Questions tagged [linear-algebra]

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Choosing unsupervised learning algorithm for analyzing the spectrum of a linear operator

I am a theoretical physicist, and new to CS.stackexchange, and have a little knowledge of CS, and in Machine Learning (only some general stuff). In physics we often analyze the spectrum of linear ...
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48 views

solve a rational equation as fast as possible

I would like to find the first positive solution(if there is one) to this equation: $$\frac{ax^2+bx+c}{dx^2+ex+f} = gx+h$$ The simplest way I fond would be to do the following: $$ax^2+bx+c = (dx^2+ex+...
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Coordinate descent for Lasso, Question about algorithm

I'm not sure why the algorithm computes $c_k$ with $\sum_{j \neq k} w_j x_{i, j}$. Why does one need to ignore the $k^{th}$ feature here? I'm not sure how this is derived. Is this the result of taking ...
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Express polynomial as sum of two lower-degree polynomials, modulo another

Suppose I have a polynomial $p(x)$, and a "modulus" polynomial $q(x)$ of degree $d$. I want to find two polynomials $r_1(x),r_2(x)$ of degree $\le d_1,d_2$ such that $$p(x) \equiv r_1(x) x^...
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Designing less than and less than or equal operators for Map with value being a Set

I am trying to come up with less than and less than or equals operators for a Map where values are Sets. These are the hints I was able to get from the Professor. ...
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38 views

fastest way to identify “singular row” of a matrix

Suppose I have a matrix that I know to be singular. This means that there is at least one row in the matrix which is a linear combination of the other rows. What is the fastest way to identify which ...
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45 views

Verifying a matrix is Copositive

A symmetric matrix $A\in \Bbb{R}^{n\times n}$ is copositive if for every vector $x\in\Bbb{R}^n$ with non-negative entries, we have $$x^TAx \ge 0.$$ What are known methods to check if a specific matrix ...
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1answer
57 views

Is a system of equations derived from mutually recursive ADTs always uniquely solvable?

After looking at Can a computer determine whether a mathematical statement is true or not? for a while, I worry we may be into incompleteness/halting problem territory with this question, so an answer ...
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1answer
34 views

Understanding the coefficient matrix of Hermite Interpolation

I was not sure whether this is a computer science question or a math question, so I posted it here, hope that it is alright. I am trying to learn the technique of Hermite interpolation. I do ...
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22 views

What is the name of the following linear solver

I am trying to understand Jos Stam's method for simulating stable fluids. Here on page 21 he explain that he want to solve the following linear system: ...
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31 views

Let M be a k × n random matrix with iid entries such that

$M$ is a $k × n$ random matrix with iid entries such that $P(m_{i,j} = +1) = P(m_{i,j} = −1) = 0.5.$ Let $k = O({1\over \epsilon^l})$ for some constant $l$. $v ∈ R_n$ is a fixed vector. Does a ...
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35 views

Complexity of a decision problem: system of linear equations over finite field with restricted solutions

I have a system of linear equations over a finite field $\mathbb F_p \cong \mathbb Z_p$, and I'm interested in the decision problem of whether there exists a solution where all of the variables $x_i$ ...
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17 views

Why is the weight matrix diagonal in weighted least squares regression?

I was going through the theory for weighted least-squares fitting and I understood its basic underlying concepts, but I couldn't understand why exactly do we keep the weights as a diagonal matrix ...
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1answer
46 views

Better way to decide if a set is a pure simplicial complex

Setup I am trying to write a function that determines if a set of vertices, edges and faces is a pure simplicial complex. A pure simplicial complex is a set where all facets have the same degree, a ...
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Randomized Assignment Problem

Given $x_1,...,x_n,y_1,...,y_n\in \mathbb{R}^d$ find a permutation matrix $P\in\mathbb{S}_d$ that minimizes $\sum_{ij}P_{ij}|x_i-y_j|$. This is an assignment problem and can be solved in $O(n^3+n^2d)$ ...
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N-dimensional generalization of map and reduce?

Is there any conceptual generalization of higher-order functions like map and reduce but for N-dimensional objects (e.g. arrays or tensors)? For mapping, I guess it would be a point-wise ...
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27 views

Seeking guidance on what to read for Feasibility Binary IP with ''almost total unimodular'' (-1, 0, 1)-Coefficient Matrix and No Obj Function

I am working on an algorithm in graph theory which I wish to prove it's polynomiality/NP-hardness. I am investigating a binary variable (0, 1) integer program which has the coefficient matrix ...
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1answer
81 views

How does the sweep line algorithm check for intersection using vector cross product?

I am trying my best to understand the sweep-line algorithm to find line intersections. I have understood most of the intuition except how it is calculating the intersection between 2 line segments ...
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78 views

When does Gaussian elimination solve exact 1-in-3 SAT?

Terms: A literal is a variable or its negation. A clause is a set of literals. An exact 3-in-1 clause is satisfied if an assignment of values to variables results in exactly 1 ...
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Computing structure tensors

As far as I understand, the structure tensor is: $$ M = \sum_{(x,y) \in W} \begin{bmatrix} I_x^2 & I_xI_y \\ I_xI_y & I_y^2 \end{bmatrix} = \begin{bmatrix} \sum_{(x,y) \in W} I_x^2 & \...
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1answer
33 views

Complexity of Matrix Inversion when $n-2$ Eigenvalues are the same

Suppose we have a symmetric matrix $A \in \mathbb{R}^{n \times n}$ that has $n-2$ equal eigenvalues and the other two are distinct. Question: What would be the complexity of its inversion? On the ...
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1answer
18 views

Math behind Multi-class linear discriminate analysis (LDA)

I have a question about Linear Discriminant Analysis (LDA) for the purpose of Dimensionality Reduction. So I understand for the algorithm to calculate for $k$ projection vector(s) you need to ...
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1answer
99 views

Applying SVD compression to integral point images

Suppose that we have an $m\times n$ matrix $A$ of rank $n$, whose entries are 8-bit unsigned integers obtained from a grayscale image. Now we want to apply SVD to $A$ and to use the first $k$ singular ...
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27 views

Minimum basis for the nullspace of sparse matrices

Let $A\in\mathbb{F}_2^{m\times n}$ and denote its nullspace as $V=\{x\in\mathbb{F}_2^m:xA=0\}$. The weight of a basis $B=\{b_1,\dots,b_l\}$ for $V$ is the total weight of vectors in the basis, denoted ...
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63 views

Calculating number of intersections of a horizontal line with line segments efficiently

I'm given an array $A = [a_1, a_2, ....a_n] $ using which I construct $n-1$ contiguous line segments by drawing a line from $(i,a_i)$ to $(i+1, a_{i+1})$. Now, I'm given $q$ queries in the form of $...
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723 views

Count number of pairs of elements whose product is a perfect square

Given two arrays whose elements lie between $[1,10^5]$ and the size of arrays is $[1,10^5]$, how can we find the total number of pairs of elements from these arrays such that their product is a ...
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1answer
46 views

Given a set of integers $D$ and a positive value$P$, find an algorithm to find set of integers satisfying a condition

Given a set of positive integers : $ \\ D = \{ D_1, D_2, ..., D_n\}$ and a non-negative integer $P$, where $P$ is divisible by every element in $D$, then find ...
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19 views

Are there practical usage of determinants in numerical simulation?

I know the historical importance of the link between linear systems and determinants. I also know that determinants have a beautiful connection with non-singular matrices, i.e., if a matrix is non-...
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43 views

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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21 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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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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30 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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15 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
59 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
50 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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46 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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80 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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58 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
50 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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55 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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38 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
43 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
258 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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43 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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172 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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125 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
90 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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50 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
97 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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2answers
243 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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