Backward pertubation analysis of least squares problems
Bibliographic record
Abstract
This thesis is concerned with backward perturbation analyses of the linear least squares (LS) and related problems. Two theoretical measures are commonly used for assessing the backward errors that arise in the approximate solution of such problems. These are called the normwise relative backward error (NRBE) and the minimal backward error (MBE). An important new relationship between these two measures is presented, which shows that the two are essentially equivalent. New upper bounds on the NRBE and MBE for the LS problem are given and related to known bounds and estimates. One important use of backward perturbation analysis is to design stopping criteria for iterative methods. In this thesis, minimum-residual iterative methods for solving LS problems are studied. Unexpected convergence behaviour in these methods is explained and applied to show that commonly used stopping criteria can in some situations be much too conservative. More reliable stopping criteria are then proposed, along with an efficient implementation in the iterative algorithm LSQR. In many applications the data in the LS problem come from a statistical linear model in which the noise follows a multivariate normal distribution whose mean is zero and whose covariance matrix is the scaled identity matrix. A description is given of typical convergence of the error that arises in minimum-residual iterative methods when the data come from such a linear model. Stopping criteria that use the information from the linear model are then proposed and compared to others that appear in the literature. Finally, some of these ideas are extended to the scaled total least squares problem.
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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 | 0.003 | 0.012 |
| 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 | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".