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Questions tagged [sparse-matrices]

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3
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0answers
83 views

Sparsest similar matrix

Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A? I guess it has to be its Jordan normal form but I am not sure. Remarks: A matrix is sparser ...
1
vote
1answer
69 views

LASSO problem but with a maximization instead of minimization

I have the following optimization problem (like the LASSO problem but with maximization instead of minimization): $\mathbf{maximize}_{\boldsymbol{\alpha}} \|\mathbf{x} -\mathbf{A}\boldsymbol{\alpha}\|...
1
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0answers
76 views

Decomposition of a Matrix by Sparse Matrices

Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over $\mathbb{F}$. $\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$ be at most ...
0
votes
0answers
92 views

Minimizing Frobenius norm with sparsity constraints

I am trying to solve the following minimization problem \begin{equation*} \begin{aligned} & \underset{X,Y \in \mathbb{R}^{n\times k}}{\text{minimize}} & & \| X Y^\top A - B \|_{\text F}^2 ...
2
votes
2answers
101 views

Spectrum of finite-band random matrices?

Let $X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that : $$ \begin{cases} &X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\ & X_{ij} \sim P_X \quad \text{otherwise} \end{cases}$$ And ...
2
votes
1answer
64 views

Maximise singular value decay by sparse matrix approximation

I have a large matrix $A \in \mathbb{R}^{n \times m}$ and would like to subtract a sparse matrix $B \in \mathbb{R}^{n \times m}$ with less than $c (n+m)$ non-zero entries, where $c > 0$ is a ...
1
vote
0answers
85 views

Matrix Sparsity Pattern

Suppose I have a matrix $H^{+} = (H^T H)^{-1} H^T$ where $H$ is a sparse matrix. Consider the case where only the sparsity pattern i.e. the zero elements of $H^{+}$ matrix is known, then would it be ...
7
votes
1answer
197 views

Square root of a large sparse symmetric positive definite matrix

I am trying to calculate $$Y = A^{\frac 12} X$$ where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any ...
0
votes
0answers
111 views

L1 norm constraint on product of 2 matrix

I want to solve below minimization problem \begin{equation*} \begin{aligned} & \underset{A, B}{\text{minimize}} & & ||Y-AB^T -D||_F^2 \\ & \text{subject to} && |A_i|_1 \leq a,...
1
vote
1answer
101 views

Upper bound on the number of non-zero entries of the product of sparse matrices

I have two sparse matrices: $A$ of dimension $m \times k$ and $B$ of dimension $k \times n$. Is there a way to know how many non-zero entries there are in $C = A B$ without computing $A B$? I can ...
1
vote
0answers
63 views

Properties of graphs with Hankel-like adjacency matrix

I am having undirected graphs with adjacency matrices which have a regular Hankel-like form, e.g., $$A=\begin{pmatrix}0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & (6\times 0 \text{ ...
3
votes
1answer
99 views

Expected spectral radius for a sparse Erdős-Rényi binary matrix with a certain density

Let $A$ be an $n \times n$ sparse matrix generated via the Erdős-Rényi method. Here, "sparse" means that $\|A\|_F = O(n)$. I am interested in the relationship between the expectation $\mathbb E(\rho(A)...
2
votes
1answer
101 views

Decomposition of rectangular matrices into a product of a sparse and a small matrices

I would like to construct a rectangular matrix which doesn't have a decomposition into a product of a sparse and a small matrices. It is easy to see that a random matrix doesn't have such a ...
5
votes
1answer
369 views

In a large sparse matrix, how many eigenvalues/eigenvectors are “spurious”?

In a large (possibly above $5000\times 5000$) matrix, the problem of finding all the eigenvalues and eigenvectors can be solved using iterative methods (Arnoldi, Lanczos etc.). However, there seems to ...
6
votes
0answers
233 views

Upper bound for $\|\textbf{D}^{-1}\|$, where $\textbf{D}$ is a matrix with specific sparse pattern

Consider the block matrix given by $$\textbf{D} = \left[ \begin{array}{ccc} \left[ \begin{array}{ccc} D & \ldots & D\\ \vdots & \ddots & \vdots\\ D &...
2
votes
0answers
74 views

Maximal “all zeros” submatrix of a sparse binary matrix

Let $A\in\left\{ 0,1\right\} ^{M\times N}$, where each row of $A$ has at most $d$ components equal to $1$, and $d\leq M\ll N\ll Md$. Question: $\forall n\leq N$, what is $m\left(n\right)$, the ...
2
votes
2answers
273 views

Sparse matrix approximation using only a few dense columns (or rows)

Given a dense matrix $A \in \mathbb{R}^{n \times m}$, with $n < m$, I am interested in finding a good approximation by choosing $s$ rows and zeroing the rest. This leads me to the following ...
2
votes
2answers
120 views

Solving linear system when one eigenvalue is known

I have a huge sparse linear system $Ax = b$ where I know that an eigenvalue/eigenvector pair is $1$ and a vector of all $1$'s. Can this knowledge help me in solving the linear system at all? It seems ...
4
votes
1answer
241 views

Finding high-dimensional correlation matrices that are both sparse and low-rank

Let $\boldsymbol{R}$ be the correlation matrix of $X_i,i=1,\dots,p$ with a large $p\gg q=\text{rank}(\boldsymbol{R})$. Is that reasonable to assume that $\boldsymbol{R}$ is both (approximately) sparse ...
4
votes
2answers
718 views

Decomposing a matrix into a product of sparse matrices

How to study the decomposition of a square matrix into a product of sparse matrices? There are no restrictions on the number of matrices in the product, but the fewer the better.
3
votes
0answers
72 views

Finding large bicliques in random bipartite graph

I want to find a $k$ by $r$ biclique hidden in an $M$ by $N$ random bipartite graph where edges are present with probability $p \in [0,1]$. I am specifically interested in $p \ll 1$, and large values ...
7
votes
2answers
932 views

How can one construct a sparse null space basis using recursive LU decomposition?

Given an $m$ by $n$ matrix $A$ I'm familiar with the standard method to compute a basis for the null space of $A$ by computing a QR factorization of $A^T$. If $A$ is large and sparse, we can use ...
1
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0answers
92 views

Spectrum of a binary markov transition matrix

I would like to know if there are any special analytic expressions or fast numerical methods to get the spectrum for the transition matrix corresponding to a Markovian binary process (Bernoulli ...
8
votes
1answer
558 views

Efficiently compute the trace of a sparse matrix times the inverse of a sparse matrix?

How can I efficiently compute $\mathrm{trace}(A(B^{-1}))$ where $A$ and $B$ are both sparse symmetric PSD $n \times n$ matrices, both with $O(n)$ non-zero entries? If it helps, the pattern of non-...
7
votes
1answer
330 views

Can I find the gap between the two least eigenvalues of this special matrix A(t)?‎

I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse ‎matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal non-...
3
votes
1answer
200 views

Decomposing large symmetric banded sparse matrices

I'm investigating 3D image deblurring and one of the approaches I'm interested in is applying spatial regularisation. To do this I have generated a matrix $A$ which encodes the 6-connectivity of each ...
1
vote
1answer
192 views

Asymptotic eigenvalue analysis for a sparse random matrix

We have an asymptotic analysis problem for the eigenvalue performance of the following random matrix: $H=\{h_{ij}\}_{N_r\times N_t}$, where each entry $h_{ij}$ is with a probability $p$ to obey the ...
6
votes
1answer
162 views

Hamiltonicity criteria for sparse graphs

Given a sparse graph, how can one go about proving that it is Hamiltonian? (Assuming it actually is, of course). There are three main classes of criteria for Hamiltonicity that I am aware of: Dirac-...
1
vote
1answer
327 views

Estimate the determinant of sparse 0-1 matrix

There is a matrix A where each entry is either 0 or 1. Each column has exactly a 1's and each row has at most b 1's. What's the upper bound of abs(|A|)? The condition is stronger than the Hadamard's ...
1
vote
1answer
109 views

Transform $a\mathbf x+\mathbf b$, then make it $k$-sparse, resulting least modification?

Consider vector $\mathbf x = (x_1,x_2,\cdots,x_n)$, $a$ a scalar, $\mathbf b = (b_0,\cdots,b_0)$ and $k < n$. I want to transform $\mathbf x$ ($\mathbf y = a\mathbf x+\mathbf b$) such that if I ...
2
votes
0answers
299 views

Inverse of sparse matrix is not generally sparse [closed]

I have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices). I encountered several times the web pages which states that the ...
3
votes
2answers
774 views

probability of having linearly independent sparse vectors over finite fields

Suppose that there are $i< N$ linearly independent $N$-dimensional vectors $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_i$ over a finite field $\mathbb{F}_q$, whose elements are denoted as $\{0,1,2,...
1
vote
1answer
355 views

sparse binary vector

I want to know if the following problem has been solved max_w w'Rw where the entries of the vector w are binary (w_i= {0,1} )
0
votes
1answer
478 views

$\ell_o$ Minimization (Minimizing the support of a vector)

I have been looking into the problem $\min: \|x \|_0$ subject to$: Ax=b$. $\|x \|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time ...
2
votes
2answers
656 views

Spectra of VERY sparse random matrices

Consider an $n\times n$ random binary matrix $M$ with i.i.d. entries $m_{ij} \sim {\rm Bernoulli}(p)$, where $p = n^{-\beta}$ with $\beta \in (1,2)$. I am interested in the behavior of the singular ...
2
votes
1answer
541 views

minimizing functions over simple matrix inequalities

I'm wondering if anything is known about minimizing convex, not necessarily linear functions subject to "simple" matrix equalities. To be precise, consider the following example: $min \Sigma x_i ln ...
0
votes
0answers
253 views

L1-regularized Least Squares on a matrix with Toeplitz Blocks (not block-Toeplitz)

I am trying to speed up a sparse signal recovery algorithms. My sensing matrix is a set of Toeplitz Blocks, M = [T1,T2,T3,...,Tk] The objective is min ||Mx - b||_2^2 + ||x||1 What I'm actually ...
6
votes
5answers
1k views

Computer algebra system for calculation of characteristic polynomial of sparse matrix

I have a $n \times n$ matrix, for which i need to calculate the characteristic polynomial. The matrix is over $GF(2)$, and $n \approx 10^4$. However the matrix is very sparse, with around $ n $ non ...
2
votes
0answers
260 views

Sparse Eigenvectors for the Discrete Fourier Transform matrix

There are many ways to choose eigenbasis for the Discrete Fourier Transform matrix since it has only $4$ distinct eigenvalues taken from $\{\pm 1,\pm i\}$. Has there been any refereed work that ...
0
votes
2answers
2k views

How to accelerate/avoid multiplication for large matrices in Matlab? [closed]

The setting is here. X: 6000x8000 non-sparse matrix B: 8000x1 sparse vector with only tens of non-zeros d: positive number M: is sparsified X'X, i.e. thresholding the elements smaller than d ...
3
votes
2answers
5k views

Computing the largest eigenvalue of a very large sparse matrix

I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter $w$, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue ...
15
votes
9answers
6k views

Exponential of large matrices

I want to make a diffusion kernel, which involves $e^{\beta A}$, where A is a large matrix (25k by 25k). It is an adjacency matrix, so it's symmetric and very sparse. Does anyone have a ...
5
votes
1answer
529 views

Are there interesting problems involving arbitrarily long time series of small matrices?

Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, ...