Bibliographic record
Abstract
The best approximation problem is of central importance in convex optimization. It is popular to use the Douglas--Rachford splitting method or the method of alternating projections to solve this problem. In this thesis, we use a classical concept, circumcenter, in Euclidean geometry to solve the best approximation problem. First, we introduce the new notion, circumcenter operator. Symmetrical and asymmetrical formulae of the circumcenter operator are provided. A sufficient condition of the existence of the circumcenter is provided. A characterization of the existence of the circumcenter of three distinct points is presented. In addition, we define the new concept: circumcenter mapping induced by operators. When we choose the operators from sets of compositions of reflectors, we obtain the circumcenter mapping induced by reflectors, which is proper, i.e., the value of the circumcenter mapping induced by reflectors is always a unique point in the space. In light of this consequence, we are able to deduce the circumcenter method induced by reflectors. We also consider the circumcenter operator induced by projectors. Both proper and improper examples are provided. Moreover, we prove that for some special sets, the circumcenter methods induced by reflectors converge at least as fast as the MAP, symmetrical MAP or some of their accelerated version to solve the best approximation problem. We also find some drawbacks of the circumcenter mapping induced by reflectors. Finally, numerical experiments are implemented to compare convergence rates of seven solvers: four circumcenter methods induced by reflectors, DRM, MAP and symmetrical MAP. As the plots of performance profiles illustrate, the experimental results are consistent with our theoretical results in the thesis. Additional comparisons with the DRM and the circumcenter method induced by reflectors are made.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".