Exact Algorithms and Lower Bounds for Stable Instances of Euclidean <i>k</i>-MEANS
Bibliographic record
Abstract
We investigate the complexity of solving stable or perturbation-resilient instances of k-means and k-median clustering in fixed dimension Euclidean metrics (or more generally doubling metrics). The notion of stable or perturbation resilient instances was introduced by Bilu and Linial [2010] and Awasthi, Blum, and Sheffet [2012]. In our context, we say a k-MEANS instance is α-stable if there is a unique optimum solution which remains unchanged if distances are (non-uniformly) stretched by a factor of at most α. Stable clustering instances have been studied to explain why heuristics such as Lloyd's algorithm perform well in practice. In this work we show that for any fixed ∊ > 0, (1 + ∊)-stable instances of k-MEANS in doubling metrics, which include fixed-dimensional Euclidean metrics, can be solved in polynomial time. More precisely, we show a natural multi-swap local-search algorithm in fact finds the optimum solution for (1 + ∊)-stable instances of k-MEANS and k-median in a polynomial number of iterations. We complement this result by showing that under a plausible PCP hypothesis this is essentially tight: that when the dimension d is part of the input, there is a fixed ∊0 > 0 such there is not even a PTAS for (1 + ∊0)-stable k-MEANS in ℝd unless NP=RP. To do this, we consider a robust property of CSPs; call an instance stable if there is a unique optimum solution x* and for any other solution x’, the number of unsatisfied clauses is proportional to the Hamming distance between x* and x’. Dinur, Goldreich, and Gur have already shown stable QSAT is hard to approximation for some constant Q [16], our hypothesis is simply that stable QSAT with bounded variable occurrence is also hard (there is in fact work in progress to prove this hypothesis). Given this hypothesis, we consider “stability-preserving” reductions to prove our hardness for stable k-MEANS. Such reductions seem to be more fragile and intricate than standard L-reductions and may be of further use to demonstrate other stable optimization problems are hard to solve.
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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 | 0.004 | 0.025 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.009 | 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".