# Space Filling Curves is All You Need: Communication-Avoiding Matrix Multiplication Made Simple

arXiv: https://arxiv.org/abs/2601.16294 · HTML: https://mlsanity.com/paper/2601.16294/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 55 references that are in the map.
- First posted on arXiv 22 Jan 2026.

## Builds on (most cited first, 18 of 18)

- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
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- [Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation](https://mlsanity.com/paper/1609.08144/) (2016)
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- [Ansor: Generating High-Performance Tensor Programs for Deep Learning](https://mlsanity.com/paper/2006.06762/) (2020)
- [Demystifying Parallel and Distributed Deep Learning: An In-Depth Concurrency Analysis](https://mlsanity.com/paper/1802.09941/) (2018)
- [Tensor Comprehensions: Framework-Agnostic High-Performance Machine Learning Abstractions](https://mlsanity.com/paper/1802.04730/) (2018)
- [Deep Learning for Computational Chemistry](https://mlsanity.com/paper/1701.04503/) (2017)
- [A Survey of Deep Learning for Scientific Discovery](https://mlsanity.com/paper/2003.11755/) (2020)
- [Red-blue pebbling revisited: near optimal parallel matrix-matrix multiplication](https://mlsanity.com/paper/1908.09606/) (2019)
- [Feature Learning in Deep Neural Networks - Studies on Speech Recognition Tasks](https://mlsanity.com/paper/1301.3605/) (2013)
- [Mathematical Foundations of the GraphBLAS](https://mlsanity.com/paper/1606.05790/) (2016)
- [Tensor Processing Primitives: A Programming Abstraction for Efficiency and Portability in Deep Learning & HPC Workloads](https://mlsanity.com/paper/2104.05755/) (2021)
- [oneDNN Graph Compiler: A Hybrid Approach for High-Performance Deep Learning Compilation](https://mlsanity.com/paper/2301.01333/) (2023)
- [Harnessing Deep Learning and HPC Kernels via High-Level Loop and Tensor Abstractions on CPU Architectures](https://mlsanity.com/paper/2304.12576/) (2023)
- [Towards a high-performance AI compiler with upstream MLIR](https://mlsanity.com/paper/2404.15204/) (2024)
- [Library Liberation: Competitive Performance Matmul Through Compiler-composed Nanokernels](https://mlsanity.com/paper/2511.13764/) (2025)
- [A Study of Energy and Locality Effects using Space-filling Curves](https://mlsanity.com/paper/1606.06133/) (2016)

## Abstract (arXiv)

General Matrix Multiplication (GEMM) is the cornerstone of HPC workloads and Deep Learning. State-of-the-art (SOTA) vendor libraries tune tensor layouts, parallelization schemes and cache blocking to minimize data movement across the memory hierarchy and maximize throughput. However, optimal settings for these parameters depend on the target platform and matrix shapes, making exhaustive tuning infeasible. In this work, we address this cumbersome scheduling search and tuning using space-filling curves (SFC). We partition the matrix multiplication using advancements in SFC, and obtain platform-oblivious and shape-oblivious matrix multiplication schemes with a high degree of data locality. We extend the SFC-based work partitioning to implement Communication-Avoiding (CA) algorithms with replication techniques in a seamless fashion. The resulting SFC-CA GEMM achieves provable asymptotic communication optimality for both square and rectangular matrix regimes. Across four x86 and Arm platforms, SFC-CA GEMM outperforms vendor libraries by up to 5.5× per shape and 1.8× in weighted harmonic mean (WHM) throughput. Last, we show the impact of our work on two real-world applications by leveraging our SFC-CA GEMM as a compute backend: i) prefill of LLM inference with speedups up to 1.85× over SOTA inference runtimes, and ii) distributed-memory matrix multiplication with speedups up to 2.3× over the SOTA distributed-memory GEMM framework with vendor-optimized compute backend.