Deterministic Reduction of Integer Nonsingular Linear System Solving to Matrix Multiplication
Bibliographic record
Abstract
We present a deterministic reduction to matrix multiplication for the problem of linear system solving: given as input a nonsingular A \in \Z^n \times n and b \in \Z^n \times 1 , compute A^-1 b. We give an algorithm that computes the minimal integer e such that all denominators of the entries in 2^eA^-1 are relatively prime to 2. Then, for a b that has entries with bitlength O(n) times as large as the bitlength of entries in A, we give an algorithm to produce the 2-adic expansion of 2^eA^-1 b up to a precision high enough such that A^-1 b over \Q can be recovered using rational number reconstruction. Both e and the 2-adic expansion can be computed in O(\MM(n,łog n + łog ||A||) \times (łog n) (łog n + łoglog ||A||)) bit operations. Here, ||A||= \max_ij |A_ij | and \MM(n,d) is the cost to multiply together, modulo 2^d, two n \times n integer matrices. Our approach is based on the previously known reductions of linear system solving to matrix multiplication which use randomization to find an integer lifting modulus X that is relatively prime to \det A. Here, we derandomize by first computing a permutation P, a unit upper triangular M, and a diagonal S with \det S a power of two, such that U := APMS^-1 is an integer matrix with 2 \perp \det U. This allows our modulus X to be chosen a power of 2.
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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 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 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".