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Questions tagged [linear-algebra]

Linear algebra deals with vector spaces and linear transformations.

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11 votes
1 answer
368 views

Theorem 2.4(i) in Valiant-Vazirani paper "NP is as easy as detecting unique solutions"

I have a question about the paper "NP is as easy as detecting unique solutions" by Valiant and Vazirani, specifically the proof of the Theorem 2.4(i). The proof starts by saying Clearly, $...
deltaepsilonnn's user avatar
3 votes
0 answers
43 views

Affine point matching in general dimensions

Fix a positive integer $d$ and consider the $d$-dimensional Euclidean space $\mathbb{R}^d$. Let $S$ and $T$ finite subsets of $\mathbb{R}^d$ of the same size $n$. Under the assumption that $S$ and $T$ ...
rr314's user avatar
  • 131
3 votes
0 answers
48 views

Name for the dimension of subspace of support of boolean function?

The support of a boolean function is defined as the set of $x$ such that $f(x) = 1$. I want to quantify the dimension of the support of the function (i.e. how large the subspace is that is made up of ...
wecanfibonacciit's user avatar
1 vote
0 answers
53 views

Find linear combination with small support

Let $v_1,\dots,v_n$ be a basis of a vector subspace of $\Bbbk^N$, say for $\Bbbk$ a finite field. I would like an algorithm to find a linear combination of the $v_i$'s with small support, i.e. with ...
grok's user avatar
  • 191
-3 votes
1 answer
108 views

Algebra in complexity theory

Recently an idea came to my mind. Suppose $V$ is vector space and $\dim V = n$. Then, since $V \simeq \mathbb{R}^n$, any conjunction of $n$ boolean formulas $\phi_1, \ldots, \phi_n$ about vectors from ...
aeet's user avatar
  • 1
4 votes
0 answers
52 views

Approximate Matrix Multiplication with approximation guarantees that ignore large elements?

Approximate matrix multiplication is a technique to replace a matrix product $A^t B$ with a smaller product $(\Pi A)^t(\Pi B)$. Intuitively, if $\Pi$ is chosen from a suitable distribution that has ...
Mark Schultz-Wu's user avatar
4 votes
1 answer
174 views

On the plausability of quantum RAM

I'm fairly new to quantum computation and quantum complexity theory, but I came across some articles that suggest that quantum RAM (QRAM) is not very realistic assumption. For example some works show ...
terett's user avatar
  • 151
1 vote
0 answers
67 views

Circuit depth of linear algebra operations

I was checking the following paper [1] about low-depth PRFs from lattices. In table 1 on page 4, there is comparison with other constructions, and it shows evaluation depths of certain PRFs. I'm not ...
terett's user avatar
  • 151
0 votes
0 answers
26 views

Linear modular equalities with $0/1$ solution

Let $Ax\equiv b\bmod q$ be a $n\times n$ modular linear system known to have $0/1$ solution where $q$ is a large prime. We can solve in $NC^2$ for general linear systems using determinant and matrix ...
Turbo's user avatar
  • 12.9k
0 votes
1 answer
203 views

Construction of a collection of subsets of $\{1,2,\ldots,n\}$ with certain properties

Let $n$ be a large positive integer. Given a collection $\mathfrak S$ of subsets of $[n] := \{1,2,\ldots,n\}$, and a vector $z=(z_1,\ldots,z_n)\in \{\pm 1\}^n$, define $$ f_{\mathfrak S}(z) := \sum_{\...
dohmatob's user avatar
  • 291
3 votes
2 answers
165 views

Worst-case complexity of computing a certain non-standard dot product + algorithms realizing this complexity

Let $n$ be a large positive integer. Give a nonempty collection $\mathcal S$ of subsets of $[n] := \{1,2,\ldots,n\}$, define an inner-product on $\mathbb R^n$ by \begin{eqnarray} \langle x,y\rangle_{\...
dohmatob's user avatar
  • 291
1 vote
1 answer
62 views

Impact HHL caveat relaxation on quantum advantage

We know that there are four caveats for the exponential speedup proven for the HHL algorithm. Could anyone answer how that exponential speedup evolves as we relax the caveats? For example, the ...
Omar Shehab's user avatar
3 votes
2 answers
868 views

Johnson-Lindenstrauss and the largest eigenvalue of a matrix

Johnson-Lindenstrauss (JL) lemma shows that for any vector $u$ in $\mathbb{R}^d$, the vector $\frac{1}{\sqrt{k}}Ru$ satisfies $(1-\epsilon)\|u\|\leq \frac{1}{k}\|Ru\|^2\leq (1+\epsilon)\|u\|$ with ...
anurag anshu's user avatar
0 votes
1 answer
76 views

An inequality about median of points in higher dimensions

Let $S$ be a set of points in $\mathbf{R^d}$ and let $m$ be the median of this set of points, i.e. $\sum_{x \in S} || x - y||$ is minimized when we have $y=m$. Now let $z$ be an arbitrary point in $\...
David's user avatar
  • 1
4 votes
0 answers
135 views

Boolean matrix $M$ with Boolean rank $r$ but real rank $2^r$

$\newcommand{\F}{\mathbb{F}}\newcommand{\R}{\mathbb{R}}$ Question is in the title basically: does there exist a Boolean matrix $M$ where $\operatorname{rank}_{\F_2}(M)=r$ but $\operatorname{rank}_{\R}(...
Ash's user avatar
  • 49
2 votes
2 answers
141 views

Are there publicly available fast Laplacian solvers?

In a much celebrated result, we know that there is a $ O(m\log \frac{1}{\epsilon}) $ time algorithm for solving laplacian systems of the form $Lx=b$ where $L$ is a laplacian of a graph $G$ with $m$ ...
user3508551's user avatar
  • 1,153
3 votes
1 answer
112 views

Strongly polynomial time algorithm for shortest convex combination

Problem: Let $S$ be a finite set of vectors. Let $C$ be their convex hull. Compute $\operatorname{argmin}_{x \in C} \|x\|$. Reference 1 gives an algorithm for this problem that is finite-time (Section ...
user76284's user avatar
  • 662
27 votes
1 answer
467 views

How many multiplications are needed to compute the determinant of a 3×3 matrix?

In a comment on this question in 2016, Jeffrey Shallit remarked: I've asked experts about this, and apparently it is not even currently known whether or not 9 multiplications are needed to compute ...
Robin Houston's user avatar
3 votes
1 answer
987 views

Complexity of matrix diagonalization

I'm probably missing a trivial answer, but somehow I can't find it. Given symmetric matrix $A \in \mathbb R^{n \times n}$, what's the complexity of diagonalizing the matrix, i.e. finding diagonal $\...
Dmitry's user avatar
  • 201
6 votes
2 answers
236 views

Reference request for linear algebra over GF(2)

I have been looking for materials on the linear algebra over $GF(2)$ but so far I haven't found any substantial textbooks or notes on this subject. In fact in one of the notes I found the introduction ...
gen's user avatar
  • 290
3 votes
1 answer
172 views

complexity class of a function - linear combinations and reductions (Fermionant, immanant, $GL_n$ representations)

The fermionant is a matrix function from physics, which is indexed by a positive integer $k$: \begin{align} \operatorname{Ferm}_k(A) = \sum_{\lambda} d_{\lambda}^{(k)} \operatorname{Imm}_{\lambda^T}(A)...
Teferi's user avatar
  • 31
1 vote
0 answers
32 views

Reference showing global optimality of local minima for matrix factorization

Consider the following matrix factorization problem: Given an $n\times m$ matrix M, find $n\times r$ and $m\times r$ matrices $U$ and $V$ such that $||UV^T - M||_F^2$ is minimized. I have heard it ...
user2684957's user avatar
1 vote
0 answers
70 views

Smallest nonzero eigenvalue of a sum of +1/-1 rank-one matrices?

Suppose we have a $k\times k$ matrix $A = \sum_{i=1}^{n} a_i a_i^T$ where $n \leq \mathrm{poly}(k)$ and each $a_i\in\{-1,1\}^{k}$. It is easy to prove that the largest eigenvalue of $A$ is at most $\...
Octopus's user avatar
  • 31
1 vote
0 answers
116 views

Is this proof of $LP$ being in $coNP$ correct?

I am referring to the natural decision version of the Linear Programming problem: given $A \in \mathbb{Q}^{m \times n}, \ b \in \mathbb{Q}^m, \ c \in \mathbb{Q}^n, \ \alpha \in \mathbb{Q}$, does there ...
reservoir's user avatar
4 votes
0 answers
69 views

Precise rank of a sparse integer matrix

Consider a large sparse rectangular integer matrix. Is there a way to compute its exact rank that is better in terms of speed and/or memory usage compared to a dense matrix?
Andrei Matveiakin's user avatar
2 votes
1 answer
120 views

Solver for uniform matroid isomorphism

I want to solve the following coNP-complete problem efficiently in practice: Given a linear matroid represented as $k \times n$ matrix over a finite field $\mathbb{F}_p$ (where $p$ is large prime), ...
Laakeri's user avatar
  • 1,786
3 votes
1 answer
71 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, ...
Mah's user avatar
  • 133
0 votes
0 answers
25 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 $\mathbf u$ ...
Maja Trela's user avatar
4 votes
0 answers
112 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 ...
relG's user avatar
  • 209
1 vote
0 answers
64 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 ...
eepperly16's user avatar
3 votes
1 answer
234 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}\...
Sudipta Roy's user avatar
2 votes
0 answers
34 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 ...
Shaull's user avatar
  • 5,646
1 vote
1 answer
125 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?
NietzscheanAI's user avatar
7 votes
0 answers
278 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 ...
Jakub Konieczny's user avatar
1 vote
1 answer
219 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 ...
M.Monet's user avatar
  • 1,429
4 votes
0 answers
91 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 ...
Naysh's user avatar
  • 686
5 votes
1 answer
369 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, ...
Yaroslav Bulatov's user avatar
1 vote
1 answer
326 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 ...
Turbo's user avatar
  • 12.9k
1 vote
0 answers
77 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 ...
user avatar
-3 votes
1 answer
131 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 $...
Will's user avatar
  • 215
10 votes
1 answer
263 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 ...
Teferi's user avatar
  • 103
7 votes
0 answers
181 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 ...
Will's user avatar
  • 215
5 votes
2 answers
296 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 ...
saxon_dr's user avatar
44 votes
3 answers
6k 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 ...
Brian's user avatar
  • 541
2 votes
1 answer
174 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 $$(\...
Magdalen Dobson's user avatar
1 vote
0 answers
54 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 ...
Root's user avatar
  • 387
5 votes
0 answers
79 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 ...
Shlw Kevin's user avatar
5 votes
0 answers
219 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 ...
newbie's user avatar
  • 235
0 votes
0 answers
35 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-...
Sudipta Roy's user avatar
1 vote
0 answers
102 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 ...
ron653's user avatar
  • 11

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