Linear Layouts: Robust Code Generation of Efficient Tensor Computation Using $\mathbb{F}_2$

TL;DR

This paper introduces Linear Layouts, a systematic method using linear algebra over $_2$ to model tensor layouts, improving flexibility, efficiency, and robustness in tensor computation for deep learning.

Contribution

It presents a novel linear algebra-based framework for tensor layouts that simplifies design, enables generic conversions, and enhances integration with Triton.

Findings

Linear layouts enable flexible tensor layout definitions.

They improve kernel optimization in Triton.

They reduce engineering effort and fix bugs in Triton's layout system.

Abstract

Efficient tensor computation is a cornerstone of modern deep learning (DL) workloads, yet existing approaches struggle to achieve flexible and performant design and implementation of tensor layouts -- mappings between logical tensors and hardware resources. The increasing complexity of DL algorithms and hardware demands a generic and systematic approach to handling tensor layouts. In this work, we introduce Linear Layouts, a novel approach that models tensor layouts using linear algebra over $\mathbb{F}_2$. By representing tensor layouts as binary matrices acting on the bits of the hardware representation, our approach enables a generic layout definition -- as opposed to the classical case-by-case approach -- and allows for generic layout-to-layout conversions, eliminating the quadratic explosion that plagues existing solutions. We integrate linear layouts with Triton and demonstrate their effectiveness in optimizing individual Triton operators as well as kernels written in Triton. We also show that linear layouts reduce engineering effort in the compiler backend while fixing several bugs in Triton's legacy layout system.