Related papers: Strassen's algorithm is not optimally accurate
The growth of big data in domains such as Earth Sciences, Social Networks, Physical Sciences, etc. has lead to an immense need for efficient and scalable linear algebra operations, e.g. Matrix inversion. Existing methods for efficient and…
Classic cache-oblivious parallel matrix multiplication algorithms achieve optimality either in time or space, but not both, which promotes lots of research on the best possible balance or tradeoff of such algorithms. We study modern…
We construct a Neural Network that approximates the matrix multiplication operator for any activation function such that there exists a Neural Network which can approximate the scalar multiplication function. In particular, we use the…
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.…
We present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of…
We propose a penalized likelihood framework for estimating multiple precision matrices from different classes. Most existing methods either incorporate no information on relationships between the precision matrices, or require this…
It is well known that Strassen and Winograd algorithms can reduce the computational costs associated with dense matrix multiplication. We have already shown that they are also very effective for software-based multiple precision…
We present an efficient numerical method for computing Hamiltonian matrix elements between non-orthogonal Slater determinants, focusing on the most time-consuming component of the calculation that involves a sparse array. In the usual case…
Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic…
In prior work, Gupta et al. (SPAA 2022) presented a distributed algorithm for multiplying sparse $n \times n$ matrices, using $n$ computers. They assumed that the input matrices are uniformly sparse--there are at most $d$ non-zeros in each…
In this paper, we present fast algorithms for the product of two multivariate polynomials in sparse representation. The bit complexity of our algorithms are studied in detail for various types of coefficients, and we derive new complexity…
Let {\alpha} be the maximal value such that the product of an n x n^{\alpha} matrix by an n^{\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic operations. In this paper we show that \alpha>0.30298, which improves the previous…
In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular…
Tensor factorization with hard and/or soft constraints has played an important role in signal processing and data analysis. However, existing algorithms for constrained tensor factorization have two drawbacks: (i) they require…
A formula is given for the propagation of errors during matrix inversion. An explicit calculation for a 2 by 2 matrix using both the formula and a Monte Carlo calculation are compared. A prescription is given to determine when a matrix with…
We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this…
We give explicit low-rank bilinear non-commutative schemes for multiplying structured $n \times n$ matrices with $2 \leq n \leq 5$, which serve as building blocks for recursive algorithms with improved multiplicative factors in asymptotic…
Algebraic matrix multiplication algorithms are designed by bounding the rank of matrix multiplication tensors, and then using a recursive method. However, designing algorithms in this way quickly leads to large constant factors: if one…
Fitting a matrix of a given rank to data in a least squares sense can be done very effectively using 2nd order methods such as Levenberg-Marquardt by explicitly optimizing over a bilinear parameterization of the matrix. In contrast, when…
In this paper, we propose and analyze a fast two-point gradient algorithm for solving nonlinear ill-posed problems, which is based on the sequential subspace optimization method. A complete convergence analysis is provided under the…