# It's All Just Vectorization: einx, a Universal Notation for Tensor Operations

arXiv: https://arxiv.org/abs/2607.27987 · HTML: https://mlsanity.com/paper/2607.27987/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 23 references that are in the map.
- First posted on arXiv 30 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Real-Time Single Image and Video Super-Resolution Using an Efficient Sub-Pixel Convolutional Neural Network](https://mlsanity.com/paper/1609.05158/) (2016)
- [Tensor Comprehensions: Framework-Agnostic High-Performance Machine Learning Abstractions](https://mlsanity.com/paper/1802.04730/) (2018)
- [Named Tensor Notation](https://mlsanity.com/paper/2102.13196/) (2021)
- [Getting to the Point. Index Sets and Parallelism-Preserving Autodiff for Pointful Array Programming](https://mlsanity.com/paper/2104.05372/) (2021)
- [Penzai + Treescope: A Toolkit for Interpreting, Visualizing, and Editing Models As Data](https://mlsanity.com/paper/2408.00211/) (2024)
- [Typesafe Abstractions for Tensor Operations](https://mlsanity.com/paper/1710.06892/) (2017)

## Abstract (arXiv)

Tensor operations represent a cornerstone of modern scientific computing. However, the Numpy-like notation adopted by predominant tensor frameworks is often difficult to read and write and prone to so-called shape errors, i.a., due to following inconsistent rules across a large, complex collection of operations. Alternatives like einsum and einops have gained popularity, but are inherently restricted to few operations and lack the generality required for a universal model of tensor programming. To derive a better paradigm, we revisit vectorization as a function for transforming tensor operations, and use it to both lift lower-order operations to higher-order operations, and conceptually decompose higher-order operations to lower-order operations and their vectorization. Building on the universal nature of vectorization, we introduce einx, a universal notation for tensor operations. It uses declarative, pointful expressions that are defined by analogy with loop notation and represent the vectorization of tensor operations. The notation reduces the large APIs of existing frameworks to a small set of elementary operations, applies consistent rules across all operations, and enables a clean, readable and writable representation in code. We provide an implementation of einx that is embedded in Python and integrates seamlessly with existing tensor frameworks: https://github.com/fferflo/einx