MatrixBandwidth.jl: Fast algorithms for matrix bandwidth minimization and recognition
Bibliographic record
Abstract
The bandwidth of an 𝑛 × 𝑛 matrix 𝐴 is the minimum non-negative integer 𝑘 ∈ {0, 1, … , 𝑛 -1} such that 𝐴 𝑖,𝑗 = 0 whenever |𝑖 -𝑗| > 𝑘.Reordering the rows and columns of a matrix to reduce its bandwidth has many practical applications in engineering and scientific computing: it can improve performance when solving linear systems, approximating partial differential equations, optimizing circuit layout, and more (Mafteiu-Scai, 2014).There are two variants of this problem: minimization, which involves finding a permutation matrix 𝑃 such that the bandwidth of 𝑃 𝐴𝑃 T is minimized, and recognition, which entails determining whether there exists a permutation matrix 𝑃 such that the bandwidth of 𝑃 𝐴𝑃 T is less than or equal to some fixed non-negative integer (an optimal permutation that fully minimizes the bandwidth of 𝐴 is not required).Accordingly, MatrixBandwidth.jloffers fast algorithms for matrix bandwidth minimization and recognition.Julia's (Bezanson et al., 2017) combination of easy syntax and high performance, along with its rapidly growing ecosystem for scientific computing, made it the ideal language of choice for this project.
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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 | 0.002 | 0.013 |
| Meta-epidemiology (narrow) | 0.004 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.035 | 0.032 |
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".