Questions tagged [linear-algebra]

Linear algebra deals with vector spaces and linear transformations.

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9
votes
1answer
595 views

Efficiently solve a system of strict linear inequalities with all coefficients equal to 1 without using a general LP solver?

Per the title, other than using a general purpose LP solver, is there an approach for solving systems of inequalities over variables $x_i, \ldots, x_k$ where inequalities have the form $\sum_{i \in I} ...
9
votes
2answers
240 views

Polynomial algorithms for UPB (Unextendable Product Bases)

Consider a Hilbert space $H = H_1 \otimes \dots \otimes H_n$. An Unextendable Product Basis (UPB) is a set of product vectors $\vert v_i \rangle = \vert v_i^1 \rangle \otimes \dots \otimes \vert v_i^n ...
1
vote
0answers
180 views

Complexity of checking whether linear equations have a positve solution [closed]

Consider a system of linear equations $Ax=0$, where $A$ is a $n\times n$ matrix with rational entries. Assume that the rank of $A$ is $<n$. What is the complexiy to check whether it has a solution $...
3
votes
0answers
152 views

Lower Bound Methods in NonDet Communication Complexity

rank+($M$) is the minimum $r$ such that the following statement holds. The statement : there exists matrices $U,V$ such that $M = UV$ and $U$ has $r$ columns and $V$ has $r$ rows. Is rank+($M$) ...
1
vote
1answer
1k views

What is the significance of abstract linear algebra in machine learning/computer vision research?

I am a computer science research student working in application of Machine Learning to solve Computer Vision problems. Since, lot of linear algebra(eigen-values, SVD etc.) comes up when reading ...
15
votes
1answer
1k views

Solving a linear diophantine equation approximately

Consider the following problem: Input: a hyperplane $H = \{ \mathbf{y} \in \mathbb{R}^n: \mathbf{a}^T\mathbf{y} = {b}\}$, given by a vector $\mathbf{a} \in \mathbb{Z}^n$ and $b \in \mathbb{Z}$ in ...
13
votes
1answer
2k views

Matrix multiplication in $O(n^2 \log n)$

I was searching about Matrix multiplication, So I first visit wiki matrix multiplication algorithms, In references I found a paper which claim that uses $O(n^2 log(n))$ algorithm , I'd going to read ...
7
votes
1answer
266 views

determining if a matrix of linear forms represents a non-degenerate matrix

Let $k$ be a field with $p$ elements. Consider the following computational problem Input: a natural number $n$, $n^2$ linear forms $M_{ij}$, $i,j=1,\ldots n$ in $n^2$ variables $X_{11}, \ldots X_{...
7
votes
0answers
130 views

Reference request: reducing rank computations to characteristic polynomials over arbitrary rings

Question. I'm looking into certain algorithms for linear algebra which lie in NC2. Does anyone know of alternative references for the proof of the proposition just below, relating rank of matrices ...
15
votes
4answers
2k views

Definition of matrix-multiplication exponent $\omega$

Colloquially, the definition of the matrix-multiplication exponent $\omega$ is the smallest value for which there is a known $n^{\omega}$ matrix-multiplication algorithm. This is not acceptable as a ...
10
votes
1answer
326 views

What is the largest gap between rank and approximate rank?

We know that the log of the rank of a 0-1 matrix is the lower bound of deterministic communication complexity, and the log of the approximate rank is the lower bound of randomized communication ...
7
votes
1answer
333 views

Complexity to calculate a full set of eigenvectors over a finite field

Let a full rank $n\times n$ matrix ${\bf A}$ with elements over $\mathbb{GF}(2)$. What is the worst case complexity to calculate $n$ linearly independent (over $\mathbb{GF}(2)$) vectors, such that ...
10
votes
1answer
534 views

Can such a matrix exist?

During my work i came up with the following problem: I am trying to find an $n \times n$ $(0,1)$-matrix $M$, for any $n > 3$, with the following properties: The determinant of $M$ is even. For ...
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 ...
13
votes
1answer
307 views

Cases of nearly linear time solvable linear systems

Let a square $n\times n$ real matrix ${\bf A}$ and two vectors ${\bf x}$ and ${\bf b}$ of length $n$, such that $${\bf A}{\bf x}={\bf b}.$$ Solving for ${\bf x}$ through standard Gaussian Elimination ...
7
votes
0answers
193 views

Are the minimal quantum and classical span programs the same?

A span program is a linear-algebraic way of specifying a boolean function introduced here which has found recent application in quantum query complexity. A span program for a function $f: \{0,1\}^n \...
10
votes
1answer
418 views

Finding a cutting plane that splits a polyhedron evenly

Say we have a polyhedron in standard form: \begin{equation*} \begin{array}{rl} \mathbf{A}\mathbf{x} = \mathbf{b} \\\\ \mathbf{x} \ge 0 \end{array} \end{equation*} Are there any known methods for ...
7
votes
0answers
133 views

What's new in sparse eigensystems solution

As a part of other work I need to solve relatively large (~1E5x1E5) and sparse (~100 non-zero elements in each raw in few blocks) hermitian eigensystems. Usually only few eigenvalues+vectors are ...
19
votes
2answers
1k views

A data structure for minimum dot product queries

Consider $\mathbb{R}^n$ equipped with the standard dot product $\langle \cdot, \cdot \rangle$ and $m$ vectors there: $v_1, v_2, \ldots, v_m$. We want to build a data structure that allows queries of ...
3
votes
0answers
323 views

Taking Square Roots of Matrices over Z/nZ

Is it easy (computationally) to take square roots of matrices over Z/nZ, if you know the factorization of n? More specifically, suppose I generate a random matrix M, and square it. Can a ...
5
votes
0answers
261 views

Hardness of finding eigenvalues?

Is there a setting in which finding eigenvalues/eigenvectors is computationally hard? Or at least, not known to be computationally easy? For example, how computationally hard or easy is it to find ...
10
votes
2answers
361 views

Restricting entries of unitary operators to real numbers and universal gate sets

In Bernstein and Vazirani's seminal paper "Quantum Complexity Theory", they show that a $d$-dimensional unitary transformation can be efficiently approximated by a product of what they call "near-...
14
votes
0answers
493 views

Approximation algorithm for Minimum Fill-In and/or minimum elimination ordering (for directed graphs)

Recently while working on a problem, I had to go through some of the literature on nested dissection. I happen to have one (maybe two?) questions related to the same. First, I will define a few ...
18
votes
1answer
553 views

What is the most general structure on which matrix product verification can be done in $O(n^2)$ time?

In 1979, Freivalds showed that verifying matrix products over any field can be done in randomized $O(n^2)$ time. More formally, given three matrices A, B, and C, with entries from a field F, the ...
2
votes
2answers
381 views

LU factorization of a 0-1 matrix

I have a rather naive question on LU factorization which probably should be easy to answer. Say I have a matrix with entries only from $\{0,1\}$. When can we expect to get an LU factorization of such ...
9
votes
0answers
536 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 ...
14
votes
1answer
3k views

What is the fastest algorithm to compute rank of a rectangular matrix?

Given an $m \times n$ matrix (assuming $m \ge n$), what is the fastest algorithm to compute its rank and basis of the columns? I am aware it can be solved through linear matroid intersection, which ...
15
votes
1answer
707 views

Sparse Walsh-Hadamard Transform

The Walsh-Hadamard transform (WHT) is a generalization of the Fourier transform, and is an orthogonal transformation on a vector of real or complex numbers of dimension $d = 2^m$. The transform is ...
9
votes
1answer
2k views

Complexity of Finding the Eigendecomposition of a *Symmetric* Matrix

This is a specialized version of a previous question: Complexity of Finding the Eigendecomposition of a Matrix . For NxN symmetric matrices, it is known that O(N^3) time suffices to compute the ...
8
votes
1answer
281 views

Transitive closure of an affine relation

I am looking for work on computing the transitive closure of an affine relation in the following sense: Let $R(x_1,\dots,x_n,x'_1,\dots,x'_n)$ be the relation defined by a system of linear ...
5
votes
3answers
362 views

Calculating a fast matrix vector product between vector of reals and a 0-1 matrix

Given: some vector $R=(r_1...r_l)$ - real numbers, and a set of distinct vectors with $0$ or $1$ coordinates $$\begin{array}{c} V_1=(c_{1,1} ... c_{1,l}),\\ V_2=(c_{2,1} ... c_{2,l}),\\ .....\...
20
votes
3answers
1k views

Survey on algorithms/complexity of linear algebra

I am looking for a good survey on algorithms and complexity of linear algebra (operations like rank, inverse, eigenvalues, ... for Boolean, $\mathbb{F}_p$, and integers/rationals matrices) with ...
18
votes
2answers
814 views

A Boolean function that is not constant on affine subspaces of large enough dimension

I'm interested in an explicit Boolean function $f \colon \\{0,1\\}^n \rightarrow \\{0,1\\}$ with the following property: if $f$ is constant on some affine subspace of $\\{0,1\\}^n$, then the dimension ...