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Sensitivity Sampling with Predictions for k-Means Clustering

T0 review · 0 major / 4 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read Predicted centers from one snapshot let sensitivity sampling build optimal-size k-means coresets on later snapshots without recomputing a bi-criteria approximation.

desk verdict Solid, usable speed-up for sensitivity coresets on sequential data; theory cleanly extends Bansal et al. to polylog-quality predictions and the experiments back it up. read the letter →

arxiv 2607.04949 v1 pith:E22USMWG submitted 2026-07-06 cs.LG cs.DS

Cristian Boldrin , Fabio Vandin This is my paper
classification cs.LGcs.DS
keywords k-meansclusteringcoresetssensitivitysamplingalgorithmswithpredictionsbi-criteriaapproximationstreamingsnapshots
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Sensitivity sampling builds small weighted coresets for k-means by sampling points with probability proportional to their importance (sensitivity). Computing those importances requires a good approximate clustering, which is the dominant cost. This paper shows that a coarser approximation than previously required is still enough to obtain the best-known coreset size, and that centers already computed on an earlier large sample from the same distribution serve as such an approximation on later samples. The resulting algorithm, PreSenS, therefore reuses historical centers as predictions, skips the expensive re-approximation step, and still produces theoretically optimal coresets. Experiments on sequences of real data snapshots confirm that the method matches the clustering quality of full sensitivity sampling while running several times faster, and far outperforms uniform sampling on uneven data.

What carries the argument

PreSenS: a sensitivity-sampling coreset algorithm whose sampling probabilities are computed from a supplied set of predicted centers rather than from a bi-criteria approximation recomputed from scratch; the size guarantee holds under the relaxed polylogarithmic approximation regime of Theorem 2, and the predictions are justified by the distributional transfer result of Corollary 1.

What would settle it

On a sequence of large i.i.d. samples from a fixed distribution with positive OPT, measure whether centers that are a constant-factor bi-criteria approximation on the first sample remain an O(1)-factor bi-criteria approximation on later samples and whether the resulting PreSenS coresets achieve the same clustering cost as full sensitivity sampling; a large, consistent gap on either measure falsifies the claim.

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Extended reading notes

Core claim

If a set of predicted centers A satisfies cost(P,A) ≤ α OPT_k(P) with k ≤ |A| ≤ βk and α,β only polylogarithmic in k/ε, then sensitivity sampling with the induced probabilities still yields an ε-coreset of the optimal size Õ(k ε^{-2} min(√k, ε^{-2})). Moreover, when successive point sets are large i.i.d. draws from the same unknown distribution, centers that form an (α,β) bi-criteria approximation on one set remain an (O(α),β) bi-criteria approximation on the next, with high probability.

Load-bearing premise

Successive data sets must be large enough independent draws from essentially the same fixed distribution; if the distribution drifts substantially, the old centers stop being a good approximation and the coreset guarantees no longer apply.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper introduces PreSenS, a sensitivity-sampling algorithm for Euclidean k-means coresets that replaces the usual bi-criteria approximation step with a predicted set of centers A. Theorem 2 shows that if cost(P,A) ≤ α OPT_k(P) and k ≤ |A| ≤ βk with α,β = O(polylog(k ε^{-1})), then m = Õ(k ε^{-2} min(√k, ε^{-2})) samples still yield an ε-coreset w.h.p., matching the optimal subset-coreset size of Bansal et al. Corollary 1 (via Ben-David’s statistical clustering framework) asserts that an (α,β)-bi-criteria solution computed on one large i.i.d. sample from an unknown distribution D remains an (O(α),β) solution on a later independent sample of comparable size. Experiments on five real snapshot sequences demonstrate that reusing centers from the first snapshot yields clustering costs competitive with recomputing k-means++ each time, while reducing total coreset-construction time by roughly 3.5–5×.

Significance. If the claims hold, the work supplies a practical and theoretically justified way to amortize the dominant cost of sensitivity sampling across a sequence of related datasets—an increasingly common setting (sensor streams, daily/monthly aggregates). The extension of the Bansal et al. coreset-size bound from constant-factor to polylog-factor approximations is a clean technical contribution that immediately justifies the use of noisy predictors. The transfer theorem is a careful, non-circular application of existing statistical-clustering machinery, and the experimental suite (means + 95 % CIs, multiple coreset sizes, distortion estimates, unique-point counts, publicly released C++ code) is thorough and reproducible. These strengths make the paper a solid addition to both the coreset literature and the algorithms-with-predictions literature.

minor comments (4)
  1. Several typographical errors appear in the main text and appendix (e.g., “unkwown”, “distorsions”, “PreSenS” inconsistently capitalized, “eO” vs. Õ). A careful proof-reading pass would improve readability.
  2. Figure 1 and Figure 2 use log-scale cost/distortion ratios; the absolute costs (or at least the absolute cost of the full-dataset baseline) would help readers judge practical relevance on the larger NYC TLC instances.
  3. Section 4.1 briefly discusses distribution drift and an additive discrepancy term, but never states an explicit quantitative bound. Adding a short corollary that makes the additive term precise would strengthen the “Relaxing the Fixed Distribution Assumption” paragraph.
  4. In Algorithm 1 the probability formula mixes four terms whose relative weights are fixed at 1/4; a one-sentence justification (or a pointer to the corresponding calculation in Bansal et al.) would help readers who have not memorized that paper.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: coreset-size and transfer guarantees are derived from first-principles concentration/nets analysis and an external statistical-clustering model; predictions are an input modeling assumption, not a quantity defined by the target coreset error.

full rationale

The paper's strongest claims (Theorem 2 on coreset size under polylog-quality predicted centers, and Corollary 1 / Theorem 3 on transfer of bi-criteria centers across i.i.d. snapshots) do not reduce by construction to their inputs. Theorem 2 adapts the sensitivity-sampling analysis of Bansal et al. [4] (external FOCS 2024 work by different authors) by explicitly tracking the dependence of coreset size on the approximation factors α, β of the predicted centers A; the probability distribution P[p] is defined from A (Alg. 1), and the subsequent concentration, ring/band/type partitioning, cost-vector nets, and Gaussian chaining arguments are standard first-principles tools that do not fit parameters to the target ε-coreset error. Theorem 3 / Corollary 1 invoke the statistical clustering framework of Ben-David [5] (external) under an explicit i.i.d. same-distribution modeling assumption; the paper itself flags the limitation under distribution drift (Section 4.1) and quantifies the price of inaccurate predictions in Theorem 2 (size multiplies by β·max(1,α^{2})). Experiments evaluate on held-out later snapshots of public datasets and report degradation precisely when the prediction quality collapses (IntelLab). There is no self-definitional loop, no fitted parameter renamed as a prediction of a closely related quantity, and no load-bearing uniqueness theorem imported from the present authors. The single minor self-referential element is ordinary reuse of the same algorithmic template as the cited external sensitivity-sampling literature; it does not force the central claims. Score 1 reflects only the ordinary presence of self-citation of the algorithmic setting, not circularity of the derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard concentration and geometric facts plus two domain modeling assumptions (i.i.d. snapshots from a fixed D with positive OPT, and the existence of a local description scheme for k-means). No free parameters are fitted to obtain the asymptotic bounds; the only free choices are the usual algorithmic parameters ε, k, m that appear in every coreset paper. No new physical or mathematical entities are postulated.

assumptions (5)
  • standard math Bernstein’s inequality and standard Gaussian max-expectation bounds (Facts 5–7)
    Used throughout Appendix B for concentration of weighted sampling estimators.
  • standard math Triangle inequality for squared Euclidean distances (Fact 4)
    Used to relate costs of points, centroids and centers in Lemmas 5–11.
  • domain assumption k-means admits a local and complete description scheme with constant multiplicative coverage (Ben-David 2007)
    Invoked in Appendix C to transfer approximation guarantees from finite samples to the underlying distribution.
  • domain assumption Successive point sets are drawn i.i.d. from a fixed unknown distribution D with OPT_{k'}(D) ≥ ν > 0
    Load-bearing modeling hypothesis of Theorem 3 / Corollary 1; without it the prediction-transfer claim fails.
  • domain assumption Distances may be assumed bounded in [0,1] (or rescaled by a constant Δ)
    Stated at the beginning of Section 4.1 to simplify concentration bounds; standard in statistical clustering.

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Pith. "Pith review of Sensitivity Sampling with Predictions for k-Means Clustering." pith.science (2026). https://pith.science/paper/E22USMWG

@misc{pith2026260704949,
  author       = {Pith},
  title        = {Pith review of: Sensitivity Sampling with Predictions for k-Means Clustering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E22USMWG}},
  note         = {Machine review of arXiv:2607.04949}
}
read the original abstract

We study the problem of k-means clustering on large datasets. The state-of-the-art for the problem is given by coresets-based approaches, which build small weighted summaries of the input and derive approximate solutions with rigorous quality guarantees from them. One of the most popular and advanced approaches to derive coresets for k-means is sensitivity sampling. However, sensitivity sampling requires to compute the importance of each input point with respect to the whole dataset over all possible choices of centers. Since the exact computation of such quantities is unfeasible, current approaches work by approximating the sensitivity values. Nevertheless, the runtime of such approaches is still impractical for large datasets. In this work, we propose to reduce the runtime of sensitivity-based approaches for k-means by leveraging predictions to approximate the importance of input points. We first formally prove that current theoretical results on coresets construction via sensitivity sampling hold for coarser approximations of sensitivities compared to the one required by existing approaches. This implies that even fairly noisy predictors can be leveraged for sensitivity-sampling approaches. We then propose a natural predictor, which applies to the common scenario where clustering is performed (over time) on a sequence of datasets from the same problem. We prove that when the datasets in the sequence come from the same (unknown) distribution, centers resulting in a low error on one dataset can be used as predictions for sensitivity sampling in subsequent datasets, with guarantees on their quality. We perform an extensive experimental evaluation showing that our approach significantly improves, in terms of clustering cost vs runtime, over uniform sampling and state-of-the-art sensitivity sampling approaches when applied to sequences of datasets.

Figures

Figures reproduced from arXiv: 2607.04949 by the authors.

Figure 1
Figure 1. k-means cost of coresets-based algorithms for coreset size m = 50k. We plot the ratio between each algorithm’s achieved cost and the cost achieved on the whole dataset, on a log scale. Black dashed line: reference cost of k-means on the whole dataset (always 1). Results are averaged over 10 independent rep￾etitions of coreset construction, reporting means and 95% confidence intervals [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 2
Figure 2. Estimated distortion on coresets with size [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Times for producing coresets across different number [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Visualization of Twitter Dataset (7 snapshots). (Left) Latitude and Lon [PITH_FULL_IMAGE:figures/full_fig_p046_4.png]
Figure 5
Figure 5. Figure 5: Visualization of Taxi Dataset (12 snapshots). (Left) Latitude and Lon [PITH_FULL_IMAGE:figures/full_fig_p047_5.png]
Figure 6
Figure 6. Figure 6: Visualization of NYC TLC (M). (Left) Plot of a [PITH_FULL_IMAGE:figures/full_fig_p047_6.png]
Figure 7
Figure 7. Figure 7: Times for producing coresets across different number [PITH_FULL_IMAGE:figures/full_fig_p048_7.png]
Figure 8
Figure 8. Figure 8: Times for producing coresets across different number [PITH_FULL_IMAGE:figures/full_fig_p048_8.png]
Figure 9
Figure 9. Figure 9: k-means cost of coresets-based algorithms for coreset size m = 200k. We plot the ratio between each algorithm’s achieved cost and the cost achieved on the whole dataset, on a log scale. Black dashed line: reference cost of k-means on the whole dataset (always 1). Resul…
Figure 10
Figure 10. Figure 10: k-means cost of coresets-based algorithms for coreset size m = 500k. We plot the ratio between each algorithm’s achieved cost and the cost achieved on the whole dataset, on a log scale. Black dashed line: reference cost of k-means on the whole dataset (always 1). Resu…
Figure 11
Figure 11. Figure 11: Estimated distortion on coresets with size [PITH_FULL_IMAGE:figures/full_fig_p051_11.png]
Figure 12
Figure 12. Figure 12: Estimated distortion on coresets with size [PITH_FULL_IMAGE:figures/full_fig_p052_12.png]

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