Related papers: Partial and weighted matrix multiplication

Tensor network diagram (graphical notation) is a useful tool that graphically represents multiplications between multiple tensors using nodes and edges. Using the graphical notation, complex multiplications between tensors can be described…

We generalize the tensor product theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a ''conformal vertex algebra'' or even more generally,…

Fast matrix-by-matrix multiplication (hereafter MM) is a highly recognized research subject. The record upper bound 3 of 1968 on the exponent of the complexity MM decreased below 2.38 by 1987, applies to celebrated problems in many areas of…

For two positive definite adjointable operators $M$ and $N$, and an adjointable operator $A$ acting on a Hilbert $C^*$-module, some properties of the weighted Moore-Penrose inverse $A^\dag_{MN}$ are established. When $A=(A_{ij})$ is…

We develop an automated framework for proving lower bounds on the bilinear complexity of matrix multiplication over finite fields. Our approach systematically combines orbit classification of the restricted first matrix and dynamic…

This educational article highlights the geometric and algebraic complexities that distinguish tensors from matrices, to supplement coverage in advanced courses on linear algebra, matrix analysis, and tensor decompositions. Using the case of…

Let $M_{\langle u,v,w\rangle}\in C^{uv}\otimes C^{vw}\otimes C^{wu}$ denote the matrix multiplication tensor (and write $M_n=M_{\langle n,n,n\rangle}$) and let $det_3\in ( C^9)^{\otimes 3}$ denote the determinant polynomial considered as a…

We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r = 6 in the space of bilinear maps C^4 x C^4 -> C^4. This…

Using a binary representation for basis elements of an algebra combined with a framework of multiplier and index functions, a connection has been established between the structure of a large class of algebras and the XOR componentwise…

As nowadays Machine Learning (ML) techniques are generating huge data collections, the problem of how to efficiently engineer their storage and operations is becoming of paramount importance. In this article we propose a new lossless…

The representation theory of tensor functions is essential to constitutive modeling of materials including both mechanical and physical behaviors. Generally, material symmetry is incorporated in the tensor functions through a structural or…

Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of…

It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…

We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed…

Obtaining superlinear lower bounds on tensor rank is a major open problem in complexity theory. In this paper we propose a generalization of the approach used by Strassen in the proof of his 3n/2 border rank lower bound. Our approach…

We propose a new method for low-rank approximation of Moore-Penrose pseudoinverses (MPPs) of large-scale matrices using tensor networks. The computed pseudoinverses can be useful for solving or preconditioning of large-scale overdetermined…

We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved exactly in polynomial time in certain situations, and we discuss how it can…

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix…

We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, or monomial algebras. We analyse them using Strassen's laser method and obtain an upper bound $2.431$ on $\omega$. We also…

Left-right and conjugation actions on matrix tuples have received considerable attention in theoretical computer science due to their connections with polynomial identity testing, group isomorphism, and tensor isomorphism. In this paper, we…