Related papers: Randomization of Approximate Bilinear Computation …
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…
Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. Unlike binary matrix factorization based on standard arithmetic, BMF employs the Boolean OR and AND operations for the…
We propose a non-commutative algorithm for multiplying 2x2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity…
We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
Debugging accumulation of floating-point errors is hard; ideally, computer should track it automatically. Here we consider twofold approximation of an exact real with value + error pair of floating-point numbers. Normally, value + error sum…
A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…
In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients…
The Boolean matrix factorization problem consists in approximating a matrix by the Boolean product of two smaller Boolean matrices. To obtain optimal solutions when the matrices to be factorized are small, we propose SAT and MaxSAT…
This paper considers a probabilistic model for floating-point computation in which the roundoff errors are represented by bounded random variables with mean zero. Using this model, a probabilistic bound is derived for the forward error of…
We present an approximate algorithm for matrix multiplication based on matrix sketching techniques. First one of the matrix is chosen and sparsified using the online matrix sketching algorithm, and then the matrix product is calculated…
We develop a novel, fundamental and surprisingly simple randomized iterative method for solving consistent linear systems. Our method has six different but equivalent interpretations: sketch-and-project, constrain-and-approximate, random…
In recent years, randomized algorithms have established themselves as fundamental tools in computational linear algebra, with applications in scientific computing, machine learning, and quantum information science. Many randomized matrix…
This text investigates relations between two well-known family of algorithms, matrix factorisations and recursive linear filters, by describing a probabilistic model in which approximate inference corresponds to a matrix factorisation…
This paper describes an approximate method for global optimization of polynomial programming problems with bounded variables. The method uses a reformulation and linearization technique to transform the original polynomial optimization…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some of the output variables are also input variables, linked by a linear dependency. Fundamental examples…
Boolean matrix factorisation aims to decompose a binary data matrix into an approximate Boolean product of two low rank, binary matrices: one containing meaningful patterns, the other quantifying how the observations can be expressed as a…
In this paper we study a worst case to average case reduction for the problem of matrix multiplication over finite fields. Suppose we have an efficient average case algorithm, that given two random matrices $A,B$ outputs a matrix that has a…
Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by…
We present a simple randomized algorithm for approximate matrix multiplication (AMM) whose error scales with the *output* norm $\|AB\|_F$. Given any $n\times n$ matrices $A,B$ and a runtime parameter $r\leq n$, the algorithm produces in…