Questions tagged [linear-algebra]

Linear algebra deals with vector spaces and linear transformations.

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2
votes
2answers
132 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 ...
39
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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
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1answer
137 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 $$(\...
0
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0answers
59 views

s,t-Graphs representing infinite number of addition chains

I am looking at directed acyclic multi-graphs $G=(V,E)$ with a single source and sink with integer labeled arcs. Each vertex has exactly two inputs except $s$. Each vertex has at least one output ...
1
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0answers
47 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 ...
8
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3answers
393 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 $v_2 \in S$ ...
5
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0answers
51 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\...
5
votes
0answers
163 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
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0answers
21 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
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0answers
69 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 ...
11
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1answer
609 views

How do database aggregations form a monoid?

On cs.stackexchange I asked about the algebird scala library on github, speculating on why they might need an abstract algebra package. The github page has some clues: Implementations of Monoids ...
10
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0answers
131 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 ...
2
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1answer
121 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{ ...
30
votes
2answers
807 views

Is there a polynomial time algorithm to determine if the span of a set of matrices contains a permutation matrix?

I would like to find a polynomial time algorithm that determines if the span of a given set of matrices contains a permutation matrix. If any one knows if this problem is of a different complexity ...
3
votes
2answers
569 views

Matrix multiplication algorithms for research

I am implementing a matrix library for use in my research. This should support 2D matrices of size 100x100 (or more perhaps later on). I am a little confused about the algorithm I should be using for ...
3
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0answers
141 views

Complexity of eigenvalue problem

Many matrix diagonalization algorithms have time complexity $\mathcal{O}(n^3)$ where $n$ is the number of columns/raws (consider only square matrices). What is the best time lower bound it is known? ...
6
votes
1answer
256 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 $\...
6
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2answers
511 views

Has anyone mixed linear algebra with formal language theory in this way?

Let $G$ be the grammar: $$ S \rightarrow aAb \\ A \rightarrow aA + a + \epsilon $$ where $\epsilon$ is the empty string, $a,b$ are terminals and $S,A$ non-terminals with $S$ the start symbol. ...
6
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0answers
267 views

Spectral Graph Theory and Matroid Theory

I have just started grad school this year and I have been into Spectral Graph Theory for some time now. Recently I got introduced to Matroid Theory and although I know the field has been around for ...
4
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1answer
655 views

A matrix rank problem over finite fields: Is that a known problem?

The following problem is simple to state, but seems quite complicated to solve to me. Any hint or reference to related work is appreciated. Let $A \odot B$ denote elementwise multiplication of ...
2
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0answers
57 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 ...
8
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1answer
3k views

What is computational complexity of calculating the Variance-Covariance Matrix?

I am using a calculation of the Variance-Covariance matrix in a program I wrote (for Principal Component Analysis), and am wondering what the complexity of it is. While obviously the Eigenvector ...
2
votes
0answers
80 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 ...
40
votes
6answers
12k views

Complexity of Finding the Eigendecomposition of a Matrix

My question is simple: What is the worst-case running time of the best known algorithm for computing an eigendecomposition of an $n \times n$ matrix? Does eigendecomposition reduce to matrix ...
11
votes
1answer
206 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
292 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
65 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
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1answer
221 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-...
5
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0answers
107 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
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1answer
61 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 $...
15
votes
2answers
470 views

Status of Raghavendra's algorithm for solving linear systems in finite fields

In 2012, Lipton wrote a blog entry about a new algorithm for solving linear systems over finite fields by Prasad Raghavendra. The link to Raghavendra's draft paper on the topic is now dead, and I can'...
1
vote
1answer
84 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
59 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
88 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
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0answers
77 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
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0answers
488 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
296 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
171 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
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0answers
95 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
123 views

Why are folded Reed Solomon Codes considered non linear?

This is for my understanding. What am I missing?
7
votes
1answer
242 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
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0answers
96 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
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0answers
80 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 $...
59
votes
4answers
3k views

Evidence that matrix multiplication can be done in quadratic time?

It is widely conjectured that $\omega$, the optimal exponent for matrix multiplication, is in fact equal to 2. My question is simple: What reasons do we have for believing that $\omega = 2$? I'm ...
14
votes
1answer
476 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, $n=5$.
3
votes
0answers
123 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 ...
12
votes
2answers
805 views

Memory requirement for fast matrix multiplication

Suppose we want to multiply $n \times n$ matrices. The slow matrix multiplication algorithm runs in time $O(n^3)$ and uses $O(n^2)$ memory. The fastest matrix multiplication runs in time $n^{\omega + ...
8
votes
2answers
681 views

Dichotomy of the spectra of directed graphs

Compared to spectra of undirected graphs, which correspond to symmetric matrices, the spectra of directed graphs is not very well known: It is known that a directed graph $G = (V,E)$ has an adjacency ...
10
votes
1answer
232 views

Complexity of reachability in linear dynamical systems over finite fields

Let $A$ be a matrix over the finite field $\mathbb{F}_2 = \{0,1\}$ and $x$, $y$ be vectors of the space $\mathbb{F}_2^n$. I am interested in the computational complexity of deciding whether there ...
6
votes
1answer
209 views

Number of solutions for a system of linear equations over a finite ring

Let $R$ be a finite ring with operations $(+,\cdot)$. Let $A \in R^{m\times n}$ and $b\in R^{m}$. Questions: What is the complexity of counting the number of solutions to the system of equations $...