Questions tagged [matrices]

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26
votes
0answers
526 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 ...
21
votes
0answers
673 views

Partial circulant matrices: Is there a non-zero vector $v\in \{-1,0,1\}^n$ such that $Mv=0$?

The following question arose as a side product of some work I have been part of recently. An $m$ by $n$ $(0,1)$-matrix $M$ is partial circulant if it can be formed by taking the first $m$ rows of a ...
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, ...
11
votes
0answers
196 views

the largest element of a matrix product

Given two matrices, I'm interested in finding the largest element of their product. I wonder if it's possible to do it significantly faster than the matrix multiplication the solution seems to require?...
11
votes
0answers
1k views

Complexity of finding the leading eigenvector of a graph Laplacian

Let ${\bf L}$ be the $n\times n$ Laplacian of a graph. What is the worst case complexity for calculating the maximum eigeinvector of ${\bf L}$? Are there any families of Laplacians for which it takes ...
9
votes
0answers
539 views

Finding SVD efficiently for $AB^T$

I have a low rank matrix given as $AB^T$ where $A,B \in \mathbb{R}^{n \times p}$ and $p \ll n$. (I know $A$ and $B$ separately) EDIT: (I have added the second question here since it was closed as a ...
8
votes
0answers
189 views

Counting small terms in a determinant calculation over polynomials (counting spanning trees by weight)

I have a $n\times n$ matrix $A$. It's terms are $a_{ij}=-x^{w_{ij}}$ if $i\neq j$ and $a_{ii}=\sum_{j=0}^{n+1} x^{w_{ij}}$ on the diagonal. The matrix is symmetric as $w_{ij}=w_{ji}$. Numbers $w_{ij}$ ...
7
votes
0answers
331 views

An algorithm to compute the number of paths of length at most k

So I had to answer the following question: Given a graph $G = (V, E)$, and two vertices $v_i, v_j$, compute the number of walks between $v_i$ and $v_j$ of length at most $k$. $G$ is not too large, ...
6
votes
0answers
835 views

Best algorithm for inversion of symmetric positive-definite matrices

As I know, the time complexity for usual inversion algorithm of general matrix is $O(n^3)$. In recent years, there is a improvement from $O(n^3)$ to $O(n^{2.8....})$. However, does there exists a ...
6
votes
0answers
180 views

Squares of random binary matrices

Are there results/techniques pertaining to the analysis of squares of random matrices ? More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with ...
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 ...
5
votes
0answers
118 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
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.
4
votes
0answers
101 views

Constructing a small (n,k)-Covering Matrices family

Let ${\cal A}$ be a set of $k\times n $ matrices over ${\mathbb F}_2$. We call ${\cal A}$ a (n,k)-covering, if for every subset of columns $I=(i_1,\ldots,i_k)\subseteq [n]$, there is a matrix $A \in {\...
4
votes
0answers
387 views

Bounding the spectral radius of a sub-stochastic matrix

Suppose that I have a "sub-stochastic" matrix, namely, for an $n\times n$ matrix $A$ with nonnegative entries such that for any $i$, $\sum_j a_{ij}\leq 1$ and there exists some $i$ with $\sum_j a_{ij}&...
4
votes
0answers
110 views

Approximating Front Size of Asymmetric Matrices

The front size of a matrix $A$ is the largest number of non-zeros below the diagonal in any column of its Cholesky factor. If $A$ is symmetric then the minimum front size of $A$ is equal to the ...
4
votes
0answers
559 views

Convergence of HITS Algorithm in the absence of Normalization

Kleinberg's HITS Algorithm (Authoritative sources in a hyperlinked environment) requires the hub (/authority) vector to be normalized after each step of the iteration $x(k+1) = A.x(k)$, in order to ...
3
votes
0answers
46 views

Reference for Nuclear Norm Relaxations

I have seen a bunch of results concerning Matrix Completion, PCA, Compressed Sensing where a common theme has been to relax the Rank constraint/objective by replacing it with Nuclear Norm. I was ...
3
votes
0answers
179 views

An interesting construction of a Tits building?

The notion of Tits building was introduced by Jacques Tits to study certain questions in group theory. The wikipedia entry gives a way to construct a Tits building from a vector space, but I would be ...
3
votes
0answers
49 views

Best complexity bound for parallel matrix-vector product?

I'm looking for the best known complexity (and a bound for the number of processors invoved) to do the calculation of a $(n,n)$ matrix-vector product. Thank you
3
votes
0answers
104 views

Delocalization of eigenvectors in Expanding Graphs

Given an adjacency matrix $A$, can we say something about whether the eigenvectors corresponding to its highest (or second-highest) eigenvalues are de-localized ? By de-localization I mean that ...
2
votes
0answers
63 views

Is there a fast algorithm for computing the Schmidt decomposition

I have a huge covariance matrix, 𝑀, with the dimension, e.g., $10^8 \times 10^8$. Luckily enough, the number of nonzero eigenpairs, $n$, is very small, i.e., $n<5$. From the computational ...
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 ...
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?
2
votes
0answers
372 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 ...
2
votes
0answers
111 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 ...
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. ...
2
votes
0answers
56 views

Parameterized Complexity of Minimum Type Selection

Consider the following problem that I call »Minimum Type Selection«: Input: $k$ sets of bit vectors, each of length $n$ and a number $l$. Question: Is it possible to pick exactly one bit vector from ...
2
votes
0answers
203 views

Computing dual of the spectral norm of tensor of order 3

It is shown in http://www.stat.uchicago.edu/~lekheng/work/jacm.pdf that computing the spectral norm (see Definition 6.6) of a $3^{rd}$ order tensor $T \in \mathbb{R}^{d_1 \times d_2 \times d_3}$ is NP-...
2
votes
0answers
92 views

Convergence of iterated stochastic matrices

(This question has been unsuccessfully asked on mathoverflow: https://mathoverflow.net/questions/161728/convergence-of-iterated-stochastic-matrices) It is well-known that for a stochastic aperiodic ...
2
votes
0answers
890 views

How is the complexity of PCA $O(\min(p^3,n^3))$?

I've been reading a paper on Sparse PCA, which is: http://stats.stanford.edu/~imj/WEBLIST/AsYetUnpub/sparse.pdf And it states that, if you have $n$ data points, each represented with $p$ features, ...
2
votes
0answers
241 views

Running time of Jacobi vs Golub-Kahan for SVD

For an $m \times n$ matrix, what is the running time for computing the Singular Value Decomposition (SVD for short) via Jacobi's method, and Golub-Kahan? The source I read mentions that Jacobi's ...
2
votes
0answers
248 views

normalized tensor rank

Is there such a thing as a "normalized tensor rank" for non-square tensors (i.e. a tensor with different sizes along each mode)? For example: If a 3rd order tensor (dimensions = 60 x 120 x 30) has ...
1
vote
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 $$ \sum_{j}(V)_{ij}=1, (V)_{ij}\ge 0 $$...
1
vote
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 ...
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 ...
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 $k<n$?
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
vote
0answers
97 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
1
vote
0answers
112 views

On Gaussian Weight matrix and rank

Suppose one has a set of $n$ points in $\mathbb{R}^d$, where the points are represented as $p_1$ through to $p_n$. Define the weight matrix $W$ as follows: let $W_{i,j}$ be $e^{-||p_i - p_j ||^2}$, ...
1
vote
0answers
64 views

Can I apply the constraint while constructing the Lagrangian?

Consider the problem: $\min_X ||XAX^T||_F$ s.t. $X^TX=I$ If A and X are real matrices, the lagrangian will be $tr(XAX^TXA^TX^T)+\sum_i\alpha_i(x_i^Tx_i-1)+\sum_i\sum_{j\neq i}\beta_{ij} q_i^Tq_j$ ...
1
vote
0answers
149 views

Row/column overlay problem for filling in a matrix

I was wondering if the following problem has been studied/has a name: Given a matrix of numbers (may assume to be integers), find the smallest number of rules which define the matrix. Three kinds of ...
0
votes
0answers
84 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 $\Bbb F_{3^t}$ ...
0
votes
0answers
100 views

Is there a diagonal matrix D such that DMD is SDD, where M is SPD matrix

Let $M$ be symmetric and positive definite matrix (SPD). It is known [1] that if $M$ is SPD and in addition satisfies $M_{ij}\leq 0$, for $i\neq j$ (called M-matrix) then there is a positive ...
-1
votes
1answer
74 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 ...
-4
votes
1answer
47 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: ...