# Very Basics of Tensors with Graphical Notations: Unfolding, Calculations, and Decompositions

arXiv: https://arxiv.org/abs/2411.16094 · HTML: https://mlsanity.com/paper/2411.16094/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 35 references that are in the map.
- First posted on arXiv 25 Nov 2024.

## Builds on (most cited first, 3 of 3)

- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [An introduction to graphical tensor notation for mechanistic interpretability](https://mlsanity.com/paper/2402.01790/) (2024)
- [Broadcast Product: Redefining Shape-aligned Element-wise Multiplication and Beyond](https://mlsanity.com/paper/2409.17502/) (2024)

## Built on by (most cited first, 1 of 1)

- [Tensor Methods for Language Models: From Token Representation to Training, Adaptation, Inference, Compression, and Interpretability](https://mlsanity.com/paper/2608.30505/) (2026)

## Abstract (arXiv)

Tensor network diagram (graphical notation) is a useful tool that graphically represents multiplications between multiple tensors using nodes and edges. Using the graphical notation, complex multiplications between tensors can be described simply and intuitively, and it also helps to understand the essence of tensor products. In fact, most of matrix/tensor products including inner product, outer product, Hadamard product, Kronecker product, and Khatri-Rao product can be written in graphical notation. These matrix/tensor operations are essential building blocks for the use of matrix/tensor decompositions in signal processing and machine learning. The purpose of this lecture note is to learn the very basics of tensors and how to represent them in mathematical symbols and graphical notation. Many papers using tensors omit these detailed definitions and explanations, which can be difficult for the reader. I hope this note will be of help to such readers.