Fast matrix multiplication via compiler‐only layered data reorganization and intrinsic lowering
Bibliographic record
Abstract
Abstract The resurgence of machine learning has increased the demand for high‐performance basic linear algebra subroutines (BLAS), which have long depended on libraries to achieve peak performance on commodity hardware. High‐performance BLAS implementations rely on a layered approach that consists of tiling and packing layers—for data (re)organization—and micro kernels that perform the actual computations. The algorithm for the tiling and packing layers is target independent but is parameterized to the memory hierarchy and register‐file size. The creation of high‐performance micro kernels requires significant development effort to write tailored assembly code for each architecture. This hand optimization task is complicated by the recent introduction of matrix engines by 's (Matrix Multiply Assist—MMA), (Advanced Matrix eXtensions—AMX), and (Matrix Extensions—ME) to deliver high‐performance matrix operations. This article presents a compiler‐only alternative to the use of high‐performance libraries by incorporating, to the best of our knowledge and for the first time, the automatic generation of the layered approach into LLVM, a production compiler. Modular design of the algorithm, such as the use of LLVM's matrix‐multiply intrinsic for a clear interface between the tiling and packing layers and the micro kernel, makes it easy to retarget the code generation to multiple accelerators. The parameterization of the tiling and packing layers is demonstrated in the generation of code for the MMA unit on IBM's POWER10. This article also describes an algorithm that lowers the matrix‐multiply intrinsic to the MMA unit. The use of intrinsics enables a comprehensive performance study. In processors without hardware matrix engines, the tiling and packing delivers performance up to (Intel)—for small matrices—and more than (POWER9)—for large matrices—faster than PLuTo, a widely used polyhedral optimizer. The performance also approaches high‐performance libraries and is only slower than OpenBLAS and on‐par with Eigen for large matrices. With MMA in POWER10 this solution is, for large matrices, over faster the vector‐extension solution, matches Eigen performance, and achieves up to ofBLASpeak performance.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".