On the Arithmetic Complexity of Strassen-Like Matrix Multiplications.
Bibliographic record
Abstract
The Strassen algorithm for multiplying 2 × 2 matrices requires seven multiplications and 18 additions. The recursive use of this algorithm for matrices of dimension n yields a total arithmetic complexity of (7n2.81 − 6n2) for n = 2k. Winograd showed that using seven multiplications for this kind of multiplications is optimal, so any algorithm for multiplying 2 × 2 matrices with seven multiplications is therefore called a Strassen-like algorithm. Winograd also discovered an additively optimal Strassen-like algorithm with 15 additions. This algorithm is called the Winograd’s variant, whose arithmetic complexity is (6n2.81 − 5n2) for n = 2k and (3.73n2.81 − 5n2) for n = 8 · 2k, which is the best-known bound for Strassen-like multiplications. This paper proposes a method that reduces the complexity of Winograd’s variant to (5n2.81 + 0.5n2.59 + 2n2.32 − 6.5n2) for n = 2k. It is also shown that the total arithmetic complexity can be improved to (3.55n2.81 + 0.148n2.59 + 1.02n2.32 − 6.5n2) for n = 8 · 2k, which, to the best of our knowledge, improves the best-known bound for a Strassen-like matrix multiplication algorithm.
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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 | 0.001 | 0.011 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.009 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 0.003 |
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".