# Questions tagged [sparse-matrices]

The sparse-matrices tag has no usage guidance.

**3**

votes

**0**answers

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

**1**answer

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**

vote

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**2**answers

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**5**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**9**answers

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

**1**answer

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, ...