Questions tagged [linear-algebra]
Linear algebra deals with vector spaces and linear transformations.
200
questions
3
votes
1answer
40 views
Problem conditions to use Laplacian solvers
I am trying to use Laplacian Solvers to solve a linear equation. I am just learning it (form here), so my question is very basic and it might not even make sense.
Suppose that we want to solve Ax=b, ...
0
votes
0answers
24 views
Looking for information about a problem of a least subset of vectors modulo 2 summing to another vector [duplicate]
I'm quite interested in the following algorithmic problem, on which I can't find any information. Phrased as a decision problem:
Given a set of vectors $V$ in $\text{GF}(2)^n$, a vector $\mathbf u$ ...
4
votes
0answers
102 views
efficiently computing a sum of products of polynomials
Let $F$ be a prime finite field. Let $d$ be a power of two dividing $p-1$. Suppose I have $d$ pairs of univariate polynomials $f_i,g_i$ over $F$ for $i=1,\ldots,d$. All have degree less than $d$.
I ...
1
vote
0answers
29 views
Complexity Lower Bounds for 3D Sparse Gaussian Elimination
I'm interested in lower bounds on the complexity in the real-RAM model of solving systems of linear equations which have the sparsity pattern of a three-dimensional cubic mesh. Specifically, consider ...
1
vote
0answers
94 views
An optimization problem
I am considering the following optimization problem. Let $P$ be a set of $n$ points in $\mathbb{R}^d$
maximize $\sum_{p\in P}\vert\langle \Vert p\Vert p, \hat{x}\rangle\vert$ subject to $\Vert\hat{x}\...
2
votes
0answers
27 views
Complexity of best folding of a 2D set (or how to optimize a sandwich)
Motivation:
I was making lunch for my son, part of which is making a sandwich from two halves of a slice of bread. In order to minimize the parts of bread that have cheese on them, and are not covered ...
1
vote
1answer
107 views
Linear regression as a hylomorphism
A hylomorphism consists of an anamorphism followed by a catamorphism.
Is it possible to express linear regression as a hylomorphism?
7
votes
0answers
132 views
Reference for computing the rank of a matrix in polynomial time
In a recent paper, I need to use the fact that computing the rank of a matrix over the integers has polynomial complexity. Given the context, I don't particularly care about the exact asymptotics, as ...
1
vote
1answer
126 views
Face-splitting product of two Vandermonde matrices: When is is invertible?
Let $A$ and $B$ be two $n^2 \times n$ Vandermonde matrices with coefficients $\alpha_1,\ldots,\alpha_{n^2}$ and $\beta_1,\ldots,\beta_{n^2}$. Let $M$ be the face-splitting product of $A$ and $B$, that ...
4
votes
0answers
79 views
Complexity of Encoding a Matroid Flow Problem in a Matrix
Context:
Take a directed graph $G$ with a specified subset of source vertices $S$ and target vertices $T$.
We say a subset $I\subseteq T$ of size $r$ is independent if there exist $r$ distinct ...
3
votes
1answer
153 views
When is it hard to invert a sparse matrix?
Are cases where numeric inversion of a sparse matrix is known to be harder than sparse matrix multiplication?
In practice, sparse matrix inversion is done with methods like Jacobi or Gauss-Seidel, ...
1
vote
1answer
150 views
What is the complexity of this submatrix selection problem?
We have a $kn\times kn$ matrix $M$ made of $n^2$ many $k\times k$ blocks.
We want to find an $n\times n$ submatrix such that each row and column is from distinct window of size $k$ such that the sum ...
1
vote
0answers
48 views
Computational complexity in linear solvers
I have recently been trying out methods of coding for solving systems of linear equations on Python. Of course, I first used the inbuilt function $\mathit{inv}$ under certain if-conditions to obtain ...
-3
votes
1answer
55 views
Finding a non-negative solution to an integer system of linear equations
Let $A$ be an $n \times m$ integer matrix and consider the system of equations $Ax = b$ where $b$ is an integer vector. I want to find a solution $x$, assuming one exists, such that the components of $...
8
votes
1answer
182 views
The complexity of the permanent of low rank matrices
I know that for an arbitrary $n \times n$ matrix, Ryser's algorithm can compute the permanent in $\mathcal{O}(2^n n^2)$ time. I'm interested in computing the permanent of $n \times n$ matrices of rank ...
7
votes
0answers
146 views
Algebraic methods for testing planarity
Mac Lane's planarity criterion states that a graph is planar if and only if it has a cycle basis such that each edge is contained in at most two cycles, we call such a basis a 2-basis. A 2-basis of a ...
4
votes
2answers
183 views
What are some good resources for strengthening my theoretical foundation for machine learning?
I'm a computer science major and I'm taking a lot of machine learning courses. I'm finding that my theoretical foundation on subjects like calculus and linear algebra are not as strong as I'd like ...
41
votes
3answers
5k views
Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time
It is commonly believed that for all $\epsilon > 0$, it is possible to multiply two $n \times n$ matrices in $O(n^{2 + \epsilon})$ time. Some discussion is here.
I have asked some people who are ...
2
votes
1answer
158 views
Finding whether $n$ polytopes have nontrivial intersection from pairwise comparisons
I have a set of $n$ convex polytopes of the form
$$\mathcal{L_i} = \{ \beta \mid C_i \beta \leq 0 \}$$
where $C$ is a matrix and $\beta$ is a vector. I know that for each pair of polytopes $$(\...
1
vote
0answers
51 views
Underlying codes in 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 ...
5
votes
0answers
57 views
Hardness result or reference for optimal Gaussian elimination process
I'm wondering if the following problem is NP-Complete or has any hardness result.
References on related problem are also welcome.
Input: integers $n\geq1,k\geq0$ and an invertible matrix $M\in\mathbb ...
5
votes
0answers
207 views
Is there a fast algorithm for inverting a sparse matrix?
I am doing research on a random-walk like problem. As a critical part of my solution, I need to invert a non-singular sparse matrix of size $n \times n$ and with $O(n)$ non-empty entries. I'm working ...
0
votes
0answers
22 views
Does optimal fitting flat must pass through the mean of the point set?
I am confused about a statement made in the paper Linear Time Algorithm for Projective Clustering, section 5.1, second paragraph, second line.
Project clustering is a natural generalization of k-...
1
vote
0answers
80 views
Missing proof in Salil Vadhan's monograph on pseudorandomness, Random Walks and S-T Connectivity
In Salil Vadhan's monograph on pseudorandomness, chapter 2, half of the proof of Lemma 2.51 is missing
http://people.seas.harvard.edu/~salil/pseudorandomness/power.pdf .
I don't state the full lemma ...
10
votes
0answers
148 views
Do Banach spaces and linear contraction maps form a model of ILL with an exponential?
Recently, I read on the nLab that the category of Banach spaces and linear contractions is small complete, small cocomplete, and monoidal closed.
This means that Banach spaces and short linear maps ...
8
votes
3answers
585 views
Non-Orthogonal Vectors Problem
Consider the following problems:
Orthogonal Vectors Problem
Input: A set $S$ of $n$ Boolean vectors each of length $d$.
Question: Do there exist distinct vectors $v_1$ and $v_2 \in S$ ...
2
votes
1answer
136 views
Recovering a rank-one matrix from its eigendecomposition after randomized rounding
Let $A = xy^T$ be a rank-$1$ matrix, and suppose every entry of $A$ is in $[0,1]$. We can create a binary matrix $A_{\rm rounded}$ by setting
$$ [A_{\rm rounded}]_{ij} = \begin{cases} 1 & \mbox{ ...
6
votes
1answer
346 views
What are the consequences of solving XOR 3-SAT in Logspace?
XOR Formulas
Consider boolean formulas with connectives $\wedge$ (AND) and $\oplus$ (XOR). Such a boolean formula is a valid instance for XOR SAT if it is a conjunction of $\oplus$-clauses. An $\...
2
votes
0answers
63 views
Which invertible linear transformations can be computed reversibly without ancilla/garbage bits just as easily as they can be computed irreversibly?
Suppose that $L:F_{2}^{n}\rightarrow F_{2}^{n}$ is an invertible linear transformation. Then define $w(L)$ to be the gate count of the smallest reversible circuit on $n$ bits without ancilla/garbage ...
2
votes
0answers
89 views
Why is triangle inequality needed for indexing?
Maybe this is a silly question but I actually can't fulfill that by myself. I'm reading some papers about similarity metrics and I always find that for a distance function $d$ the triangle inequality ...
11
votes
1answer
240 views
Binary vector $t$ in $span(S)$ over $\mathbb{Z}/q\mathbb{Z}$ for all prime powers $q$ $\Rightarrow$ $t$ in $span(S)$ over $\mathbb{Z}$?
I have a set of $n$ binary vectors $S = \{s_1, \ldots, s_n \} \subseteq \{0,1\}^k \setminus \{1^k\}$ and a target vector $t = 1^k$ which is the all-ones vector.
Conjecture: If $t$ can be written as ...
11
votes
1answer
326 views
Is there a P-complete problem on diophantine equations?
In general deciding whether a diophantine equation has any integer solutions is equivalent to the halting problem. I believe that deciding if a quadratic diophantine equation has any solution is NP-...
3
votes
0answers
66 views
Minimizing a sum of thresholded quadratics
Let $W_1, \ldots, W_k$ be positive semi-definite matrices, $b_1, \ldots, b_k$ be vectors, and $a_1, \ldots a_k$, $c_1, \ldots, c_k$ be scalars. How difficult is it to find an approximate minimum of
...
6
votes
1answer
234 views
Algebraic account of Gaussian elimination?
For fun, I've been looking at the interpretation of linear logic in terms of finite-dimensional vector spaces, and ran into an interesting
question about the interpretation of double-negation-...
6
votes
0answers
157 views
An optimal subspace projection problem
Suppose we have a $k$-dimensional subspace $V$ in $\mathbb{R}^n$ given by a basis $\{v_1,\cdots,v_k\in \mathbb{R}^n\}$, find an index set $I\subset [n]$ with $|I|=m$ where $k\le m\le n$, such that
$$\...
-1
votes
1answer
81 views
given a set of $n$ points in $d$-dimensional space and the basis vectors of some subspace, how to find all the points on that space?
given a set $A$ of $n$ points with integer coordinates in $\mathbb{R}^d$, and $k<d$ basis vectors of a subspace $K$ of $\mathbb{R}^d$, is there an efficient algorithm that returns all points from $...
2
votes
1answer
89 views
Some consequences of the Roychowdhury-Orlitsky-Siu result from 1994
This pertains to the proof of theorem 1.1 in this paper, http://dl.acm.org/citation.cfm?id=2897636
So Roychowdhury-Orlitsky-Siu had shown that the number of depth $2$ linear threshold gate circuits ...
2
votes
0answers
62 views
A random ensemble of sparse boundary operators
The following question arises from the study of quantum error correction, and high-dimensional expanders:
Is there an algorithm that for given numbers $n>0,d≤n,r≤n$ samples uniformly a linear ...
3
votes
1answer
92 views
About the sign-rank of the Minsky-Pappert function
Apologies this might be a very trivial thing I am getting confused by!
Firstly in corollary 1.1 (page 3) in this paper, https://eccc.weizmann.ac.il/report/2016/075/ the authors claim that they have ...
4
votes
0answers
82 views
About Boolean functions with a high sign-rank
Recently in this beautiful paper, https://arxiv.org/pdf/1705.02397.pdf it has been shown that there is an explicit $Th \circ Th$ function with sign-rank scaling exponentially in dimension. I wanted to ...
26
votes
0answers
553 views
Rank mod 6 vs rank over the reals
Let $A$ be a boolean matrix (eg with $0,1$ entries). Assume that $A$ has rank $\le r$ both over $\mathbb{F}_2$ and over $\mathbb{F}_3$. Does this imply that $A$ has low rank over the reals? This seems ...
1
vote
1answer
328 views
“Linear” hashing function
Say we have two chunks of data $X$ and $Y$, which may be of different sizes, is there a non-trivial function $hash$, and operation $*$, such that:
$$hash(X+Y) = hash(X) * hash(Y)$$
...where $+$ is ...
7
votes
1answer
173 views
Rank-robustness of the parallel complexity of linear algebra problems
It is known that most computational problems related to linear algebra
can be computed in $NC^2$ - i.e. for an $n\times n$ matrix $A$, over the reals
or a finite field, we can compute the rank of $A$, ...
1
vote
0answers
112 views
Can we make a matrix stable by changing its upper-left submatrix?
A matrix $A$ is called strictly stable if its eigenvalues have negative real parts. Given a matrix $A \in \mathbb{R}^{n \times n}$, suppose we can change its upper-left $k \times k$ submatrix at will (...
1
vote
1answer
125 views
Why are folded Reed Solomon Codes considered non linear?
This is for my understanding. What am I missing?
7
votes
1answer
249 views
Using an oracle to find a vector $b$ for which $Ax=b$ has a solution
There is an oracle built around a hidden $m\times n$ matrix $A$ all of whose entries are 0 or 1, where $m>n$. The oracle takes as input an integer vector $b$ with positive entries, and answers as ...
1
vote
0answers
121 views
Non-backtracking paths and the zeta function of graphs
This question has also been posted on mathSE here: https://math.stackexchange.com/questions/2215888/non-backtracking-paths-and-the-ihara-zeta-function
For a connected $d$-regular graph $G=(V,E)$ with ...
3
votes
0answers
87 views
Complexity of Underdetermined Systems [closed]
Given a field $\mathbb{F}$ and a consistent underdetermined system $Ax=b$ over $\mathbb{F},$ $A\in \mathbb{F}^{m \times N}$ and $b \in \mathbb{F}^m,$ finding a vector $z \in \mathbb{F}^N$ such that $...
3
votes
0answers
127 views
Given a matrix $A$ maximize the number of positive elements in $Ax$ under specific constraints for $x$
Let $A = [a_{ij}]$ be a symmetric matrix with nonnegative values and $k << n/2$ a given constant. We want to rearrange the columns of the matrix such that the number of rows with the following ...
14
votes
1answer
1k views
The minimum number of arithmetic operations to compute the determinant
Has there been any work on finding the minimum number of elementary arithmetic operations needed to compute the determinant of an $n$ by $n$ matrix for small and fixed $n$? For example, $n=5$.
