Questions tagged [linear-algebra]

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3
votes
1answer
546 views

Positive Definiteness Constraint

I want to add a constraint to a convex program, to guarantee some matrix $A$ to be positive semidefinite. How should I do it? The library I am working with can cope with linear/ quadratic ...
3
votes
1answer
237 views

Derandomization of an approximation algorithm for solving a linear system

I was given a HW assignment that asks me the following: Given a system of $m$ linear equations in variables $x_1,x_2,...,x_n$ over $\mathbb{F_p}$, find a randomized algorithm that find an assignment ...
3
votes
0answers
53 views

Heuristic for making set of indexes in an array/matrix with generating functions/patterns

I am trying to find a lead on how to solve or find a heuristic the following kind of problem: Given an array/matrix with entries of only 1s and 0s, using a set of looping functions/patterns of a ...
2
votes
1answer
328 views

Can the unbounded knapsack problem be described as a matrix exponentiation?

It seems that the general approach to a dynamic programming problem is to formulate a recurrence relation and then either implement a top down recursive solution or a bottom up iterative solution. ...
2
votes
1answer
932 views

Locality-sensitive hashing random projection

I'm trying to understand how the LSH works for Cosine Similarity metric. For instance, let's say you have $\vec{v} \in \mathbb{R}^d$ and the random vectors $\vec{r_{i}} \sim \mathcal{N}(0, 1)^d$ that ...
4
votes
1answer
992 views

Closed form solution for optimization problem

Consider the problem of finding the real-valued matrix $C$ so that $$\|S-AC\|_F^2\qquad(1)$$ is minimal. ($S$ and $A$ are real valued matrices and $_F$ denotes the Frobenius norm). This problem has ...
2
votes
1answer
42 views

Proving complexity of computing product of matrices

If $A$ is a non-singular $n\times n$ matrix, $B$ is an $n\times p$ matrix, and $C$ is a $p\times n$ matrix (where $1\le p \ll n$), how does one prove that the complexity of $$D=A^{-1}(BC)$$ is $\frac{...
1
vote
1answer
1k views

Complexity of matrix inverse via Gaussian elimination

I'm trying to determine the exact complexity of finding an $n\times n$ matrix inverse of $A$. If it is known that the complexity of Gaussian elimination is $\frac{2}{3}n^3 + \frac{1}{2}n^2+O(n)$, then ...
2
votes
0answers
148 views

Computing $\mathrm{tr}(X^{-1}Y)$ efficiently

I know that one can compute the expression $X^{-1}\mathbf{v}$ quickly with conjugate gradient method. Is there a similar approach for computing $\mathrm{tr}(X^{-1}Y)$? Similarly interesting to me are ...
-1
votes
1answer
69 views

A max-even subset problem

I want to know if there is any polynomial algorithm for the problem, or any NP-completeness result. Given a set $S$ and $m$ subsets $C_1, \dots, C_m$ of $S$, we want to find a non-empty set $X\...
2
votes
1answer
315 views

Permutation on matrix to fill main diagonal with non-zero values

I am currently working on some sparse non-singular matrices. One of the algorithms I use requires divisions by the elements on the main diagonal so I have to ensure that my main diagonal is filled ...
4
votes
0answers
67 views

Is this a known question in matrix sketching?

Say one has a $D \times n$ matrix $A$ all of whose entries are non-zero. One wants a method which will look at each of the columns of $A$ one by one and create new $m \ll D $ dimensional columns and ...
3
votes
1answer
214 views

How to construct a running kd-tree?

I have a stream of 3-tuples of type (x,y,t) where x and y are in the range ...
5
votes
1answer
980 views

Computing Von Neumann Entropy Efficiently

The Von Neumann entropy $S$ of a density matrix $\rho$ is defined to be $S(\rho)= -\text{tr}(\rho \lg \rho)$. Equivalently, $S$ is the classical entropy of the eigenvalues $\lambda_k$ treated as ...
4
votes
0answers
204 views

Intuitive self-contained proof of Farkas' Lemma

I've been studying the proof of Farkas' Lemma, and given my rather fuzzy memory of Linear Algebra, am having some trouble with it. One version of Farkas' lemma states: For any convex cone generated ...
7
votes
1answer
776 views

It is possible to implement a *greater than* function using only addition, substractions and multiplications?

All values are from a finite field $Z_t$. I want to write a function greater than like this $GT(x,y) = \begin{cases} 1, & \text{if } x > y, \\ 0, & \text{otherwise}. \end{cases}$ ...
5
votes
1answer
275 views

Solving systems of linear equations over semirings

So I have come across an issue where it would be very nice to solve systems of linear equations over semirings but I have no clue how to do that. Over a field I would use Gaussian elimination but I'm ...
0
votes
1answer
40 views

Using Taylor series in 1D Grayscale Image

Could someone point me in the direction of how to solve this? I = [I1, . . . , In] is a 1D grayscale image and D = [D1, . . . , Dn] represents the second derivative of I. I am given the four pixel ...
3
votes
1answer
76 views

Rank of random binary matrix subset

I have a problem that smells like it is NP-complete, but at the same time it feels like maybe you can solve it by just keeping track of column-wise Hamming distance or something, or that it's ...
1
vote
1answer
82 views

Properties of coset

Let $C$ be a linear code with minimum distance $2k$. I want to show that there is a coset of $C$ that contains at least two vectors of weight $k$. Firstly, it holds that the minimum distance of the ...
6
votes
1answer
124 views

How to compute $\mathbf{X}^T \mathbf{X}$ efficiently for large $\mathbf{X}$?

Let $\mathbf{X}$ be a $n \times n$ matrix. Given that we can only keep $k$ rows ($k << n$) or columns of the matrix in memory, how can we compute $\mathbf{X}^T \mathbf{X}$ while minimizing the ...
2
votes
1answer
126 views

Linear equation solving with special sparse coefficient matrix

Given a linear equation system of $n$ equations with unknowns $a_1,a_2,...,a_n\in [0,1]$, where the left hand side of each equation consists of not more than $k$ variables (so there are at least $n-k$ ...
8
votes
1answer
672 views

Minimal basis for set of binary vectors using XOR

I would be surprised if this isn't a well-studied problem, but I'm not sure what else to search for at this point: you're given a set of binary $n$-vectors $S \subset \{0,1\}^n$. The problem is to ...
1
vote
0answers
77 views

Relation between determinant and matrix multiplication

I remember reading somewhere that if matrix multiplication can be done in $O(n^\omega)$ time then determinants can be computed in $O(n^\omega)$ time. I am unable to find the reduction and would it be ...
3
votes
2answers
351 views

Solving/Optimizing a linear system in a finite field (Z/2Z)

I'm trying to solve the following optimization problem. A is a rectangular matrix with coefficients in the finite field Z/2Z (size less than 1000 X 1000). I have a system of the form A.X = Y (X and Y ...
4
votes
1answer
311 views

Number of solutions to linear system of equations over GF(2)

Linear systems of equations over the reals have either 0, 1 or infinitely many solutions. However, when applied to finite fields (specifically GF(2)), infinitely many is not an option. Is there a ...
1
vote
1answer
134 views

Scheduling distributed computational graph

I work in computational fluid dynamics. And I spend most of my time waiting for simulation to complete. The common way to improve simulation performance is to use a suitable distributed linear ...
9
votes
1answer
2k views

Inverting a band matrix

I have a band matrix -- a sparse, square, symmetric $N \times N$ matrix whose structure looks like the following: Here, the area under the blue stripes is the non-zero elements; everything else is ...
1
vote
0answers
41 views

2 qubits, correct way CNOT a 3rd qubit with the 1st? [duplicate]

Suppose I have 2 qubits in the state a|00>+b|01>+c|10>+d|11>. And suppose I want to perform some operation between only the 1st qubit and a 3rd qubit - for example a CNOT operation. What would be ...
4
votes
1answer
79 views

Unfeasible linear program becomes feasible if a variable is removed

Apologies, not a computer scientist by trade but I'm playing with linear programming these days. Let $\{x_i\}$ be $N$ optimization variables with bounds $$l_i \leq x_i \leq u_i$$ I'm interested in ...
5
votes
1answer
73 views

Efficient algorithms for dealing with linear algebra over the rationals

I'm an academic mathematician. While trying to verify a counterexample to something in my research, the following computational problem has arisen: Fix a vector space $V = \mathbb{Q}^n$. Here $n$ ...
1
vote
0answers
35 views

Matrix whose eigenvectors are Hermite polynomials [closed]

I first constructed a symmetric matrix as the Laplacian operator, and its eigenvectors are a series of harmonics functions as expected. I programmed it and convinced myself. The matrix looks like: $$ \...
22
votes
4answers
1k views

Automated optimization of 0-1 matrix vector multiplication

Question: Is there established procedure or theory for generating code that efficiently applies a matrix-vector multiplication, when the matrix is dense and filled with only zeros and ones? Ideally, ...
0
votes
1answer
153 views

Algorithm for projection of polytope

Let a convex bounded polytope be given by an intersection of half planes: $Ax \leq b$. Let $z=Cx$ be a vector (in my case $z$ is 2-dimensional, while $x$ has a higher dimension). How can I compute $D$...
0
votes
0answers
415 views

Most stable algorithm to solve a system of linear equations?

I am doing some image processing involving solving a system of linear equations. I am getting some errors and bits of the image looks corrupted. I would like to know what is the most stable way to ...
3
votes
1answer
258 views

Devising an Algorithm for Linear Combination with Column Restrictions

Application: We intend to factor an integer $N$ using a variation of the rational sieve. This involves constructing a congruence of squares modulo $N$ from a set of linear relations $$x - N = y$$ ...
5
votes
1answer
279 views

Intuitive idea/proof behind Kirchhoff's Matrix Tree Theorem using as little matrices/linear algebra as possible?

could someone provide me/refer me to a intuitive idea/proof behind Kirchhoff's Matrix Tree Theorem that uses as little technical details involving matrices/linear algebra as possible? I'm trying to ...
6
votes
1answer
228 views

Are there any non-naive parallel sparse matrix multiplication algorithms?

I was wondering about a problem in analyzing a social network (counting friends-in-common between all pairs of members) that requires squaring its adjacency matrix, and started reading up on ...
5
votes
2answers
454 views

solving large nonlinear systems in parallel

I am solving a large (~1e5 equations & unknowns) set of nonlinear equations using Newton-Raphson iterations. Currently I am using the GPU accelerated Krylov methods implemented in ViennaCL to ...
1
vote
1answer
38 views

Minimise simultaneous equation remainder

I'm not sure if this is the right place to ask, or what the right terminology to use is. The problem I have is this: I have a vector for example: v = [1,4,5,6,3,1,4,5,6,7,...7] and I have a set of ...
1
vote
0answers
146 views

Comparison of matrix determinant in less than $O(n^2)$

I was reading this question and I think maybe somebody here could help. This is the idea: Given a matrix $M$ of integers, and a number $d$ is there a way to compare the determinant of $M$ and $d$ in ...
2
votes
1answer
1k views

How to solve a Simple Linear Equation using a binary tree data structure

i am currently working on a school project that takes in a simple linear equation and has to return the value of x, the code i have transforms x + 3 = 3x - 2 into a binary tree format like so: ...
1
vote
1answer
175 views

Creating a single layer perceptron for the OR problem

I am working on the following problem Find the linear least squares unit weights for the `OR' problem, ie. $v_1^T = (0,0), v_2^T = (1,0), v_3^T = (0,1), v_4^T = (1,1)$ and $u_1 = 0, u_2 = u_3 = u_4 = ...
15
votes
6answers
6k views

What parts of linear algebra are used in computer science?

I've been reading Linear Algebra and its Applications to help understand computer science material (mainly machine learning), but I'm concerned that a lot of the information isn't useful to CS. For ...
3
votes
1answer
2k views

Efficient computation of Kronecker product

Given matrices $A \in \mathbb{C}^{n_1,m_1}, B \in \mathbb{C}^{n_2,m_2}$ a naive way to computer the Kronecker product would be as such: $M = \operatorname{zeros}(n_1n_2,m_1m_2)$ (initialize an empty ...
0
votes
1answer
41 views

What is the exact definition of a dictionary in the concept of linear algebra and computer science?

I am reading this paper for my master thesis and trying to understand every concept in it. The following sentence confuses me as to what is the exact definition of a dictionary in this concept except ...
2
votes
1answer
474 views

Implementing the Schur decomposition of a matrix

I'm trying do implement the Schur decomposition of a matrix, but I can't find any good articles for the theory. Could someone share one?
8
votes
1answer
318 views

Generate algorithmically all grid points inside a hypercube

$\def\R{\mathbb{R}}\def\Z{\mathbb{Z}}\def\n#1{\|#1\|_\infty}$The problem comes directly from computational mathematics, and can be stated as follows: Given a regular matrix $M\in\R^{d\times d}$, find ...
1
vote
1answer
412 views

Partially diagonalizing a matrix

Suppose I have an $n\times n$ matrix. I would like to keep an $m\times m$ subblock undiagonized, while keeping the rest diagonized. Is there any algorithm for this? More precisely, I mean using ...
5
votes
1answer
300 views

Complexity of Pythagorean triples

We define a Pythagorean triple as a triple $\langle a,b,c\rangle$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$. In order to avoid duplicates, we say that a triple $\langle a,b,c\rangle$ is legit ...