Residual Symmetry for Linear Algebraic Equations and the Lorentz-Group Algorithm
Bibliographic record
Abstract
In the iterative solution of $n$ linear algebraic equations ${\bf B}{\bf x}={\bf b}$ by using the steepest descent method, i.e., ${\bf x}_{k+1}={\bf x}_k-\alpha_k {\bf B}^{\mbox{\scriptsize T}}{\bf r}_k$, it is known that the steplength $\alpha_k:={\bf r}_k^{\mbox{\scriptsize T}} {\bf A}{\bf r}_k/\|{\bf A}{\bf r}_k\|^2$ causes a slow convergence, where ${\bf r}={\bf B}{\bf x}-{\bf b}$ is the residual vector and ${\bf A}={\bf B}{\bf B}^{\mbox{\scriptsize T}}$. In this paper we study the residual symmetry of the residual dynamics for a scaled residual vector ${\bf y}\in {\mathbb S}^{n-1}_{\|{\bf r}_0\|}$, which as expressed in the augmented space is a nonlinear Lorentzian dynamical system, and is endowed with a cone structure in the Minkowski space with the Lorentz group $SO_o(n,1)$ being the internal symmetry group. Consequently, we can modify the steplength to $\alpha_k={\bf y}_k^{\mbox{\scriptsize T}} {\bf A}{\bf y}_k/\|{\bf A}{\bf y}_k\|^2$ with ${\bf y}_k$ being computed by a Lorentz group algorithm (LGA) based on $SO_o(n,1)$, which can significantly improve the convergence speed and enhance the stability. Several linear inverse problems are used to assess the numerical performance of the LGA.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".