Share Your Experience: Take the 2024 Developer Survey

# Questions tagged [matrix-product]

The tag has no usage guidance.

45 questions
Filter by
Sorted by
Tagged with
77 views

### Uniform lower bounds in terms of the matrix multiplication exponent $\omega$?

Let $f(n)$ denote the minimum number of arithmetic operations needed for multiplying two $n\times n$ matrices, and $\omega = \inf\{p \ge 0: f(n) = O(n^p)\}$ be the matrix multiplication exponent. Is ...
55 views

### Matrix Multiplication $C\cdot (S\otimes S)$ where one product-and is a tensor square?

Let $D\in\mathbb{Z}^{1\times m}$. Let $A, A'\in\mathbb{Z}^{\sqrt{m}\times n}$, so $A\otimes A'\in\mathbb{Z}^{m\times n^2}$. Let $S\in\mathbb{Z}^{n\times 1}$. I am in a regime where $m \gg n$. I am ...
63 views

### computational complexity of sparse matrix powers

Given a sparse matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in it. What is the computational complexity of computing $A^k$, for some positive integer $k$? As $k$ gets larger, I ...
• 209
84 views

### Has there been any research on faster tensor inner products?

Matrix multiplication is a well studied problem which is recently back in the news due to deepmind. That got me wondering has anyone looked at the more general problem of faster tensor multiplication? ...
225 views

### Exact lower bound on matrix multiplication

The recent publication in Nature of "Discovering faster matrix multiplication algorithms with reinforcement learning" by Fawzi et al. has shown a method for discovering fewer element ...
• 905
86 views

### Flipping one bit to maximize BMM output

Consider a boolean matrix $A$ of size $N \times N$ and let $A^\top$ be its transpose. Let $C = AA^\top$ be the boolean matrix multiplication (BMM) result and let $c$ be the number of non-negative ...
• 1,177
1 vote
267 views

### An $O(n^2\log^c{n})$ algorithm for matrix multiplication in this paper?

In the newest version of this paper by Yijie Han, the author claims that matrix multiplication can be solved by an $\tilde{O}(n^2)$ algorithm. It should be a big result, but it is still in arxiv and ...
• 513
289 views

### Is the exponent in the rectangular matrix multiplication convex?

My question is regarding the paper "Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor". In the paper, the authors show an algorithm for multiplying a ...
• 630
140 views

### Triangle counting using approximate matrix multiplication -- suspicious paper

This paper [1] claims that for matrices with entries in $O(1)$, one can approximately multiply them in time $O(n^2 \log 1/\delta)$ to within error delta in the Frobenius norm (Theorem 1 in that paper)....
• 630
1 vote
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 ...
• 1,429
148 views

### Dynamic matrix-matrix multiplication

Suppose A and B are initial Boolean matrices. Let C = A*B. Suppose one can perform the sequence of the next operations: "set A[i,j] = 1", "set B[i,j] = 1". The result of each ...
• 421
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 ...
• 541
127 views

### Fast Finding Main Diagonal of Matrix Multiplication

Suppose we have two matrices $A_{m\times n}$ and $B_{m\times m}$. Such that $B$ is a symmetric positive definite matrix. Is it possible to compute main diganoal of $A^TBA$ in $O(n\times m)$?
• 189
147 views

### Trying to find Warren D. Smith's Matrix Multiplication article

One of my favorite articles on the complexity of matrix multiplication has gone dark. Warren Smith, Fast Matrix Algorithms And Multiplication Formulae, https://math.cst.temple.edu/~wds/matgrant.ps ...
• 2,359
116 views

344 views

### "Matrix complexity" - is it possible?

While browsing old CStheory.se posts, I ran across a fascinating blog post on the matrix mortality problem. Unless I've misinterpreted the problem, it states that given a finite collection of 3 x 3 ...
459 views

### state-of-the-art bit complexity of the determinant

I'm trying to understand the full bit-complexity of computing the determinant of an $n\times n$ integer matrix, with each entry represented by $M$ bits. I would like to know what is the state-of-the-...
• 1,224
317 views

### Commutative matrix multiplication algorithms

What is known about commutative algorithms like Winograd algorithm and its variants for Matrix Multiplication? Why is there not much study on them? Can they be asymptotically as efficient as Non-...
• 12.9k
325 views

### the product of a matrix and a permutation matrix [closed]

Can a permutation matrix (P) be used to change the rank of another matrix (M)? Is there any literature to this effect, or to the contrary? I've tried a few small examples and the resulting matrix (M2)...
• 17
650 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 ...
• 101
282 views

### the largest element of a matrix product

Given two matrices, I'm interested in finding the largest element of their product. I wonder if it's possible to do it significantly faster than the matrix multiplication the solution seems to require?...
• 259
1k views

### Determinants and Matrix Multiplication - Similarity and differences in algorithmic complexity and arithmetic circuit size

I am trying to understand the relation between algorithmic complexity and circuit complexity of Determinants and Matrix Multiplication. It is known that the determinant of an $n\times n$ matrix can ...
• 12.9k
772 views

### Bigger picture behind the choice of matrices in the Strassen algorithm

In the Strassen algorithm, to compute the product of two matrices $\mathbf{A}$ and $\mathbf{B}$, the matrices $\mathbf{A}$ and $\mathbf{B}$ are divided into $2 \times 2$ block matrices and the ...
402 views

### All Pairs Shortest Path - Directed graph with integer weights

I don't understand how Distance Product works (or Min Plus Product). If we replace each argument in $A$ from $a_{i,j}$ to $x^{a_{i,j}}$ and each argument in $B$ from $b_{i,j}$ to $x^{b_{i,j}}$ and ...
• 241
666 views

### What about apply maxplus algebra for all-pairs shortest paths?

I didn't find deep informations on Wikipedia about all-pairs shortest path, in particular I do not know what is the best algorithm to solve this problem beyond Floyd-Warshall's one, then I do not know ...
473 views

### Capacity of Uniquely Solvable Puzzle (USP)

In their seminal paper Group-theoretic algorithms for matrix multiplications, Cohn, Kleinberg, Szegedy and Umans introduce the concept of uniquely solvable puzzle (defined below) and USP capacity. ...
• 14.5k
8k views

### The computational complexity of matrix multiplication

I am looking for information about the computational complexity of matrix multiplication of rectangular matrices. Wikipedia states that the complexity of multiplying $A \in \mathbb{R}^{m \times n}$ by ...
• 149
657 views

### How can a problem have complexity $O(n^{2+\epsilon})$ for all $\epsilon > 0$?

For instance, it is believed that for any $\epsilon>0$ there is an algorithm for matrix multiplication that runs in $O(n^{2+\epsilon})$, but possibly no algorithm that runs in $O(n^2)$. How is this ...
• 390
1k views

### Multiplication of circulant matrices with a diagonal matrix

Let $A_{i}$, $B_{i}$ be a sequence of circulant matrices of size $n \times n$. We know that $\sum_{i=1}^{n}A_{i}B_{i}$ can be calculated in quadratic time (use FFT to diagonalize and add the diagonal ...
• 12.9k
228 views

• 3,440
5k views

### Quantum matrix multiplication?

It doesn't seem like this is known - but are there any interesting lower bounds on the complexity of matrix multiplication in the quantum computing model? Do we have any intuition that we can beat ...
• 3,798
2k views

### Fast sparse boolean matrix product with possible preprocessing

What are the most practically efficient algorithms for multiplying two very sparse boolean matrices (say, N=200 and there are just some 100-200 non-zero elements)? Actually, I have the advantage that ...
• 8,951