O(N^2 k) implementation of non-random truncated SVD

I am looking for an implementation that calculates a truncated singular value decomposition (SVD) of a dense NxN matrix in O(N^2 k) steps with a deterministic rather than randomized method. (Here, 0&...
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2 answers
174 views

Given a symmetric matrix, is it ok to apply Cholesky decomposition to see if it has negative eigenvalues?

I intend to check the diagonal of L, where A = L'L, for negative elements. However, I don't know if Cholesky is meaningful in theoretical / computational sense if there are some negative eigenvalues.
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How to compute Singular value decomposition of a large matrix with Python

Language: Python3 Problem: I have a matrix Q of shape [51200 rows x 51200 cols] stored in a binary file, each of the element in this matrix has a data type of complex64. To load the data into memory I ...
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1 answer
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Asymptotic complexity of fixed-rank SVD

According to the Wikipedia article on Singular Value Decomposition, the asymptotic complexity of computing the SVD of an arbitrary m×n matrix M with m>n by the popular Householder QR methods is O(...
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updating the matrix Adjugate/Cofactor

I would like to calculate the Adjugate matrix of a given matrix $A$, and its updates in the diagonal: $B=A-\lambda I$, where $I$ is the identity matrix, $\lambda$ is a scalar. To this end, I am using ...
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Formula for overdetermined logical matrix pseudoinverse not requiring SVD?

In https://commons.wikimedia.org/wiki/File:YI_%3D_PI.png, you will find a formula-based solution for an overdetermined logical matrix pseudoinverse. This simple formula gives the same result as the ...
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Rank of a double-precision augmented matrix

Let $A$ be a matrix with real entries, and let $A_+$ be $A$ augmented by a single column. From linear algebra we know \begin{equation} \operatorname{rank}(A_+) = \operatorname{rank}(A) \hspace{10pt} ...
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Computing Singular Value Decomposition of small ($4\times 4$) matrices

I need to compute the Singular Value Decomposition (SVD) of many $4 \times 4$ matrices. I'm looking for SVD algorithms specialized for small matrices. I've read that the ...
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159 views

Explanation of Givens rotation in Jacobi Rotation SVD

I'm trying to implement Singular Value Decomposition (homework of sorts) via the Jacobi Rotation method (more info here, pages 11 and 12). I am stuck at the bullet saying (sorry for the picture, but I'...
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3 votes
1 answer
256 views

Scaling/Performance of Matlab's svds function (Lanczos bidiagonalization)

I have a simple Matlab script which aims to compute $k$ singular values of a matrix $A$. $A$ is a random dense square matrix of size $5000\times5000$, with 100 of its singular values constrained to ...
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1 vote
1 answer
318 views

Implementation of sparse matrix SVD for GPU

I have a sparse matrix $W$ which is almost-squared ($N+1 \times N$) and I would like to know the eigenvalues of $A = W^T W$. $A$ is Hermitian so the eigenvalues are real-positive valued. The usual ...
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2 votes
1 answer
458 views

Matlab - Fast Computation of Truncated SVD / PCA

I'm working with a Matlab codebase wherein I'm attempting to solve A*c = b by approximating the (square) matrix A with its ...
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3 votes
1 answer
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Singular vectors of s1 for tiny dense matrices

I have a function whose main bottleneck is finding a(ny) singular vector pair in the space of the largest singular value, along with the singular value itself. This is done a huge number of times. ...
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1 answer
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Analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^T A z$, where $A=uu^T+vv^T$

Let $u$ and $v$ be column vectors of size $n \gg 1$ (not both zero), and consider the matrix $A:=uu^T+vv^T$ Question What is an analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^TAz=\arg\max_{\|z\...
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1 answer
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Analytic formula for leading eigenvector of $uu^T + vv^T$?

Let $u$ and $v$ be nonzero column vectors of size $n$ and consider the $n \times n$ positive-definite matrix $A:=uu^T + vv^T$. In this post https://math.stackexchange.com/a/112201/168758, the ...
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1 answer
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Whitening transformation does NOT return a unit covariance matrix

For this question, I am using the following Wiki definition of Matrix whitening: Suppose $X$ is a random (column) vector with non-singular covariance matrix $\Sigma$ and mean 0. Then the ...
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Stability of SVD, Eigendecompositions, and pseudoinverse procedures in modern LAPACK routines

I have proposed an optimisation algorithm which I claim has improved upon the previous algorithm in a number of ways. One of these claims is that my proposed solution requires no explicit SVD and ...
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8 votes
1 answer
187 views

Finding the $i$-th largest eigenvalue of a matrix

Given a large matrix $A$ with eigenvalues $\sigma_1\ge \sigma_2 \ge \dotsc $, I want to determine only a subset of these values, say $\sigma_5,\sigma_8$ and $\sigma_{19}$. Is there an algorithm that ...
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Golub-Kahan-Lanczos Bidiagonalization Procedure implementation doesn't produce bidiagonal matrix

I'm trying to implement the aforementioned procedure using this website as a reference. At the end of the page the algorithm is described as follows: I think I've mapped the given algorithm to code ...
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2 votes
0 answers
621 views

finding null space to a complex matrix

I need to solve the following equation: $$ \begin{pmatrix} \frac{\omega^2}{c^2}\varepsilon_x-\mu_z^{-1}k_y^2-\mu_y^{-1}k_z^2 & \mu_z^{-1}k_xk_y & \mu_y^{-1}k_xk_z\\ \mu_z^{-1}k_xk_y &\...
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2 votes
1 answer
627 views

How we can use CUR decomposition in place of SVD decomposition?

I have understood how CUR and SVD works, but have not been able to understand the following. How can we use CUR in place of the SVD decomposition? Do the $C$ and $R$ matrices in the CUR follow the ...
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4 votes
1 answer
567 views

Nystrom approximation of SVD for asymmetric matrices

Suppose I have a symmetric matrix $K$. Subdivide $K$ into pieces as $$K=\begin{pmatrix} K_{11} & K_{12} \\ K_{21} & K_{22}\end{pmatrix},$$ where $K_{21}=K_{12}^\top$. Then, the Nystrom ...
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1 vote
2 answers
288 views

Inverting big symmetric and singular matrices

In this post I found a very similar probem to the one I have, but not a satisfactory answer for my purposes. I have a set of matrices $C_\ell$. They are exactly symmetric by construction. ...
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4 votes
1 answer
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Poor SVD reconstruction of singular matrix

I am trying to calculate the singular value decomposition of this matrix using numpy.linalg.svd . However, reconstructing the matrix from the SVD gives a poor ...
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Best algorithm for inversion of matrix spanning many orders of magnitude [duplicate]

I have a very similar problem to the one described in Calculate inverse of dense matrix with entries of very different magnitude. The reason why I am opening a new question is because as far as I ...
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10 votes
2 answers
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How much regularization to add to make SVD stable?

I've been using Intel MKL's SVD (dgesvd through SciPy) and noticed that results are are significantly different when I change precision between ...
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1 vote
0 answers
543 views

Pseudoinverse of a large sparse matrix in r

This question was moved from Cross-Validated: https://stats.stackexchange.com/questions/274042/pseudoinverse-of-large-sparse-matrix-in-r I am trying to calculate the pseudoinverse of a large sparse ...
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4 votes
1 answer
5k views

Block-matrix SVD and rank bounds

Assume, we have an $m\times n$ block matrix $M=\left[\begin{array}{c c}A&C\\B&D \end{array}\right]$, where $A$ is an $m_1 \times n_1$ matrix of rank $k_A$. $B$ is an $m_2 \times n_1$ matrix ...
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2 votes
0 answers
175 views

Generalized eigenvalue with null space

Define $S\in\mathbb{R}^{n\times n}$ as $$S:=H+Q^\top V^{-1} Q.$$ $H,V$ are positive semidefinite. Here, $H$, $Q$, and $V$ are large, dense matrices but they are structured: I can write code for ...
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7 votes
1 answer
341 views

What is the fastest way to compute the sum of the singular values of a matrix?

Is there a faster way to compute the nuclear norm (trace norm, sum of singular values) of a matrix A than computing SVD(A) directly (or diagonalizing A^*A)? I am particularly interested in the case ...
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2 votes
0 answers
72 views

Under what circumstances does Elemental's distributed SVD not work? [closed]

I am playing around with Elemental's distributed singular value decomposition and am running into two particular issues. Building the test at tests/lapack_like/SVD.cpp, and running with ...
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7 votes
2 answers
280 views

Lanczos algorithms for Hermitian system with Toeplitz kernel

Basically, I am trying to compute the SVD of a large Hermitian matrix $H$ using Lanczos iteration, while $H$ consists if a Toeplitz kernel $K$, which should be able to help speed up the matrix-vector ...
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2 votes
0 answers
59 views

Optimal ordering in Jacobi SVD algorithm

In Jacobi SVD algorithm as given here every pair of columns of the matrix is orthogonalized until convergence. I want to know that how does the order of selection of the pair of columns affect the ...
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2 votes
1 answer
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Condition Number of Rectangular Matrices

The 2-norm condition number can be easily extended to rectangular matrices. I'm wondering if the inequality for the product of matrices still holds in that case, i.e., $\operatorname{cond}(AB) \leq \...
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1 vote
2 answers
276 views

Efficient methods to solve large dense singular least square problem (linear system)

I am trying to solve a singular linear least square problem: $$minimize: \phantom{2} ||Ax-b||^2 \\ subject \phantom{2} to: \phantom{2} x \ge 0$$ Here $ A \in R^{n \times m} $, and $ n\lt m$. here m ...
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2 votes
0 answers
27 views

Choosing suitable polynomial degree based on information in advection stencil

I'm working on a finite volume advection scheme for unstructured meshes which uses a multidimensional polynomial weighted least squares fit for interpolating from cell centres onto faces. In 2D, the ...
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1 vote
1 answer
759 views

Iterative Closest Point Algorithm

I am currently working on an iterative closest point algorithm (in C++, see here). I understand the basic premise of an ICP algorithm. You have two point clouds (a target and a reference) and you ...
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4 votes
1 answer
474 views

Obtaining column vectors of pseudo-inverse of a matrix

I need to compute the pseudo-inverse of a very large rectangular dense matrix without any special structure or properties. I run out of memory/computing power and have no access to a large parallel ...
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1 vote
1 answer
244 views

Compare reconstruction of matrices using SVD

I'm interested in how much 'signal' is retained from including k singular values in a Singular Value Decomposition, but I'm having trouble conceptualizing (or ...
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1 vote
1 answer
683 views

On the fly/matrix free SVD of large sparse matrix

I am trying to apply SVD to large sparse matrices. I already compared the performances of Propack and irlba to those of the matlab svd and ...
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3 votes
2 answers
2k views

SVD of large block-hankel matrix

I am trying to do SVD of a large block-hankel matrix for model order reduction (Low rank approximation). However, I quickly run into memory issues in forming the large Block-Hankel matrix and CPU ...
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1 vote
1 answer
338 views

Fast algorithms for computing only the generalized singular values (but not the vectors)

I am interested in computing only the generalized singular values, and was wondering if this was faster (and by how much?) than computing the full GSVD. In particular, I was wondering what the ...
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2 votes
0 answers
204 views

Using Centroid decomposition instead of SVD

This paper says centroid decomposition (CD) is an approximation to singular value decomposition (SVD). First I do not understand CD yet, since code is available I just want to try it out how it works ...
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  • 201
6 votes
3 answers
1k views

Why is Matlab's SVD faster on skinny matrices than on fat matrices?

I noticed something odd today. I have a matrix X that is very skinny (20800 x 200), double precision real numbers, not sparse, ...
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  • 201
8 votes
4 answers
5k views

Incremental SVD implementation in MATLAB

Is there any library/toolbox which has implementation of incremental SVD in MATLAB. I have implemented this paper, it is fast but does not work well. I tried this but in this also error propagates ...
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  • 201
4 votes
1 answer
522 views

Using SVD to biorthogonalize left and right eigenvectors?

I have a set of left and right eigenvectors from an nonsymmetric eigenproblem, and I'd like to biorthogonalize them. I tried Gram-Schmidt, but this fails for most cases. I then read that the SVD is ...
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