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30

There's an algorithm for multiplying an $N \times N^{0.172}$ matrix with an $N^{0.172} \times N$ matrix in $N^2 \operatorname{polylog}\left(N\right)$ arithmetic operations. The main identity used for it comes from Coppersmith's paper "Rapid multiplication of rectangular matrices", but the explanation for why it leads to $N^2 \operatorname{polylog}\left(N\... 22 No such matrix exists. The Desnanot-Jacobi identity says that for$i \neq j$, $$\det M_{ij}^{ij} \det M = \det M_i^i \det M_j^j -\det M_i^j \det M_j^i$$ so using this, we get $$\det M_{12}^{12} \det M = \det M_{1}^{1} \det M_{2}^{2} - \det M_{1}^{2} \det M_{2}^{1}$$ But your requirements force the left-hand-side to be 0 (mod 2) and the right-hand-... 20 The matrix multiplication exponent being$\omega$does not guarantee that there is an algorithm that runs in time$O(n^\omega)$, but only that for each$\epsilon > 0$, there is an algorithm that runs in$O(n^{\omega+\epsilon})$. Indeed if you can find an algorithm that runs in time$O(n^2 \mathrm{polylog}(n))$, then this shows that$\omega = 2$. You can ... 20 The decision versions of many common problems in linear algebra over the integers (or rationals) are in the class$\mathsf{DET}$, see the paper Gerhard Buntrock, Carsten Damm, Ulrich Hertrampf, Christoph Meinel: Structure and Importance of Logspace-MOD Class. Mathematical Systems Theory 25(3): 223-237 (1992)$\mathsf{DET}$is contained in$\mathsf{DSPACE}(...

18

It is equivalent to deciding whether two given multigraphs (or edge-labelled graphs) are isomorphic or not, which is known to be equivalent to the usual graph isomorphism problem.

17

There is a combinatorial algorithm by Mahajan and Vinay that works over commutative rings: http://cjtcs.cs.uchicago.edu/articles/1997/5/contents.html

17

Your question is equivalent to whether $A_1, \dotsc, A_k$ generate a nilpotent algebra, which in turn is equivalent to each of the $A_i$ being nilpotent. Hence not only is it decidable, but in $\tilde O(n^{2 \omega})$ time where $\omega$ is the exponent of matrix multiplication. Let $\mathcal{A}$ be the associative algebra generated by the $A_i$: that is, ...

16

There will generally not exist a set of $n$ linearly independent vectors $x$ such that $Ax = x$; this can only happen for $A$ being the identity matrix. On the other extreme, there may be no such vectors at all. For example, the matrix $$A = \left(\begin{array}{cc} 0 & 1\\ 1 & 1 \end{array}\right)$$ has full rank, but there are no nonzero vectors ...

16

The space usage is at most $O(n^2)$ for all Strassen-like algorithms (i.e. those based on upper bounding the rank of matrix multiplication algebraically). See Space complexity of Coppersmith–Winograd algorithm However, I realized in my previous answer that I did not explain why the space usage is $O(n^2)$... so here goes something hand-wavy. Consider what a ...

15

If you know the factorization of $m = p_1^{e_1} \cdots p_n^{e_n}$ you can compute modulo each $p_i^{e_i}$ separately and then combine the results using Chinese remaindering. If $e_i = 1$, then computing modulo $p_i^{e_i}$ is easy, since this is a field. For larger $e_i$, you can use Hensel lifting.

15

De Groote (On Varieties of Optimal Algorithms for the Computation of Bilinear Mappings. II. Optimal Algorithms for 2x2-Matrix Multiplication. Theor. Comput. Sci. 7: 127-148, 1978) proves that there is only one algorithm to multiply $2 \times 2$-matrices with 7 multiplications up to equivalence. This might be a unique feature of $2 \times 2$-matrix ...

14

You ask why database aggregations have monoidal structure. Say we want to combine data values $a$ and $b$, but want to keep things general -- these may be integers, strings, floating point numbers, vectors, matrices, probability distributions, sets, or anything else we want to store and manipulate. So we denote the "aggregation" of $a$ and $b$ by $a.b$. ...

14

The conjecture fails over $\mathbb{F}_2$. Look at $M(x, y) = \langle x, y \rangle \bmod 2$, and $x, y \in \{0, 1\}^n$. The communication complexity is $\Omega(n)$, but the rank of $M$ over $\mathbb{F}_2$ is $n$, by the linearity of inner product.

13

I think the answer to your first question is also $\widetilde O(n^3 \log( \|A\| + \|b\|))$ due to the following arguments: Edmonds' paper does not describe a variant of Gaussian elimination but it proves that any number computed in a step of the algorithm is a determinant of some submatrix of A. By Schrijver's book on Theory of Linear and Integer Programming ...

13

Classical work of Coppersmith shows that for some $\alpha > 0$, one can multiply an $n \times n^\alpha$ matrix with an $n^\alpha \times n$ matrix in $\tilde{O}(n^2)$ arithmetic operations. This is a crucial ingredient of Ryan Williams's recent celebrated result. François le Gall recently improved on Coppersmith's work, and his paper has just been ...

13

The paper Vandermonde Matrices, NP-Completeness, and Transversal Subspaces [ps] by Alexander Chistov, Hervé Fournier, Leonid Gurvits and Pascal Koiran may be relevant to your question (though it does not answer it). They prove the $\mathsf{NP}$-completeness of the following problem: Given an $n\times m$ matrix over $\mathbb Z$ ($n\le m$), decide whether ...

12

Isn't the following a counter-example? Let $f(x)$ be the majority of $x_1, \ldots, x_{1/\epsilon^2}$, which is an indicator of a set of size $2^n/2$, so $d = 1$. However, $\hat{f}(\{i\}) = \Theta(\epsilon)$ for $1 \le i \le 1/\epsilon^2$, so you have $1/\epsilon^2$ linearly independent large Fourier coefficients.

12

For matrices of sizes $k = 2,3$ the Matrix Powering Positivity Problem is in $\mathsf{P}$ (cf. this paper to appear in STACS 2015)

12

The exponent of computing a basis of the kernel is the same as the exponent of matrix multiplication, see the book Algebraic Complexity Theory by Bürgisser, Clausen & Shokrollahi. So it can be done in time $O(n^{2.38})$.

12

I cannot say for sure if you will consider the following approach as better, but from a complexity-theoretic point of view there is a more efficient solution. The idea is to rephrase your question in the first-order theory of the rationals with addition and order. You have that $P_1$ is included in $P_2$ if and only if \begin{align*} \Phi := \forall \vec{x}....

12

Consider the problem $\text{MAX-LIN}(R)$ of maximizing the number of satisfied linear equations over some ring $R$, which is often NP-hard, for example in the case $R=\mathbb{Z}$ Take an instance of this problem, $Ax=b$ where $A$ is a $n\times m$ matrix. Let $k=m+1$. Construct a new linear system $\tilde{A}\tilde{x} = \tilde{b}$, where $\tilde{A}$ is a $kn ... 12 There is a very neat randomized algorithm by Cheung, Kwok, and Lau, which computes the rank of an$n\times m$matrix$A$,$n\le m$, in time$O(\mathrm{nnz}(A) + \min\{n^\omega, n\cdot\mathrm{nnz}(A)\})$. Here$\mathrm{nnz}(A)$is the number of nonzero elements of$A$, and$\omega$is the matrix multiplication exponent. Not quite what you are asking for, but ... 12 Take a look at https://www.sciencedirect.com/science/article/pii/S0022000008001141 "The complexity of satisfiability problems: Refining Schaefer's theorem" by Allender et al. which answer your questions: (1) Open (2)$\oplus L = L$(3) Yes 12 Well, one thing is I think that all the constructions we know of - and even the families of potential constructions that people have proposed (e.g., Cohn-Umans approaches, generalizations of Coppersmith-Winograd) - would "simply" produce a family of algorithms$A_\epsilon$running in time$O(n^{2+\epsilon})$. So to have a single algorithm which ran in$O(n^2 ...

11

To solve this problem there is a fast deterministic algorithm based on Smith normal forms whose worst-case complexity is upper-bounded by the cost of matrix-multiplication over the integers modulo $m$. For any matrix $A$, the algorithm outputs its Smith normal form, from where $\text{det}(A)$ can be easily computed. More concretely, define $\omega$ so that ...

11

Matrices A and B whose elements are in a field F are similar (in F) if and only if they have the same Frobenius normal form. According to a quick search, it seems that the Frobenius normal form of an n×n matrix can be computed with O(n3) field operations [Sto98], and that this can be improved to something comparable to the complexity of matrix ...

11

There is some sort of explanation in the book Algebraic Complexity Theory by Bürgisser, Clausen and Shokrollahi (p. 11-12). The idea is to start with two bases $A_0,A_1,A_2,A_3$ and $B_0,B_1,B_2,B_3$ of the space of $2\times 2$ real matrices which satisfy the following property: $A_iB_j \in \{0,A_0,A_1,A_2,A_3,B_0,B_1,B_2,B_3\}$. Furthermore, $A_0 = B_0$. To ...

11

This is a paper that seems to improve upon the result. https://arxiv.org/pdf/1209.3995.pdf

11

Josh Alman showed some cool lower bound results of MM, which won CCC 2019 best student paper award! http://drops.dagstuhl.de/opus/volltexte/2019/10834/pdf/LIPIcs-CCC-2019-12.pdf

10

The fastest way to multiply dense matrices on a modern computer is to call BLAS.

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