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Questions tagged [linear-algebra]

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49 views

How to quickly solve a linear equation for 7000 times?

I need to solve a linear equation Ax=b for 7000 times (A is sparse and complex square matrix), and at each time only 4 elements (A(i,k), A(i,m), A(j,k) and A(j,m)) are changed while all other elements ...
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0answers
28 views

How to make sure matrix completion can generate a matrix with values in expected range?

I am doing a matrix completion project. Assume that I have an incomplete matrix like ...
2
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1answer
18 views

Determine image of hypercube under linear map

Let $A$ be an $3\times N$ matrix (where $N$ is large) with nonnegative real entries. I'd like an algorithm for determining when a vector $v\in\Bbb R^3$ can be written as $Aw$ for some vector $w\in\Bbb ...
3
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1answer
89 views

Can this equation be solved in polynomial time?

I came across a more general form of this question. Can we find the value of variables in polynomial time ? Let $m = n^{2}$, there are $m$ variables ($x,y,z\ldots$) in the equation and these $m$ ...
2
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0answers
37 views

Check if a matrix over finite fields is superregular

Is there any practical, efficient algorithm to check if a matrix over $\mathbf{F}_{p^n}$ is superregular? It need not be theoretically polynomial, just roughly be implementable for $n=32$ and for ...
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2answers
144 views

Can this system of polynomial equations be solved in polynomial time?

I have these $n$ equations, with $n$ variables. Variables are first $n$ positive integers, constants can be any rational number including zero. Given that there is always a solution, how do we find a ...
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0answers
50 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
47 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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0answers
43 views

An efficient algorithm to find a linear transformation between two ternary quadratic forms

Let $\mathbb{F}_p$ be a prime finite field for $p > 2$. Consider two ternary quadratic forms $$Q_1\!: x^2 - a_1(t)y^2 - b_1(t)z^2,\\ Q_2\!: x^2 - a_2(t)y^2 - b_2(t)z^2$$ over the field $\mathbb{F}...
1
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1answer
75 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 ...
3
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2answers
236 views

Transforming a byte with a subset of a small, fixed set of values and xor into any other value

If I have some collection of bits, -- a byte, say -- of arbitrary value then I can transform it into some other value by means of exclusive-oring it with a subset of (in this case) eight fixed values, ...
1
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1answer
16 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 ...
2
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1answer
30 views

Sublinear Homomorphism Property Testing Counter Example

This is a homework question, so I'm not looking for answers, just general guidance. I'm looking at a Sublinear Algorithms survey where (Group) Homomorphism property testing is discussed. The case of ...
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2answers
48 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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0answers
26 views

Equivalent Algorithm with Sharman Morrison inversion

I am trying to invert a matrix using Woodbury identity. The inversion using Cholesky decomposition has the following pseudo-code: For $t=1,2,...$ $(1)\;\; \text{Read}\;x_t\in\mathbb{R}^n$ ...
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0answers
32 views

Randomly choose matrices $A_{j}B = C_{j}$ with elements between 0 and 1

Problem I have $J$ matrices $C_{j}$, which are $K \times M$. Elements of each matrix $C_{j}$ are between 0 and 1. I want to randomly choose $J$ matrices $A_{j}$ and one matrix $B$ such that: ...
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1answer
53 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
18 views

Avoiding underflow when identifying neighbours of a cell in a grid by using modulo

I'm going through a tutorial that is using the Game of Life as example code. It has a function in it that finds the neighbor of a given cell. It is explained quickly that "When applying a delta of -1, ...
2
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1answer
26 views

What's the connection between the two “Fast Walsh Transform”?

First Let's take a look at the convolution $\displaystyle C _ { i } = \sum _ { j \oplus k = i } A _ { j } * B _ { k }$, and the $\oplus$represents any boolean operation. And we are able to evaluate $C$...
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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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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 ...
2
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1answer
39 views

Randomly construct linear combination that is within bounds over two basis

Given rank deficient matrix $A$, I want to randomly construct vectors $\vec{x}$ such that: $0 \le x_{j} \le 1$ $0 \le b_{j} \le 1$ where $\vec{b} = A\vec{x}$ Matrix $A$ is about 10 x 15. I want to ...
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0answers
12 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 ...
1
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1answer
96 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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0answers
21 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=...
4
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1answer
174 views

Minimize sum of squares of rows in matrix when sum of columns have some constraint

I'm looking for an algorithm that can find any matrix $a_{j,i}$ such that $$ \sum_{i \in I} \left(\sum_{j\in J} a_{j,i}\right)^2 $$ is minimal, while also for each $j\in J$ satisfying the constraint ...
4
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0answers
110 views

Imbalance of variables in Mixing Newton's method and Linear solver for a Non-linear system

Problem Solving a non-linear system of equations. The number of variables is the same as the number of equations. When I fix a set of variables (say $\vec{y}$) and keep another set free (say $\vec{...
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0answers
43 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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1answer
47 views

Compute unknown matrices that minimize a sum

This problem is about working with smart-phone accelerometers. To calibrate accelerometer, I need to find three unknown matrices T, K and B that minimize this sum: $$\sum_{i=0}^N(|g|^2 - |TK(a_i + B)|...
2
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1answer
12 views

Strassen's algorithm on unit vectors?

I am trying to do a dot product of two vectors of each 128 dimension. I am just looping each member and calculating the sum. ...
4
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1answer
61 views

Interpreting camera matrix

I'm having some trouble interpreting the camera matrix $K = \begin{bmatrix} f_x & s & x_0 \\ 0 & f_y & y_0 \\ 0 & 0 & 1 \end{bmatrix}$ after it multiplies some 3D vector. ...
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1answer
200 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 ...
2
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1answer
57 views

Is it possible to transfer a point from one camera to another, given n corresponding points?

I have 2 images of a scene taken at one moment by two identical cameras (similar cameras intrinsic parameters) by to arbitrary locations and at two arbitrary orientations (different cameras poses). On ...
3
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1answer
78 views

Eigenvalue computation for large graph

Consider a large graph, minimum 1 000 vertices but it can easily go up to 50 000 vertices depending the case. The graph is the result of social relationships (followers, following, friendship) so it ...
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2answers
56 views

Among $k$ unit vectors, find odd set with sum length less than 1

I have $k$ unit vectors in $\mathbb{R}^k$. Can I efficiently identify a set of $2n+1$ vectors $v_1, \dots v_{2n+1}$ such that $\sum_{i< j} v_i\cdot v_j < -n$ for any $n$ -- or determine that no ...
0
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1answer
359 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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2answers
110 views

Given matrix $A$, find vector $x$ such that every entry of $Ax$ is nonzero

Given a matrix $A \in \mathbb{R}^{n \times n}$ with no zero rows, what is the complexity of deterministically finding a vector $x \in \mathbb{R}^n$ such that every entry of $Ax$ is nonzero? It is ...
0
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1answer
185 views

LP formulation and integer solution existance

I’m trying to prove that the following problem has an integer optimal solution. This will hold if the corresponding linear program would have totally unimodular constraint matrix. We have $m$ pieces ...
2
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0answers
56 views

Complexity of a submatrix rank problem

Given a matrix $M \in \mathbb{R}^{n \times m}$ and a set $S \subset \{1, \ldots, n\}$, let $M_{S, {\rm row}}$ be the matrix obtained by picking the rows of $M$ from the set $S$. Similarly, given $S' \...
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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 ...
2
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2answers
310 views

Is the Tomasi-Kanade factorization still commonly used as a modern computer vision technique?

My understanding is that, in very rough terms, the Tomasi-Kanade algorithm published in 1992 describes a way to reconstruct the 3D structure of an object from multiple images of that object, given ...
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1answer
215 views

maximizing inner product of vectors in an ellipsoid and a given vector

I have been wrestling with this for quite a long time but couldn't convince myself that the following is true: What I do understand: $\theta_a$ denotes the set of points that are within the ellipsoid....
2
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0answers
71 views

Vandermonde matrix and its binary representation

Say one is given a Vandermonde matrix (https://en.wikipedia.org/wiki/Vandermonde_matrix) of dimension $2^q \times k$ such that the elements of the first column of it are $\{0,1,2,..,-1+2^q\}$. (This ...
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0answers
111 views

Why solutions for Normal form Ordinary least squares and Linear Regression are different

I am trying to apply Linear Regression method for a dataset of 9 sample with around 50 features using python. I have tried different methodology for Linear Regression i.e Closed form OLS(Ordinary ...
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0answers
32 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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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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0answers
25 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 ...
2
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0answers
44 views

How many iterations of Lanczos bidiagonalization are required in order to obtain the first k singular values/vectors of a matrix?

I am trying to implement a fast SVD algorithm for obtaining the first $k$ singular values/vectors of an $M\times N$ matrix ($k < \min(M,N)$) using the following 2-step process: 1) bidiagonalize ...
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1answer
50 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 ...