Bibliographic record
Abstract
Krylov-based algorithms have recently been used, in combination with other methods, to solve systems of linear equations and to perform related matrix computations over finite fields. For example, large and sparse systems of linear equations over F 2 are formed during the use of the number field sieve for integer factorization, and elements of the null space of these systems are sampled. Block Lanczos algorithms have been used to perform this computation with considerable success. However, the algorithms that are currently in use do not appear to be reliable in the worst case. This report presents a block Lanczos algorithm that is somewhat simpler than block algorithms that are presently in use and provably reliable for computations over large fields. This can be implemented, using a field extension, in order to produce several uniformly and independently selected elements from the null space at once. The amortized cost to produce each vector closely matches the cost to generate such a vector with the methods currently in use. An algorithm is also given to compute the rank of a matrix A E F m x n over a small finite field F. The expected number of matrix-vector products by A or A t used by this algorithm is in O (r), where r is the rank of A. The expected number of additional field operations used by this algorithm is within a polylog factor of r (n+m), and the expected storage space is within a polylog factor of n + m. This is asymptotically more efficient than existing black box algorithms to compute the rank of a matrix over a small field, assuming that the cost of matrix-vector products dominates the cost of other operations.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".