Questions tagged [matrices]

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-6
votes
0answers
60 views

Complexity of finding largest non-vanishing minor?

Rank of a matrix is given by $SVD$. What is the complexity of finding largest submatrix whose dimensions agree with rank? This is the largest non-vanishing minor. What is the complexity of deciding ...
-3
votes
1answer
49 views

Finding a non-negative solution to an integer system of linear equations

Let $A$ be an $n \times m$ integer matrix and consider the system of equations $Ax = b$ where $b$ is an integer vector. I want to find a solution $x$, assuming one exists, such that the components of $...
0
votes
1answer
166 views

Time complexity for multiplying two lower triangular matrices

I was wondering, if multiplication of two $n \times n$ lower (or upper) triangular matrices has a more efficient algorithm than multiplication of two general $n \times n$ matrices? $$ \begin{bmatrix} ...
-4
votes
1answer
39 views

Distinguish Graph from Tree using Adjacency Matrix

Given an adjacency matrix, is there a way to determine if the graph will be a tree or a graph (whether or not there is a cycle). For example, given the adjacency matrix: ...
7
votes
1answer
135 views

The complexity of the permanent of low rank matrices

I know that for an arbitrary $n \times n$ matrix, Ryser's algorithm can compute the permanent in $\mathcal{O}(2^n n^2)$ time. I'm interested in computing the permanent of $n \times n$ matrices of rank ...
40
votes
3answers
5k views

Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

It is commonly believed that for all $\epsilon > 0$, it is possible to multiply two $n \times n$ matrices in $O(n^{2 + \epsilon})$ time. Some discussion is here. I have asked some people who are ...
23
votes
2answers
644 views

Question about two matrices: Hadamard v. “the magical one” in the proof of the sensitivity conjecture

The recent and incredibly slick proof of the sensitivity conjecture relies on the explicit* construction of a matrix $A_n\in\{-1,0,1\}^{2^n\times 2^n}$, defined recursively as follows: $$A_1 = \begin{...
2
votes
0answers
39 views

Minimize The Number of Connected Components in Hit-map of A Boolean Matrix

Suppose there is a matrix with the value of 0 and 1. The hit-map of the matrix (0 is blue and 1 is red) create some connected component (see the following figure as an instance): Is there any ...
1
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0answers
90 views

Complexity of low-rank matrix factorizations with rows in a simplex and outliers

Our goal is to obtain a matrix factorization in form of $M = U V'$, where $U\in\mathbb{R}^{d\times r}, V \in\mathbb{R}^{N\times r}$ and each row of $V$ satisfies $$ \sum_{j}(V)_{ij}=1, (V)_{ij}\ge 0 $$...
-4
votes
1answer
100 views

Ordering of sub problems in dynamic programming

1) Can every dynamic programming question be solved using 3 different orderings or can there be more than 3 or less than 3 ( like unique ordering )? My understanding is that a) it might have a unique ...
4
votes
0answers
150 views

research problem for undergraduate majoring in Computer Science and Mathematics

I am wondering if somebody can suggest a small research problem for undergraduate majoring in Computer Science and Mathematics. Would love to work with random matrices or wavelets.
1
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0answers
59 views

Finding a Minimal k-Subgraph

Given a complete, positively weighted, bidirectional graph with $n$ nodes without self-loops. Hence the corresponding adjacency matrix $A$ is positive, symmetric, and has zero main diagonal. I am ...
0
votes
1answer
221 views

Open/unsolved problems in (computational) random matrix theory / matrix completion?

I was wondering what some open problems are in random matrix theory (especially those of interest to TCS people/so mainly non-asymptotic things, I imagine). Also, and relatedly, what are remaining/...
2
votes
0answers
62 views

Efficient Online Algorithms for Matrix rank

Suppose, we have a matrix with known rank. Then, in each step one element of this matrix will be changed. Is there any efficient online algorithm to find rank of each matrix after each element update?
6
votes
1answer
227 views

Algebraic account of Gaussian elimination?

For fun, I've been looking at the interpretation of linear logic in terms of finite-dimensional vector spaces, and ran into an interesting question about the interpretation of double-negation-...
1
vote
0answers
128 views

Connection between diamond norm and output purity norm

Setting of the problem: Given a quantum channel $\mathcal{E}: \mathcal{H}_A\rightarrow \mathcal{H}_B$ (where $\mathcal{H}$ refers to a Hilbert space and subscript refers to the quantum register ...
0
votes
0answers
83 views

On permanent mod $3^t$ computation

By $'$ I mean transpose. I gather the info here from rjlipton.wordpress.com/2014/12/21/modulating-the-permanent. We know that if $U\in\Bbb F_{3^t}^{n\times n}$ satisfies $UU'=I_n$ in $\Bbb F_{3^t}$ ...
2
votes
1answer
164 views

Complexity of $\{0,\pm1\}$ determinant in sparse cases?

If $M\in\{-1,0,+1\}^{n\times n}$ be a matrix with only $O(n)$ non-zero entries and hadamard product $M\odot M$ being symmetric can we compute $Det(M)$ in $O(n)$ bit complexity? Assume that the matrix ...
2
votes
0answers
346 views

Is there any algorithm to find just the largest eigenvalue with subquadratic time complexity?

SVD or PCA can be used find the largest eigenvalue, but at a cost of $O(n^3)$ complexity. Lanczos algorithm runs much faster on a sparse matrix with complexity $O(dn^2)$ where $d$ is the average ...
26
votes
0answers
520 views

Rank mod 6 vs rank over the reals

Let $A$ be a boolean matrix (eg with $0,1$ entries). Assume that $A$ has rank $\le r$ both over $\mathbb{F}_2$ and over $\mathbb{F}_3$. Does this imply that $A$ has low rank over the reals? This seems ...
6
votes
1answer
132 views

What's the complexity of factoring over a set of generators (say in $GL_2$)?

In particular, if I have some char-0 field $k$ (let's take $\mathbb C$ for now) and I consider $G = GL_2(k)$ with arbitrary nontrivial distinct $A, B \in G$. Then for some $C \in GL_2(k)$ do we know ...
7
votes
1answer
243 views

Using an oracle to find a vector $b$ for which $Ax=b$ has a solution

There is an oracle built around a hidden $m\times n$ matrix $A$ all of whose entries are 0 or 1, where $m>n$. The oracle takes as input an integer vector $b$ with positive entries, and answers as ...
5
votes
0answers
202 views

On permanent of $\{\pm1,0\}$ matrices

Consider the problem of computing the permanent $Per(M)$ of a matrix $M\in\{0,-1,1\}^{n\times n}$ such that the result is bounded in absolute value, $|Per(M)|<B$ where $B$ is part of input. Is ...
1
vote
1answer
94 views

Properties of convex polytope of 0-1 matrices

Problem setting Consider a set $ S = \big\{ 1,2,\cdots,n \big\}$. Now consider $k$ equal-sized subsets $S_i \subset S$ s.t of size $\big|S_i\big|=n' \;\forall i$. Consider a $k\times k$ matrix $M$ ...
2
votes
1answer
253 views

Complexity involving connected components of 0/1 matrix

Assume a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where each step you take you ...
1
vote
1answer
94 views

reference request- property of subset of rows in a matrix

I am interested in the following quantity. Suppose we are given a matrix $M\in \mathbb{F}_2^{m\times n}$ and a string $z\in \{0,1\}^n$. I am interested in finding the largest subset $S$ of rows in M ...
2
votes
1answer
95 views

Low-depth arithmetic complexity of the product of $k$ matrices

Is anything known about the size of $\Sigma\Pi\Sigma$ (or other constant depth) circuits for the product of $k$ matrices? Trivial upper bounds (up to small factors) are: if $k=2$, then there are $\...
3
votes
3answers
191 views

Canonisation of boolean matrices under row and column permutations

Consider the equivalence relation $\sim$ on boolean matrices $A,B\in\{0,1\}^{m\times n}$ which is defined as follows: $A\sim B$ :iff there are permutation matrices $P\in\{0,1\}^{n\times n}, Q\in\{0,1\...
2
votes
0answers
109 views

On FFT and trigonometric matrix eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots ...
6
votes
1answer
217 views

Checking properties of matrices

Given a sparse matrix $A$ in $\mathbb{Z}^{n\times n}$, how easily could one check whether a coefficient $\alpha_k$ of the characteristic polynomial $P_A$ of $A$ is equal to $0$ (without the need to ...
1
vote
0answers
146 views

How to efficiently generate a random 0-1 matrix of a given rank

How to efficiently generate a random $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$?
2
votes
1answer
396 views

Matrix multiplication with transpose

Let $A,B\in\mathbb{F}^{n\times n}$ be two $n\times n$ matrices over the underlying field $\mathbb{F}$. In addition, $A$ is guaranteed to be a symmetric matrix, i.e, $A=A^{T}$. We assume complexity ...
1
vote
1answer
135 views

Low rank approximation of matrix under $l_2$ norm

Theorem 14 of this paper by Tam´as Sarl´os gives a relative error rank-$k$ approximation of a given matrix $A$ under the frobenius norm. I am looking for reference of a similar result (relative error ...
13
votes
4answers
913 views

Finding the sparsest solution to a system of linear equations

How hard is it to find the sparsest solution to a system of linear equations? More formally, consider the following decision problem: Instance: A system of linear equations with integer coefficients ...
3
votes
2answers
264 views

Submatrix of small rank

Let $G=(V,E)$ be a graph with adjacency matrix $M=(m_{ij};i,j \in V )$ over $\mathbb{F}_2$ and $k \in \mathbb{Z^+}$. How can we find in polynomial time a subset $A \subseteq V$ such that The rank of ...
14
votes
0answers
591 views

Does solving matrix multiplication in quadratic time imply that SETH is false?

I have a little conjecture that if you could perform matrix multiplication (or solve 3-clique) in $O(n^2 \log(n))$ time, then you could solve CNF-SAT in $O(2^{(1-\epsilon)n})$ time. In other words, ...
5
votes
2answers
1k views

Matrix permanent is 0

Valiant's theorem says that computing the permanent of an $n\times n$ matrix is #P-hard. Is the problem of determining if a permanent is 0 any easier? This arises in the context of sequence A006063 in ...
0
votes
1answer
185 views

Graph isomorphism problem with invertible adjacency matrices

This question is supplementary to the question asked here. One of the answers give a class of graphs for which the adjacency matrices are invertible which is defined as follows. Given a ...
10
votes
2answers
268 views

Matrix vector multiplication algorithm using minimal number of additions

Consider the following problem: Given a matrix $M$ we want to optimize the number of additions in the multiplication algorithm for computing $v \mapsto Mv$. I find this problem interesting ...
6
votes
1answer
218 views

Arithmetic complexity of matrix powering

Assume $M\in\Bbb Z_{\geq0}[x_1,\dots,x_n]^{m\times m}$ be an $m\times m$ matrix in $n$ variables. We know that size of smallest formula that computes $\mathsf{Tr}(M^d)$ where $d\in\Bbb N$ could be ...
3
votes
1answer
321 views

“How much diagonal” a matrix is

I have a (biological) computational system that outputs squared matrices. These matrices will sometimes have a tendency to be diagonal-like, with higher values at and around the diagonal. I would ...
9
votes
1answer
487 views

Complexity of k-clique for hypergraphs

Classic Problem: Let a number $k$ be given. The $k$-clique problem is as follows. Given a graph $G$, does there exist a subset $S$ of $k$ vertices so that any two vertices of $S$ are adjacent? ...
2
votes
0answers
121 views

Complexity of a particular determinant

Suppose we have an $n\times n$ matrix $A$ with non-negative integer entries such that $\mathsf{Tr}(A^i)=0$ at every $i\in\{1,2,\dots,n-2,n-1\}$ and $\mathsf{Tr}(A^n)\neq0$, then from Trace-Determinant ...
2
votes
0answers
42 views

Products of PSD matrices that equal an orthogonal matrix

You have k symmetric real n-by-n matrices A_1...A_k, each bounded by some parameter eps < 1/2 in spectral norm. You know that the product (Id+A_1)(Id+A_2)...(Id+A_k) equals an orthogonal matrix Y. ...
10
votes
2answers
349 views

Exact formula for the number of spanning trees of a rectangle

This blog talks about generating "twisty little mazes" using a computer an enumerating them. The enumeration can be done using Wilson's algorithm to get the UST, but I don't remember the formula for ...
3
votes
1answer
127 views

Minimal set of Hyperrectangles covering an n-dimensional binary matrix with row permutations

My input is a n-dimensional binary matrix. My goal is to find the set of Hyperrectangles that covers every '1' at least once and covers not a single '0', which has minimal cardinality (the least ...
1
vote
0answers
72 views

Matrix of ranking

In any graph of n nodes in any dimension, define matrix $M_r$ of ranking as $\forall r_{ij}\in M_r$, $r_{ij}$ is the ranking of j to i. That is, j is the $r_{ij}$th nearest node to i. Therefore, any ...
-1
votes
1answer
73 views

Find non-intersecting submatrices

I have a rectangular boolean matrix and I'd like to have an efficient algorithm to find non-intersecting submatrices. I'll to demonstrate that in the example below. The ideal case is when all ...
3
votes
1answer
290 views

Rate of convergence for the Perron–Frobenius theorem

The Perron–Frobenius Theorem states the following. Let $A = (a_{ij})$ be an $n \times n$ irreducible, non-negative matrix ($a_{ij} \geq 0, \forall i,j: 1\leq i,j \leq n$). Then the following ...
3
votes
1answer
168 views

Low rank Log rank conjecture

What is known about log rank conjecture in special situations of $O(\log N)$ rank $0/1$ matrix of size $N\times N$? Is there at least a conditional result showing better than $O(\sqrt{\log N})$ bound?