Bibliographic record
Abstract
In this work, we study fairness in centroid clustering. In this problem, k cluster centers must be placed given n points in a metric space, and the cost to each point is its distance to the nearest cluster center. Recent work of Chen et al. [Chen et al., 2019] introduces the notion of a proportionally fair clustering, in which no group of at least n/k points can find a new cluster center which provides lower cost to each member of the group. They propose a greedy capture algorithm which provides a 1+√2 approximation of proportional fairness for any metric space, and derive generalization bounds for learning proportionally fair clustering from samples in the case where a cluster center can only be placed at one of finitely many given locations in the metric space. \nWe focus on the case where cluster centers can be placed anywhere in the (usually infinite) metric space. In case of the L² distance metric over ℝ^t, we show that the approximation ratio of greedy capture improves to 2. We also show that this is due to a special property of the L² distance; for the L¹ and L^∞ distances, the approximation ratio remains 1+√2. We provide universal lower bounds which apply to all algorithms. \nWe also consider metric spaces defined on graphs. For trees, we show that an exact proportionally fair clustering always exists and provide an efficient algorithm to find one. The corresponding question for general graph remains an interesting open question. \nFinally, we show that for the L² distance, checking whether a proportionally fair clustering exists and implementing greedy capture over an infinite metric space are NP-hard problems, but (approximately) solvable in special cases. We also derive generalization bounds which show that an approximately proportionally fair clustering for a large number of points can be learned from a small number of samples. Our work advances the understanding of proportional fairness in clustering, and points out many avenues for future work.
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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 | 0.005 | 0.017 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.003 | 0.004 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.004 | 0.004 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 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".