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

2025-05-28

AI Summary

Existing tensor layout design methodologies struggle to balance flexibility and performance, limiting their adaptability to complex deep learning algorithms and heterogeneous hardware. This paper introduces the first unified modeling framework grounded in $mathbb{F}_2$ linear algebra, formally representing layouts as binary linear transformation matrices acting on hardware address bits. Our approach enables verifiable, constant-time conversions between arbitrary layouts—marking the first such capability, and overcoming the limitations of case-by-case layout definitions and $O(n^2)$ conversion schemes prevalent in prior work. Integrated end-to-end into the Triton compiler, it supports bit-level address mapping analysis and automated layout optimization. Experimental evaluation demonstrates significant performance improvements across multiple operators, simplifies backend development, and resolves several longstanding defects in legacy layout systems.

Technology Category

• Machine Learning: Hardware-aware ML
• Planning, Routing, and Scheduling: Planning with Language Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

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.

Problem

• Achieving flexible and performant tensor layout design
• Handling increasing complexity of DL algorithms and hardware
• Eliminating quadratic explosion in layout-to-layout conversions

Innovation

• Models tensor layouts using linear algebra over F2
• Represents layouts as binary matrices for flexibility
• Integrates with Triton for optimized computation