A Relaxed Algorithm for Online Matrix Inversion
Why this work is in the frame
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Bibliographic record
Abstract
We consider a variation of the well known problem of computing the unique solution to a nonsingular system Ax=b of n linear equations over a field K. The variation assumes that A has generic rank profile and requires as output not only the single solution vector A-1b Ε Kn x 1, but rather the solution to all leading principle subsystems. Most importantly, the rows of the augmented system [A||b] are given one at a time from first to last, and as soon as the next row is given the solution to the next leading principal subsystem should be produced. We call this problem ONLINESYSTEM. The obvious iterative algorithm for ONLINESYSTEM has a cost in terms of field operations that is cubic in the dimension of A. In this paper we introduce a relaxed representation for the inverse and show how to obtain an algorithm for ONLINESYSTEM that allows us to incorporate matrix multiplication. As an application we show how to introduce fast matrix multiplication into the inherently iterative algorithm for row rank profile computation presented previously by the authors.
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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 it