All Questions
Tagged with linear-algebra algorithms
127 questions
5
votes
1
answer
303
views
Efficiently computing $\prod_{i=1}^{n} A_i$
Let $k$ be a nonnegative integer, how to compute $\prod\limits_{i=1}^{n} A_i$ quickly and accurately, where $$A_i=\begin{bmatrix}
0 & 1\\
i^k & 1
\end{bmatrix}?$$
I know if $k=0$, we can use ...
- 696
5
votes
2
answers
189
views
Bisymmetric Hadamard matrices
Definitions: An $n\times n$ Hadamard matrix is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal.
A symmetric matrix is a square matrix that is equal to its own ...
- 696
1
vote
0
answers
138
views
Questions on integer matrix multiplication
Question 1:
Given two integer matrices $A$ and $B$, and let $C$ be $AB$.
$C$ can be very big in pratice, so what is the fastest way to compute the statistical data of $C$?
For example,
$$A=\begin{...
- 696
5
votes
1
answer
304
views
Recover unknown vector through shifted argmax queries
$\DeclareMathOperator*{\argmax}{arg\,max}$
I am interested in finding an efficient algorithm for the following problem:
Let $x \in [0,1]^n$ be some vector, with $x_n = 1$.
We want to recover $x$, ...
- 167
2
votes
0
answers
81
views
Degeneracy and the "Linear Degeneracy Testing" problem
The Affine Degeneracy problem is about deciding whether $n$ given points in $\mathbb{R}^d$ (or $\mathbb{Q}^d$) are "in general position". i.e. there is no $d+1$ tuple of points which lies in ...
- 153
0
votes
0
answers
68
views
Optimal top-k column subset
Let $V$ be a set of vectors over $\mathbb{R}^l$, $l\ge 1$, $\pi_i(V)$ be the permutation of vectors in $V$ such that they are ordered by their $i$th component (descending) in order for $\pi_i(V)(\...
- 101
1
vote
2
answers
137
views
Methods to solve for a matrix whose entries satisfy certain properties
(This question is a repost of a deleted question I asked, because the previous version had several elements missing)
Setting
For fixed $N \in \mathbb{N}$, I wish to compute the entries of a matrix $...
- 135
1
vote
0
answers
35
views
Are there any known lower complexity bounds on solving positive semidefinite or positive semidefinite feasibility problems?
I've been trying to attack the problem posted here, about quickly checking if a matrix has any positive semidefinite completions. I suspect that the answer to the question is "no", because ...
- 111
34
votes
3
answers
3k
views
Quickly determining if a matrix has any PSD completion
Given $m$ entries of an $n \times n$ matrix, is it possible to determine in $O(m n)$ time whether there is any positive semidefinite completion?
Slightly more precisely: for simplicity let's assume ...
- 777
2
votes
1
answer
68
views
Generating a random matrix with large spark (i.e., each $k$-tuple of columns is linearly independent)
Let $F$ be a field, and let $m, n, k$ be positive integers. Is there an efficient algorithm to compute a uniformly random $m \times n$ matrix $A$ over $k$ such that each $k$-tuple of columns of $A$ is ...
- 21
3
votes
0
answers
120
views
Checking the generic rank of a matrix
Suppose that $A,B\in M_{p,q}(\mathbb{Z})$ are two rectangular integer matrices of the same size. Suppose that one has a conjecture stating that the rank of the matrix $A+tB$ for Zariski generic values ...
- 16.9k
1
vote
0
answers
59
views
Resconstructing finite planar point sets from projections
What is the smallest length $m$ of a sequence $u_1,\ldots,u_m$ of $d$-dimensional vectors with real entries such that every finite set $X$ of $d$-dimensional vectors with real entries can be ...
- 3,873
1
vote
1
answer
68
views
Given a set of vectors how to pick $M$ such that sum of maximums of coordinates is maximized?
I asked the same on math.Stackexchange.
I have $n$ (say $n = 300$) vectors $v_1,\dots,v_n$. Each of them has $K$ coordinates (say $K = 30$). For vector $v_j$ I denote it's coordinates as $v_{j1},\...
- 1,048
0
votes
1
answer
116
views
Enumerating (i.e. generating one by one) matrices of given rank over a finite field
Let be given positive integers $m,n,r$, with $r \leq \min(m, n)$, and a finite field of $q$ elements $\mathbb{F}_q$.
I'm looking for an efficient algorithm to enumerate (i.e., generate one by one) all ...
- 43
1
vote
0
answers
38
views
Efficient solution to linear matrix equations
A general form for a linear matrix equation can be written as
$$AX + XB + \sum C_iXD_i$$
If $C_i$ and $D_i$ are all 0, then this simplifies into a well known and studied matrix equation that has an ...
- 11
1
vote
0
answers
134
views
Non-negative least squares: how bad is this heuristic?
The non-negative least squares (NNLS) optimization problem is as follows: for given $A \in \mathbb{R}^{n \times m}$, $y \in \mathbb{R}^n$, find $x \in \mathbb{R}_{\geq 0}^m$ that minimizes $||Ax - y||...
- 5,590
2
votes
3
answers
400
views
Matrix-free linear solve for nullspace
I'm looking for an algorithm to solve for the classic:
$$A\mathbf{x} = \mathbf{b}$$
I cannot compute $A$ directly, but rather can compute matrix-vector products $A\mathbf{v}$ for any $\mathbf{v}$.
At ...
- 141
2
votes
2
answers
330
views
Polynomial time algorithm for rigid graph isomorphism
We found, implemented and tested algorithm
for graph isomorphism and it appears to be polynomial
time if the graph is rigid.
Q1 Is the algorithm below correct and polynomial time for rigid graphs?
A ...
- 25.4k
3
votes
0
answers
148
views
Linear combinations of special matrices
I am a hobby computer scientist and I have a problem to which I am searching an efficient algorithm.
Given an integer n, we want to combine some square input-matrices of size n in a way that is ...
- 73
4
votes
1
answer
209
views
Finding a binary variable assignment to make a matrix with variables singular (over F_p)
Consider a square matrix defined over a finite field $M\in\mathbb{F}_p^{n\times n}$ having the following form
$$M=\begin{bmatrix}a_{11}+b_{11}x_1&a_{12}+b_{12}x_1&\dots&a_{1n}+b_{1n}x_1\\...
- 153
1
vote
0
answers
209
views
Solution to system of linear equations
Input: System of linear equations $$A[x_1,\dots,x_{t}]=b$$ where number of equations is at least number of variables but independence is not guaranteed. However there is atmost one non-negative ...
- 13.9k
2
votes
1
answer
164
views
The "best way" to order unknowns in linear systems
Start with a linear system of the form
\begin{equation*}
Ax + Bt + C = 0,
\end{equation*}
where $x = (x_1, \dots, x_n) \in \mathbb R^n$ is the vector of unknowns, $t \in \mathbb R^m$ is a vector of ...
- 41
2
votes
1
answer
241
views
How to solve this set of equations as efficiently as possible (with "efficiently" measured in FLOPS)?
The system of equations is the following:
$$
\Gamma_i^{\ -1} = \sum_{i=1}^nA_{ij}\Gamma_j,
$$
where $\Gamma = (\Gamma_i)$ is a vector of size $n$ and $A$ is a matrix of size $n\times n$, with $n \gt ...
- 21
0
votes
0
answers
41
views
Iterative algorithm for obtaining similarity
Let $x_1,x_2,\ldots,x_M$ be $M$ non-negative variables. Moreover, assume that $f_m(x_m)=\frac{x_m}{1+\sum_{n}\beta_{n}^{(m)}x_n}$ be $M$ fractional functions with non-negative constants $\beta_{n}^{(m)...
- 287
1
vote
0
answers
121
views
Algorithm for the nilpotence of matrix polynomials
Let $P$ be a multivariate polynomial of real-valued $N \times N$ matrices. Given $X_1, X_2, ..., X_M \in \mathcal{M}_N\{\mathbb{R}\}$, is there any optimal algorithm to determine whether the result of ...
- 19
1
vote
1
answer
52
views
Reference Request: Randomly Generated Contraction
Let $n_1>n_2\geq 1$ be integers. Are there a known algorithms for generating $n_2\times n_1$-dimensional random matrices $A$ such that
$$
\|Ax - Ay\|<\|x-y\| \mbox{ if $x\neq y$}?
$$
- 5,405
5
votes
0
answers
149
views
Iterative method for solving certain systems of linear equations
I've noticed that a method for calculating the stationary distribution of a finite-state rational-transition-probabilities Markov chain introduced by Arthur Engel many years ago generalizes to give an ...
- 19.7k
1
vote
0
answers
78
views
Reference for the algorithm to find the intersection between a subspace and positive orthant
I came across this algorithm, in this question Algorithm for the intersection of a vector subspace with a cone of non-negative vectors ;
Is there any reference for the algorithm described in the ...
2
votes
1
answer
591
views
Intersection of a vector subspace with a cone
Given a set of vectors $S=\{v_1, v_2,...,v_d\} \subset \mathbb{R}^{N}, \, N>d$, is there any algorithm to decide if there exist a vector with all coordinates strictly positive in the generating ...
3
votes
1
answer
836
views
Solving multilinear equations
Let $N=\{1,2,\ldots,n\}$. Suppose we are given $n$ equations, with each equation taking the form $\sum_{A\subseteq N}\left(c_A \prod_{i\in A}x_i \right) = 0$, where each $c_A$ is a real number ...
- 239
2
votes
1
answer
456
views
Integrality certification for product of two matrices $A B^{-1}$
Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...
- 149
0
votes
0
answers
81
views
Constructing set with maximal independent subset
What is the minimal $m$ such that there exists a set $A = \{a_1,...a_n\}$ of vectors : $a_i \in \{0,1\}^m$ ($n$ is given) such that every subset of vectors of size $k$ is independent, but only with ...
- 51
2
votes
0
answers
288
views
3-uniform hypergraphs and their circuit space
So, I'll break this post into two questions. Both concern 3-uniform hypergraphs. A 3-uniform hypergraph $H=(V,E)$ consists of a set of vertices $V$ and a set of edges $E$, where each edge $e\in E$ is ...
- 21
2
votes
0
answers
84
views
Given positions find the symmetry group
Given a finite set of vectors in $\mathbb{R}^n$ ($n=2,3$),
is there any algorithm to find its symmetry group?
For example, if the input is {(1,0),(0,1),(-1,0),(0,-1)}, then the output is the dihedral ...
1
vote
1
answer
221
views
How to find all minimal dependent sets of a set of vectors effectively?
In my research, I need to find the set of all minimal dependent sets of a given set of vectors. One method is to check every subset of the given set. But this method is very slow when the set of ...
- 6,201
1
vote
0
answers
111
views
Hutchinson-type algorithm for efficient computation of trace of inverse of non symmetric matrix
Let $A$ be an invertible $N$-by-$N$ matrix, for some large $N$ (say $N = 10^6$). Suppose the only thing we know how to do is apply $A$ to a vector, i.e compute matrix-vector products $Az$.
Question....
- 6,853
5
votes
1
answer
839
views
Row-based iterative algorithms for computing the kernel of a matrix
Suppose $A$ is an $m \times n$ matrix in the form
$$A=\begin{pmatrix} — a_1 —\\ — a_2 —\\ \vdots \\ — a_m — \end{pmatrix}$$
where $a_i \in R^n$ is the $i$-th row of $A$. I know that it is possible ...
- 51
0
votes
0
answers
52
views
How do I test two square matrices are transpose to each other if only the column vector summations are known?
Given two secret square matrices, say $\left( {\begin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\
{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\
{{a_{31}}}&{{a_{32}}}&{{a_{33}}}
\...
- 33
7
votes
1
answer
503
views
Cycle types of permutations from affine group
Let $V$ be a vector space of dimension $n$ over the field $F=\mathrm{GF}(2)$. We identify $V$ with the set of columns of length $n$ over $F$. Let $G = \mathrm{AGL}(V)$ be a group of affine ...
4
votes
1
answer
96
views
Separate the trivial partition by a linear hyperspace
Let $e=[1,1,\ldots,1]\in\mathbb{Z}^n$. I am looking for a way to find a vector $a\in\mathbb{Z}^n$ such that:
$\langle a,e\rangle=0$ and
for every nonnegative $v\in\mathbb{Z}^n$ such that $\langle e,v\...
- 3,041
4
votes
2
answers
734
views
Fast projection onto a subspace
Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{...
- 1,677
2
votes
0
answers
99
views
Recovering a rank-one matrix from its eigendecomposition after randomized rounding [closed]
Let $A = xy^T$ be a rank-$1$ matrix, and suppose every entry of $A$ is in $[0,1]$. We can create a binary matrix $A_{\rm rounded}$ by setting
$$ [A_{\rm rounded}]_{ij} = \begin{cases} 1 & \mbox{ ...
- 121
11
votes
0
answers
764
views
Fast computation of matrix product $AXA^T$ with fixed $A$?
Suppose we have two $n$-by-$n$ matrices $X$ and $A$, where $A$ is known and $X$ may change in different invocations, and we want to compute $AXA^T$. Is there an algorithm that beats the naive one of ...
- 75
3
votes
1
answer
99
views
Lattice basis reduction over rings of number fields
Can one use lattice basis reduction algorithms, such as LLL over (low-rank) module lattices over rings of number fields of degree greater than 1? Is there any work on lattice reductions over Euclidean ...
- 1,099
0
votes
1
answer
169
views
Kernel vectors with given number of non-zero entries?
Let $A$ be a $n\times n$ real matrix. Is there a vector $\vec x \in \mathbb{R}^n$ with exactly $0 \le k < n$ zero entries such that $A \vec x = 0$?
Is there an efficient algorithm to tackle this ...
- 884
10
votes
1
answer
1k
views
Cryptographic Secret Santa
Is there a protocol for conducting a Secret Santa without a central authority? Precisely, we want to sample uniformly a permutation that has no one-cycles and reveal to each member his or her ...
- 101
0
votes
1
answer
917
views
What is the most accurate and efficient method of finding an inverse of a hessian matrix?
For any hessian matrix, of say 300 by 300, and may or may not necessarily be positive semi-definite, thus algorithms such as Cholesky decomposition may not be used.
I've found that some algorithms ...
- 17
1
vote
0
answers
174
views
Approximating symmetric matrices by symmetrized low rank matrices
Fix an integer $k$, and suppose $M$ is a real symmetric $n\times n$ matrix admitting a decomposition:
$$
M = A + A^t + B
$$
with $\mathrm{rank}(A)=k$ and:
$$
\|B\|_2 \ll \lambda_{1}(M_{|\mathrm{range}(...
- 121
2
votes
0
answers
113
views
Complexity of tensor decomposition vector over $\Bbb F_q$ or $\Bbb Z$
Suppose we have a matrix $$T\in\Bbb K^{n^k\times m}$$ and a target vector $v\in\Bbb F_q^m$ where $m<n^k$ and $1<k$ holds.
We need to find $k$ vectors $u_1,\dots,u_k\in\Bbb K^n$ such that $$v=...
- 13.9k
1
vote
1
answer
216
views
Matrix golf puzzle: enumerate a series by matrix multiplication
It is super easy to find matrices $X_0$, $F$ and $H$ such that $H F^n X_0$ is equal to $n$-nth element of the sequence $0,1,0,1,0,1,0,1,0,1,0,1,...$
Maybe it is a slightly harder challenge to find ...
- 67