Improving matrix-vector multiplication via lossless grammar-compressed matrices
Bibliographic record
Abstract
As nowadays Machine Learning (ML) techniques are generating huge data collections, the problem of how to efficiently engineer their storage and operations is becoming of paramount importance. In this article we propose a new lossless compression scheme for real-valued matrices which achieves efficient performance in terms of compression ratio and time for linear-algebra operations. Experiments show that, as a compressor, our tool is clearly superior to gzip and it is usually within 20% of xz in terms of compression ratio. In addition, our compressed format supports matrix-vector multiplications in time and space proportional to the size of the compressed representation, unlike gzip and xz that require the full decompression of the compressed matrix. To our knowledge our lossless compressor is the first one achieving time and space complexities which match the theoretical limit expressed by the k -th order statistical entropy of the input. To achieve further time/space reductions, we propose column-reordering algorithms hinging on a novel column-similarity score. Our experiments on various data sets of ML matrices show that our column reordering can yield a further reduction of up to 16% in the peak memory usage during matrix-vector multiplication. Finally, we compare our proposal against the state-of-the-art Compressed Linear Algebra (CLA) approach showing that ours runs always at least twice faster (in a multi-thread setting), and achieves better compressed space occupancy and peak memory usage. This experimentally confirms the provably effective theoretical bounds we show for our compressed-matrix approach.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".