# Multilinear algebra

Outer products, tensors, einstein summation

October 8, 2021 — October 8, 2021

Tensors and stuff. Like linear algebra, but linear in multiple arguments.

Keywords: Ricci calculus, Einstein summation notation, index notation, subscript notation.

- Here is a compact explanation of Einstein summation, which turns out to be as simple as it can be, but no simpler.
- Tai-Danae Bradley
- Jeremy Kun:
- Ilan Ben-Yaacov and Francesc Roig, Index Notation for Vector Calculus
- Dan Fleisch’s Student’s Guide to Vectors and Tensors
- J.Pearson, Index Notation
- John Crimaldi, A Primer on Index Notation
- John D. Cook’s Tensor expo, Parts 2, 3

## 1 Tensor calculus

When some of the vectors are differentials.

Should this be smushed into differential geometry?

If we crack open a tensor textbook we get a lot of guff about general relativity and tensor fields and such, which is all very nice but not germane to typical machine learning applications. We want to start with the immediately-needed thing, which is some tidy notation conventions for dealing with multilinear operations without too many squiggles in our notation.

Soeren Laue, Matthias Mitterreiter, Joachim Giesen and Jens K. Mueller have been popularising such an approach recently. In their paper (Laue, Mitterreiter, and Giesen 2018), they argue that derivation of matrix differential results can be greatly simplified with Ricci calculus, and P.S. it can induce faster code. They have a website. MatrixCalculus.org which showcases this trick to do the useful special case of matrix calculus; Here are tasty readings on relevant bits of tensor machinery.

- generously multimediafied intro
- John D. Cook again
- Kees Dullemond & Kasper Peeters, Introduction to Tensor Calculus

## 2 Tensor decomposition

See tensor decomposition.

## 3 References

*Introduction to Vectors and Tensors, Vol 1: Linear and Multilinear Algebra*.

*Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis*.

*Differential Forms with Applications to the Physical Sciences*.

*Clifford Algebra to Geometric Calculus a Unified Language for Mathematics and Physics*.

*Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences*.

*Advances in Neural Information Processing Systems 31*.

*Acta Numerica*.