Valiant showed that Boolean matrix multiplication (BMM) can be used for CFG parsing. We prove a dual result: CFG parsers running in time O([Gl[w[ 3-e) on a grammar G and a string w can be used to multiply m x m Boolean matrices in time O(m3-e/3). In the process we also provide a formal definition of parsing motivated by an informal notion due to Lang. Our result establishes one of the first lim...

We present and compare several methods for multiplying banded square matrices. Various storage schemes and their implementations are discussed. Of particular interest is an algorithm for multiplying matrices by diagonals, which always references contiguous matrix elements. Two blocked implementations also are presented. These specialized routines are attractive for multiplying matrices whose ba...

The polar decomposition of an m x n matrix A of full rank, where rn n, can be computed using a quadratically convergent algorithm of Higham SIAMJ. Sci. Statist. Comput., 7 (1986), pp. 1160-1174]. The algorithm is based on a Newton iteration involving a matrix inverse. It is shown how, with the use of a preliminary complete orthogonal decomposition, the algorithm can be extended to arbitrary A. ...

Decomposing a given matrix into two factor matrices is a frequently used technique in data mining for uncovering underlying latent patterns in the data. Unlike in pure mathematics, the emphasis is put on obtaining results that are interpretable, rather than necessarily having a small reconstruction error. One approach to increase interpretability is to pose constraints on the factors. For examp...

A three-dimensional (3D) matrix multiplication algorithm for massively parallel processing systems is presented. The P processors are configured as a "virtual" processing cube with dimensions pl, p2, and p3 proportional to the matrices' dimensions-M, N, and K. Each processor performs a single local matrix multiplication of size Mlp, x Nlp, x Wp,. Before the local computation can be carried out,...

We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]/(f). In particular, the structure of the null space of the multiplicat...

Valiant showed that Boolean matrix multiplication (BMM) can be used for CFG parsing. We prove a dual result: CFG parsers running in time O([Gl[w[ 3-e) on a grammar G and a string w can be used to multiply m x m Boolean matrices in time O(m3-e/3). In the process we also provide a formal definition of parsing motivated by an informal notion due to Lang. Our result establishes one of the first lim...

Today, we will continue with our discussion of scalar and matrix concentration, with a discussion of the matrix analogues of Markov’s, Chebychev’s, and Chernoff’s Inequalities. Then, we will return to bounding the error for our approximating matrix multiplication algorithm. We will start with using Hoeffding-Azuma bounds from last class to get improved Frobenius norm bounds, and then (next time...

This paper addresses the problem of rectangular matrix multiplication on bidirectional linear systolic arrays (SAs). We analyze all bidirectional linear SAs in terms of efficiency. We conclude that the efficiency depends on the relation between the loop boundaries in the systolic algorithm (i.e. matrix dimensions). We point out which SA is the best choice depending on the relation between matri...

New matrix multiplication algorithms are proposed for massively parallel supercomputers with 2D/3D, all-port torus interconnection networks. The proposed algorithms are based on the traditional row-by-column multiplication matrix product model and employ a special routing pattern for better scalability. They compare favorably to the variants of Cannon’s and DNS algorithms since they allow matri...