Enhanced Matrix Chain Multiplication

Fast Matrix Multiplication

We give an overview of the history of fast algorithms for matrix multiplication. Along the way, we look at some other fundamental problems in algebraic complexity like polynomial evaluation. This exposition is self-contained. To make it accessible to a broad audience, we only assume a minimal mathematical background: basic linear algebra, familiarity with polynomials in several variables over r...

Matrix Multiplication I

These notes are based on a lecture given at the Toronto Student Seminar on February 2, 2012. The material is taken mostly from the book Algebraic Complexity Theory [ACT] and the lecture notes by Bläser and Bendun [Blä]. Starred sections are the ones I didn't have time to cover. This lecture discusses the problem of multiplying two square matrices. We will be working in the algebraic complexity ...

Coded Sparse Matrix Multiplication

In a large-scale and distributed matrix multiplication problem C = AB, where C ∈ Rr×t, the coded computation plays an important role to effectively deal with “stragglers” (distributed computations that may get delayed due to few slow or faulty processors). However, existing coded schemes could destroy the significant sparsity that exists in large-scale machine learning problems, and could resul...

Compressed Modular Matrix Multiplication

Matrices of integers modulo a small prime can be compressed by storing several entries into a single machine word. Modular addition is performed by addition and possibly subtraction of a word containing several times the modulus. We show how modular multiplication can also be performed. In terms of arithmetic operations, the gain over classical matrix multiplication is equal to the number of in...

Fast matrix multiplication