Related papers: Unit triangular factorization of the matrix symple…
The idea of decomposing a matrix into a product of structured matrices such as triangular, orthogonal, diagonal matrices is a milestone of numerical computations. In this paper, we describe six new classes of matrix decompositions,…
Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, since they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as…
The Clustered Factor (CF) model induces a block structure on the correlation matrix and is commonly used to parameterize correlation matrices. Our results reveal that the CF model imposes superfluous restrictions on the correlation matrix.…
Matrix multiplication optimization remains a fundamental challenge in computational mathematics. This work introduces a novel approach that discovers matrix multiplication schemes whose coefficients are restricted to the set $\{-1, 0, 1\}$…
Constraint programming (CP) is a powerful tool for modeling mathematical concepts and objects and finding both solutions or counter examples. One of the major strengths of CP is that problems can easily be combined or expanded. In this…
Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut…
The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group $G$ satisfying a simple combinatorial condition (the Triple Product Property).…
We show that a "mate'' $B$ of a set $A$ in a near-factorization $(A,B)$ of a finite group $G$ is unique. Further, we describe how to compute the mate $B$ very efficiently using an explicit formula for $B$. We use this approach to give an…
The decomposition of a matrix, as a product of factors with particular properties, is a much used tool in numerical analysis. Here we develop methods for decomposing a matrix $C$ into a product $X Y$, where the factors $X$ and $Y$ are…
Previously work of the author with Meier and Starkston showed that every closed symplectic manifold $(X,\omega)$ with a rational symplectic form admits a trisection compatible with the symplectic topology. In this paper, we describe the…
A new method involving particle diagrams is introduced and developed into a rigorous framework for carrying out embedded random matrix calculations. Using particle diagrams and the attendant methodology including loop counting it becomes…
In a recent paper, an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices -- the so called `matrix properties'. In this paper we extend this algorithm to…
We propose a new geometric method of IR factorization in sector decomposition. The problem is converted into a set of problems in convex geometry. The latter problems are solved using algorithms in combinatorial geometry. This method…
The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering in the mapping class group of the total surface. We prove that the balanced superelliptic mapping class…
We consider a refinement of triangular factorization for unitary matrix valued loops.
Matrix factorization is an important mathematical problem encountered in the context of dictionary learning, recommendation systems and machine learning. We introduce a new `decimation' scheme that maps it to neural network models of…
This paper concerns the enumeration of simultaneous conjugacy classes of $k$-tuples of commuting matrices in the upper triangular group $GT_n(\mathbf F_q)$ and unitriangular group $UT_m(\mathbf F_q)$ over the finite field $\mathbf F_q$ of…
This paper is dedicated to the factorizations of the symmetric group. Introducing a new bijection for partitioned 3-cacti, we derive an el- egant formula for the number of factorizations of a long cycle into a product of three permutations.…
In this article, circular arcs are considered both individually and as elements of a piecewise circular curve. The endpoint parameterization proves to be quite advantageous here. The perspective of symplectic geometry provides new vectorial…
Matrix extensions have emerged as an essential feature in modern CPUs to address the surging demands of AI workloads. However, existing designs often incur substantial hardware and software design overhead. Tight coupling with the CPU…