Related papers: Symmetries of matrix multiplication algorithms. I
The well known algorithm of Volker Strassen for matrix multiplication can only be used for $(m2^k \times m2^k)$ matrices. For arbitrary $(n \times n)$ matrices one has to add zero rows and columns to the given matrices to use Strassen's…
A new method to represent and approximate rotation matrices is introduced. The method represents approximations of a rotation matrix $Q$ with linearithmic complexity, i.e. with $\frac{1}{2}n\lg(n)$ rotations over pairs of coordinates,…
We find a recursive algorithm for computing the precise centralizers of the complex orthogonal and symplectic groups, and hence the isotropy groups, with respect to the similarity transformation on the spaces of skew-symmetric and…
Many complex questions in biology, physics, and mathematics can be mapped to the graph isomorphism problem and the closely related graph automorphism problem. In particular, these problems appear in the context of network visualization,…
The complexity of matrix multiplication is a central topic in computer science. While the focus has traditionally been on exact algorithms, a long line of literature also considers randomized algorithms, which return an approximate solution…
A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation…
We explore new approaches for finding matrix multiplication algorithms in the commutative setting by adapting the flip graph technique: a method previously shown to be effective for discovering fast algorithms in the non-commutative case.…
It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…
The fastest quantum algorithms (for the solution of classical computational tasks) known so far are basically variations of the hidden subgroup problem with {$f(U[x])=f(x)$}. Following a discussion regarding which tasks might be solved…
Parallel transport is a fundamental tool to perform statistics on Rie-mannian manifolds. Since closed formulae don't exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms…
An open-source C++ framework for discovering fast matrix multiplication schemes using the flip graph approach is presented. The framework supports multiple coefficient rings -- binary ($\mathbb{Z}_2$), modular ternary ($\mathbb{Z}_3$) and…
Matrix multiplication is a fundamental kernel in high performance computing. Many algorithms for fast matrix multiplication can only be applied to enormous matrices ($n>10^{100}$) and thus cannot be used in practice. Of all algorithms…
We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph $H$ to $n$-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new…
A matrix algorithm is said to be superfast (that is, runs at sublinear cost) if it involves much fewer scalars and flops than the input matrix has entries. Such algorithms have been extensively studied and widely applied in modern…
This study proposes the "adaptive flip graph algorithm", which combines adaptive searches with the flip graph algorithm for finding fast and efficient methods for matrix multiplication. The adaptive flip graph algorithm addresses the…
The fast assembling of stiffness and mass matrices is a key issue in isogeometric analysis, particularly if the spline degree is increased. We present two algorithms based on the idea of sum factorization, one for matrix assembling and one…
This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with…
We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc…
Sparse matrix multiplication is an important component of linear algebra computations. Implementing sparse matrix multiplication on an associative processor (AP) enables high level of parallelism, where a row of one matrix is multiplied in…
Matrix multiplication is a fundamental building block for large scale computations arising in various applications, including machine learning. There has been significant recent interest in using coding to speed up distributed matrix…