Questions tagged [linear-algebra]

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What is the complexity of finding e^(A) for a Hermitian matrix A?

If A is a hermitian matrix of size NxN .What is the order of no. of steps required to compute e^(A).How to prove it?
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Looking for good book similar Stability/Conditioning in Numerical Linear Algebra,

I am currently reading Numerical Linear Algebra by Trefethen and Bau and I am finding it quite difficult to read. In particular, I have been trying to read the sections on Floating Point Arithmetic, ...
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23 views

Check the Complexity time of Power method

Hi i have write a function in Matlab to calculate the power method and i wont to find the time complexity. ...
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5 views

How to prove 3rd order polynomial kernel used for support vector machine positive semi definite?

I have a 3rd order polynomial $K(\bar{\textbf{x}},\bar{\textbf{y}}) = (\bar{\textbf{x}}.\bar{\textbf{y}}+c)^3$ that I need to prove it is positive semidifinite. I see that I need to show $\sum_{i=1}^n ...
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20 views

Minimal subset of rows that generate smaller polyhedron

Given a matrix $[A|B]$ I want to find a minimal matrix $[A'|B'] \subseteq [A|B]$ (i.e. the rows in $[A'|B']$ are also in $[A|B]$) such that $A'x < B' \Rightarrow Ax < B$. Geometrically, I want ...
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32 views

Given a system in $\mathbb{F}_2$ in RREF, how do I find a solution of minimal norm?

I have a $12 \times 12$ (so not really large) system of linear equations in $\mathbb{F}_2$ which I got to RREF through the usual row reduction. Suppose the system has multiple solutions, and call the ...
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23 views

How do you solve a general linear diophantine equation in polynomial time (with minimization constraint)?

Given $$ a_1 X_1 + \dots + a_n X_n = b $$ where $a_i, b \in \Bbb{Z}$. How do you come up with a clearer picture of the solution set in polynomial time. Also, what I really want is to do the above,...
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14 views

Convergence of Conjugate gradient method

I have implemented my own matrix library in Java to solve fluid simulations. So I have also implemented the conjugate gradient method and I got a little bit confused. What I have done to test my CG-...
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1answer
28 views

Recover boolean vector from dot products

Question: I want to determine a boolean vector $b \in \{0,1\}^n$ consisting of zeros and ones, but cannot access it directly. I can only call a black-box computer code which will take the dot product ...
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1answer
23 views

Underlying codes for Niederreiter cryptosystems

Niederreiter cryptosystem is usually described by a parity check matrix $H$ over $\mathbb{F}_{2^n}$. The minimum distance $d$ is given by $d := min\lbrace k \text{ such that there are $k$ linearly ...
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31 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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24 views

Calculating diagonal of inverse of sparse band-like matrix

I'm trying find an optimization for an equation related to theorem 3.5.7 from "Finite Markov Chains" by Kemeny and Snell (1976). The theorem is: $$H=(N-I)N_{dg}^{-1}$$ Where $N_{dg}$ is a diagonal ...
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32 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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1answer
28 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 ...
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53 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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32 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 ...
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1answer
27 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 ...
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1answer
102 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 $m$ ...
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39 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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154 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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86 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
62 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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45 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}...
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1answer
89 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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240 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, ...
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1answer
20 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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1answer
36 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
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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27 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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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
96 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, ...
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1answer
28 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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34 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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1answer
42 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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22 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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1answer
176 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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28 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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1answer
270 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 ...
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121 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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63 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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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
51 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)|...
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1answer
13 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. ...
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1answer
84 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
344 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
60 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 ...
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1answer
88 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
60 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 ...