A Global Optimization Algorithm for <i>K</i> -Center Clustering of One Billion Samples
Bibliographic record
Abstract
This paper presents a practical global optimization algorithm for the K-center clustering problem, which aims to select K samples as the cluster centers to minimize the maximum within-cluster distance. Specifically, we propose a reduced-space branch and bound scheme that guarantees convergence to the global optimum in a finite number of steps by only branching on the regions of centers. To improve efficiency, we design a two-stage decomposable lower bound, the solution of which can be derived in a closed form. In addition, we also propose several structure-exploiting acceleration techniques to narrow down the region of centers, including bounds tightening, sample reduction, and parallelization. Extensive studies on synthetic and real-world data sets have demonstrated that our algorithm can solve the K-center problems to global optimal within four hours for 10 million samples in the serial mode and 1 billion samples in the parallel mode, whereas existing studies can only address small-scale problems (e.g., thousands of samples). Moreover, compared with the state-of-the-art heuristic methods, the global optimum obtained by our algorithm reduces the objective function by an average of 25.8% on these synthetic and real-world data sets. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2019-05499]. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2023.00218 .
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How this classification was reachedexpand
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".