Questions tagged [linear-algebra]

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Geometric interpretations of midpoint algorithm, homogeneous linear least squares and nonlinear least squares method in 3D reconstruction?

In "Multiple View Geometry in Computer Vision" Chapter 12. Structure Computation, page 310-313, triangulation is used for point 3D reconstruction. There are three methods mentioned: Midpoint method ...
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0answers
127 views

Why is my Forrest-Tomlin update worse than recomputing LU?

I wrote a simple C++ implementation of the revised simplex method that recomputes the LU decomposition of the basis from scratch on each iteration. I have to solve problems with many variables but few ...
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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 ...
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144 views

Maximum feasible subsystem problem (MaxFS) in 2 variables

Topic: The maximum feasible subsystem problem, which is generally NP-hard [1]. Question: Are there special algorithms in case of only 2 variables (2D linear constraints)? The problem seems to be a ...
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1answer
425 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?
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1answer
266 views

Inputting a superposition into a cNOT gate

I am unsure how to calculate what happens when you put a superposition as the control qubit into a cNOT gate or what happens when you put a superposition as the target qubit into a cNOT gate. I know ...
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1answer
83 views

What is the fastest algorithm to establish whether a linear system in $\mathbb{R}$ has a solution?

I know the best algorithm to solve a linear system in $\mathbb{R}$ with $n$ variables is Coppersmith-Winograd's algorithm, which has a complexity of $$ O\left(n^{2.376}\right). $$ How much easier is ...
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1answer
182 views

Calculating eigenvalue gap of a torus graph

Consider a two-dimensional grid with wrap-around edges (a doughnut-shaped graph). I need to calculate the second-largest eigenvalue of the adjacency matrix. Is there a faster way of computing it for ...
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1answer
358 views

Significance of parameters in Tiny Mersenne Twister algorithm

I am trying to implement and optimize the Tiny Mersenne Twister (TinyMT) algorithm as required by an API I am developing with my team at work. The algorithm utilizes a C structure with 32-bit unsigned ...
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2answers
103 views

Finding the required value of an algebric expression

I have an expression $$Ax+By+Cz.$$ where $A$, $B$ and $C$ are positive constants $\ge1$. The variables $x$, $y$ and $z$ are non-negative integers. I am also given a number $T$. I want to find the ...
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2answers
49 views

Algorithm to solve $L_1$ optimization of $\sum_i ||\mathbf{A_i x} - \mathbf{b_i}||_1$

Is there is an efficient algorithm to solve the following optimization: $\mathbf{x}^* = \arg\min_\mathbf{x}\sum_i ||\mathbf{A_i x} - \mathbf{b_i}||_1$ for given $\mathbf{b_i}, \mathbf{A_i}\ \forall ...
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1answer
54 views

Computational Complexity of Integer Linear Program [with Fixed number of 'Pure Constants']

This is a follow up to the previous Question: Conditions for Linear Diophantine Equations to always have a solution It was established in the above's answer that obtaining or testing for the ...
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1answer
61 views

small size and small depth circuit for set intersections

Input: Given sets $S_i \subseteq \{1,2,3,4,\cdots,n\}$ for $1 \leq i \leq n$. Output: sets intersection with restriction (pick first set $S_1$. If $a \in S_1$ such that $a$ is the least element then ...
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1answer
31 views

Finding dimensions of a point cloud - floating point issues

I have $N$ data points of some dimension $D$. I want to know the dimension of the shape that those data points represent - for instance they might be a 2 dimensional triangle, even though they are in ...
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1answer
72 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 ...
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1answer
37 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 ...
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1answer
282 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 ...
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1answer
50 views

What is the intuition behind the way of reading off a dual optimal solution from simplex primal tabular in CLRS?

Section 29.4 "Duality" of CLRS (3rd Edition) describes the way of reading off an optimal dual solution from the last slack form of the primal as follows: Suppose that the last slack form of the ...
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1answer
67 views

Representing chained XOR operations as linear inequalities

I'm trying to solve an integer linear program (ILP) in which a constraint of the following kind must be met: $x_1 \oplus x_2 \oplus \cdots \oplus x_n = 1$ where $\oplus$ is the binary xor operator. ...
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1answer
38 views

Randomly choose vector b in range such that $\vec{a} \cdot \vec{b} = 1$

Given I have a $n$ dimensional $\vec{a}$. All elements of $\vec{a}$ are between 0 and a positive number $K$. $n$ is about 15 to 20. Problem I want to randomly and unbiasedly choose a vector $\vec{b}...
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1answer
126 views

In most locality sensitive hashing implemensions of SimHash, why is the cosine distance used and not the euclidean distance?

In Chapter 3 of Mining of Massive Datasets, the basis of locality sensitive hashing is explained. They notably mention simhash for the cosine distance, where random hyperplanes are generated, and for ...
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1answer
241 views

Applications of the LU factorization in computer science [closed]

I have been searching for the applications of the LU factorization/decomposition in computer science. From Wikipedia, I have found some of the applications, but these don't seem to be relevant to ...
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1answer
31 views

What is the bit complexity of Gaussian eliminaton over $\Bbb F_q$?

Given matrix $M\in\Bbb F_q^{n\times n}$ with rank $r$ what is the complexity of converting to row-echelon form? Is it $O(n^3\log q)$ or $O(n^3q)$ bit operations? Technically $O(n^3)$ row ...
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1answer
244 views

Approximation of a gaussian function

I want to approximate a Gaussian function as shown below $$ e^{\frac{-\|x-c\|^2}{2\sigma^2}} \approx \sum_{i=1}^{N}\alpha(c,c_i)e^{\frac{-\|x-c_i\|^2}{2\sigma^2}} \forall x $$ Here c, $\sigma$ and $...
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1answer
660 views

Absorbing Markov Chains: An efficient algorithmic approach

Following this procedure I have successfully written a program to calculate the probability of ending in a given absorbing state given the initial state. The procedure is as follows: Given the ...
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1answer
745 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 ...
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1answer
127 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 ...
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1answer
166 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 = ...
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0answers
27 views

Minimize number of math operation of a specific matrix vector multiplication?

Let's say we have a Matrix M and a column vector v like below multiply equals Assume we can only perform multiplication, addition and substraction operation. With normal approach we need 3 ...
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0answers
30 views

How is the modular multiplication matrix unitary in Shor's Algorithm?

I have been reading papers about the construction of this matrix in Shor's Algorithm all night. The behavior of the controlled modular multiplication matrix is described as $$C U_{a^{2}}(|c\rangle|y\...
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56 views

Confusion about the geometric interpretation of the simplex method for linear programming

In Section 7.6.2 of the textbook "Algorithms" by Sanjoy Dasgupta, Christos Papadimitriou, and Umesh Vazirani, the authors provide a geometric interpretation of the two main tasks of each iteration of ...
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1answer
17 views

Smallest Circuit for Square of Sparse Symmetric Matrix

I have an n by n symmetric matrix, and I would like to compute its square in as small a circuit complexity as possible. It's sparse: there are sqrt(n) nonzero entries in each row/column, so the input ...
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0answers
31 views

Check if no linear combination is within a hypercube

Shapes Let $C$ be the unit hypercube in $\mathbb{R}^{n}$. Let $\vec{o}$ be a point in $\mathbb{R}^{n}$. Let $B$ be a $n \times m$ matrix. The columns of $B$ are a set of linearly independent vectors ...
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0answers
18 views

Sparse feasible solution $|x|_0\le k$ for system of linear inequalities $A x \le b$

Suppose the set of linear inequalities $Ax\le b$, in which $A\in\mathbb{R}^{m\times n},x,b\in\mathbb{R}^n$ is given. Is it possible to determine in polynomial time with regard to $m$ and $n$ if there ...
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0answers
22 views

Need help implementing an algorithm to solve roots of a transcendental equation

I'm trying to implement this algorithm but I'm having problems reproducing the exemple that it gives a solution to. The general method that I tried is: Make a grid $\theta \in [0,2\pi)$, with say N=...
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0answers
50 views

Sparse Matrix inversion without actual inversion

I want to know what are the efficient way to invert a Sparse Matrix? Are there any algorithm,linear algebra or expansions that make this task easier with out actually inverting the matrix? Thank you ...
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0answers
25 views

Construct unitary $Q$ such that $span\{q_1,q_2\} = span\{v_1,v_2\}$ where $v_1,v_2 \in \mathbb{R}^n$ are given as input

Assume also that $v_1,v_2$ are linearly independent, and $q_i \in \mathbb{R}^n$ denotes the $i$-th column of $Q$. This is what I've got so far. First obtain unit vectors $w_1,w_2$ which are orthogonal ...
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0answers
24 views

Want to do 3D reconstruction via simple matching

I have 2 images, called left and right images. I have some matched points $[c_l,r_l]$ and $[c_r,r_r]$ in both of them (these points are in pixel coordinates). For a 3D point in the real world, they ...
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0answers
33 views

Scalability of factor-solve vs. pseudoinverse + product

I’ve always read advice and warnings about the poor scalability and time spent trying to invert a matrix and why it’s better to solve a system of equations whenever the inverted matrix will be used ...
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26 views

Testing whether a set of integers can be written as a combination of module basis elements

Input We are given a set of basis elements, $\ v_1$,$\ v_2$ ,...,$\ v_n$ of a $\mathbb Z^m$- module and a multiset of integers $\ B :=$ {$\ b_1, ..., b_m$} Desired Output Return true if there ...
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188 views

How to solve linear system with modulus operation?

I came across linear equation $G(x,y) = g_k(x,y) l_k(x,y)$ mod $(y^{2^{k}})$ while reading factoring algorithm see section 3 for bivariate polynomials. I need to find the $G(x,y)$ and $l_k(x,y)$. ...
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0answers
76 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 ...
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36 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 ...
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34 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: $$ \...
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140 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 ...
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109 views

Laplace's Approximation for graphical models

A question about Laplace's approximation: In Laplace's method, we need to find the mode of a function and take second order Taylor's expansion. The first order term will vanish (since the gradient is ...
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34 views

Is this problem in P: Finding a common key for a collection of systems of equations?

Let $B=\{b_1=g_1,\cdots,b_n=g_n\}$ be a set of binary variables $b_i$ and their corresponding values $g_i \in \{0,1\}$. Let $M=\{\sum_{e \in A}e \;:\; A \subset B\}$, i.e., $M$ is the set of all ...
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1answer
675 views

How to use different size features in SVM?

I want to train a support vector machine with some features. The problem is, one of the features is 1-dimensional (only an angle) and the other is an LBP Histogram, an 58-dimensional vector. ...
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1answer
386 views

Cache efficient matrix multiplication

Consider these matrices: $A=\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ $B=\begin{bmatrix}-1 & -2\\-3 & -4\end{bmatrix}$ Using standard algorithm: $C=\begin{bmatrix}1*-1+2*-3 & 1*...
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1answer
19 views

Geometric median of two disjoint sets of points lies on line between their respective medians

I was working on a problem about geometric medians and I had an idea for a divide and conquer solution, but it would only work if a set of points, when split into two disjoint sets, and those sets ...