Questions tagged [linear-algebra]
Linear algebra deals with vector spaces and linear transformations.
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63 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
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0answers
115 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
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2answers
252 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
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1answer
111 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
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1answer
208 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
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0answers
52 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
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0answers
72 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
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1answer
203 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 ...
12
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1answer
275 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-...
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0answers
67 views
Why do spectral ultrasparsifiers need to be trees
Given a graph $G$ with $n$ vertices and $m$ edges, a $(\kappa,h)$-ultra-sparsifier $H$ is a graph on the same set of vertices such that: [1]
$H$ consists of a spanning tree $T$ along with $\frac{hm}{\...
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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
204 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
104 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
$$\...
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1answer
59 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 $...
1
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1answer
78 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
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0answers
58 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
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1answer
84 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 ...
5
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0answers
73 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
442 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 ...
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1answer
248 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 ...
8
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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 $A$, ...
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0answers
92 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 (...
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1answer
114 views
Why are folded Reed Solomon Codes considered non linear?
This is for my understanding. What am I missing?
8
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1answer
238 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 ...
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0answers
83 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 ...
4
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0answers
76 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 $...
4
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0answers
122 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 ...
15
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1answer
435 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, $n=5$.
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2answers
420 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'...
8
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2answers
571 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
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1answer
230 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
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1answer
191 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 $...
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0answers
63 views
Fourier expansion of boolean functions and affine subspaces
Let $f:\mathbb{F}^n_2\rightarrow \{0,1\}$ be a function constant on an affine subspace $V$ of co-dimension $t$. Assume that that $V$ is a linear subspace, by replacing $f(x)$ with $f(x+v)$ for some $v ...
7
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2answers
342 views
Estimating the rank of a large sparse matrix
Consider a large sparse n by n matrix. Are there any methods to estimate its rank in time roughly proportional the number of elements in the matrix?
12
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2answers
642 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 + ...
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0answers
106 views
Optimization of function on the Eigenvector of symmetric Matrices
My problem is like that,
I want to use some function, say $f$ on the principal eigenvector(PEV), $X_1 \in R^n$ of an adjacency matrix, $A \in R^{n \times n}$, of a simple undirected and unweighted ...
4
votes
0answers
134 views
Testing for satisfiability of a system of linear equations over GF(2)
Consider a system linear equations in $x$, $Ax =b$, where A is an $n\times n$ matrix, and $b$ is a column vector, and all operations are over $GF(2)$.
Is it easier to check satisfiability of the ...
4
votes
1answer
150 views
How well do subspaces hit sets
Let $S\subset F_2^n$ be a subset of size $\epsilon\cdot 2^n$.
Say I choose a random subspace $V$ of dimension $k$ in $F_2^n$.
I want to know what is the smallest $k$ such that $V$ `hits' $S$, i.e., $V\...
3
votes
2answers
430 views
Quantum complexity of maximum inner product search
Given two matrices $X \in \mathbb{R}^{m \times k}$, $Y \in \mathbb{R}^{n \times k}$, maximum inner product search (MIPS) asks for the largest $l$ entries of $X Y^T$. Typically $k \ll m, n$ (many ...
7
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1answer
214 views
Checking properties of matrices
Given a sparse matrix $A$ in $\mathbb{Z}^{n\times n}$, how easily could one check whether a coefficient $\alpha_k$ of the characteristic polynomial $P_A$ of $A$ is equal to $0$ (without the need to ...
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1answer
65 views
How the hardness of hidden subgroup problem in $S_n$ changes as the order of the subgroup grows?
In Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. In this paper it is showed that no hidden subgroup algorithm can distinguish the trivial ...
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0answers
359 views
Complexity of approximating the range of a matrix
Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define:
$$S_M = |\{Mx : x \in \{-1,1\}^n\}|.$$
I believe that it is NP-hard to compute $S_M$ exactly, by applying ...
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0answers
123 views
How to efficiently generate a random 0-1 matrix of a given rank
How to efficiently generate a random $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$?
3
votes
2answers
116 views
Is there any hidden subgroup of a symmetric group which can be efficiently determined?
There have been a number of cases where efficient hidden subgroup algorithms have been found for specific non-Abelian groups with very specific structures. Why haven't we found any efficient quantum ...
3
votes
0answers
88 views
Why hidden subgroup problem is easy for very large subgroup?
I am going through QUANTUM MECHANICAL ALGORITHMS FOR THE
NONABELIAN HIDDEN SUBGROUP PROBLEM by Grigni et al. On page 2, it is said that solving the hidden subgroup problem becomes very easy when the ...
3
votes
1answer
336 views
Matrix multiplication with transpose
Let $A,B\in\mathbb{F}^{n\times n}$ be two $n\times n$ matrices over
the underlying field $\mathbb{F}$. In addition, $A$ is guaranteed
to be a symmetric matrix, i.e, $A=A^{T}$. We assume complexity ...
1
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1answer
118 views
Low rank approximation of matrix under $l_2$ norm
Theorem 14 of this paper by Tam´as Sarl´os gives a relative error rank-$k$ approximation of a given matrix $A$ under the frobenius norm. I am looking for reference of a similar result (relative error ...
13
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4answers
763 views
Finding the sparsest solution to a system of linear equations
How hard is it to find the sparsest solution to a system of linear equations?
More formally, consider the following decision problem:
Instance: A system of linear equations with integer coefficients ...
7
votes
1answer
389 views
Finding the number of independent rows of a matrix
There is a $n\times n$ matrix $A$, and we are asked to find the number $N(A)$ of independent rows in it, i.e. rows that are not a linear combination of the other rows. Clearly, if $rank(A)=n$, then $N(...
3
votes
2answers
725 views
Min Hamming distance of a given string from substrings of another string
Let $U$ be a small finite set.
Consider the following problem:
Input: two strings $u \in U^k$ and $v\in U^n$ with $k \leq n$.
Output: a (contiguous) substring of $v$ of length $k$
with the minimum ...
