Stable Coresets: Methods and Applications

• Stable coresets are concise, weighted data summaries that ensure robust approximation guarantees even when candidate solutions, constraints, or noise models vary.
• They are applied in clustering, statistics, and deep learning to maintain relative solution quality, preserve minimax risk, and align optimization dynamics with full datasets.
• Key algorithmic motifs include sensitivity sampling, random projections, and merge-and-reduce schemes that enable efficient and stable data summarization across diverse applications.

Stable coresets are small weighted summaries whose guarantees are designed to remain meaningful under changes in candidate solutions, constraints, noise models, or training dynamics. The literature does not use the term in a single uniform sense. In clustering, it is closely tied to strong query-wise approximation and to intermediate notions that preserve the relative quality of solutions; in statistics, it refers to minimax-risk preservation under compression; and in deep learning, it refers to subsets whose induced optimization dynamics or loss landscape stay aligned with those of the full dataset (Sohler et al., 2018, Turner et al., 2020, Chang et al., 21 Nov 2025, Carmel et al., 29 Jun 2026).

1. Core definitions and interpretive scope

The classical strong coreset guarantee is query-uniform. For a clustering cost function $\operatorname{cost}(P,S)$, a strong coreset $\Omega$ satisfies

$$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$

for every solution $S$ (Prabhu et al., 15 Jun 2026). In the terminology of strong coresets for $k$-median and subspace approximation, this is precisely the guarantee that a summary is valid for all feasible queries simultaneously, not only for an optimum or near-optimum (Sohler et al., 2018).

Weak coresets relax this requirement. In the approximation-preserving coreset literature, weak coresets are described as a moral predecessor that only requires that one can use the coreset to extract a near-optimal solution; by contrast, approximation-preserving coresets ask that any $\alpha$-approximate solution on the coreset can be converted into an $\alpha\beta$-approximate solution on the full data (Prabhu et al., 15 Jun 2026). This places them between strong and weak guarantees.

The recent stable-coreset definition for geometric median makes this intermediate position explicit. Writing

$\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$

a weighted subset $Q \subseteq P$ is a stable $(\varepsilon,\eta)$-coreset if for every pair of candidate solutions $\Omega$0,

$\Omega$1

This preserves relative quality comparisons rather than absolute costs for all points, and was introduced precisely to handle constrained variants that weak coresets do not cover (Carmel et al., 29 Jun 2026).

A different relaxation appears in learned coresets. For a measurable query space $\Omega$2, an $\Omega$3-coreset for the average loss satisfies

$\Omega$4

Here the guarantee is distributional over queries rather than uniform over all queries (Maalouf et al., 2021).

Notion	Guarantee	Representative source
Strong coreset	Uniform $\Omega$5 cost preservation for all queries	(Sohler et al., 2018)
Stable coreset	Relative ordering of solutions on coreset transfers to full data	(Carmel et al., 29 Jun 2026)
Approximation-preserving coreset	$\Omega$6-approximate on coreset implies $\Omega$7-approximate on full data	(Prabhu et al., 15 Jun 2026)
Average-loss coreset	Expected loss under query distribution $\Omega$8 is preserved	(Maalouf et al., 2021)

These notions are not interchangeable. A plausible implication is that “stability” in the coreset literature names a family of guarantees that trade absolute uniformity for robustness under the kinds of variations that matter in the downstream task.

2. Clustering, projective geometry, and query stability

Strong coresets remain the canonical stability notion for geometric clustering. For $\Omega$9-median and subspace approximation with sum-of-distances objectives, strong coresets of size $$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$0 independent of both $$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$1 and $$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$2 were obtained together with algorithms running in $$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$3 time (Sohler et al., 2018). The key interpretation in that line of work is that a strong coreset is stable with respect to the query space: changing centers or subspaces does not invalidate the summary.

For $$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$4-means, random projections provide a different route to stability. An offline coreset of size

$$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$5

and a streaming coreset storing

$$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$6

points were obtained by combining Johnson–Lindenstrauss type embeddings with online-algorithmic ideas, thereby avoiding a direct reliance on merge-and-reduce (Bury et al., 2015). This suggests a notion of stability against ambient dimension and stream length: the summary is maintained through a fixed low-dimensional embedding rather than through an ever-deepening merge tree.

For general $$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$7-clustering, a nearly linear-time algorithm constructs $$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$8-coresets of size

$$\left|\operatorname{cost}(P,S)-\operatorname{cost}(\Omega,S)\right| \le \varepsilon \cdot \operatorname{cost}(P,S)$$9

with runtime $S$0 up to the stated logarithmic and $S$1-dependent factors (Deng et al., 2022). The guarantee is still strong, namely uniform over all sets of $S$2 centers, so the coreset is stable under arbitrary downstream center selection.

Sensitivity sampling yields the sharpest known interaction between worst-case optimality and stability assumptions. For Euclidean $S$3-means, it produces worst-case optimal coresets of size

$S$4

and, uniquely among known coreset algorithms, improves to

$S$5

on $S$6-stable instances with $S$7, without knowing $S$8 in advance (Bansal et al., 2024). The same work also proves that any coreset for stable instances consisting only of input points must have size $S$9, so the stability-dependent bound is optimal in that model (Bansal et al., 2024).

The most explicit stable-coreset result currently available is for geometric median. A uniform sample of size

$k$0

is a stable $k$1-coreset with high constant probability, and the $k$2 lower bound shows that this is tight up to the logarithmic factor (Carmel et al., 29 Jun 2026). The significance of this result is not merely sample complexity. Stable coresets in that setting are presented as the intermediate notion that simultaneously handles constrained variants, where weak coresets are insufficient and strong coresets remain comparatively expensive (Carmel et al., 29 Jun 2026).

Projective clustering extends the same theme. For $k$3-projective clustering, an $k$4 coreset of size polynomial in $k$5 was obtained, together with strong coreset constructions for general M-estimator regression, including Cauchy, Welsch, Huber, Geman-McClure, Tukey, $k$6, and Fair regression (Tukan et al., 2022). The salient stability property is again uniformity over all queries: one geometric summary supports many objective functions and many downstream models.

3. Statistical and noise-aware notions of stability

A statistical notion of stable coresets is developed for nonparametric density estimation. There the central question is not only whether a small subset approximates an optimization objective, but whether a coreset-based estimator preserves minimax risk. The full-data minimax rate over Hölder-smooth densities is

$k$7

and the minimal coreset size for retaining this rate is

$k$8

Weighted Carathéodory coreset KDEs attain the minimax rate with

$k$9

while uniformly weighted coresets require substantially larger subsets in the one-dimensional lower bounds presented there (Turner et al., 2020). In this framework, “stable” means that risk remains uniformly controlled over the full function class, and the paper makes explicit that high dimension destroys aggressive compressibility: as $\alpha$0, $\alpha$1 (Turner et al., 2020).

Stability under noise is treated explicitly for $\alpha$2-clustering with stochastic corruption. Because the clean dataset $\alpha$3 is unobserved and only a noisy version $\alpha$4 is available, the paper replaces the classical error metric

$\alpha$5

with the approximation-ratio metric

$\alpha$6

which directly controls

$\alpha$7

(Huang et al., 27 Oct 2025). Under the stated structural assumptions, enforcing an $\alpha$8-bound under $\alpha$9 yields smaller coresets and tighter guarantees on the true clustering cost than the classical metric, with error terms scaling with $\alpha\beta$0 rather than $\alpha\beta$1 in the regime emphasized by the paper (Huang et al., 27 Oct 2025). The abstract further states that the coreset size can improve by a factor of up to $\alpha\beta$2 (Huang et al., 27 Oct 2025).

A generic stability-oriented sensitivity framework also appears in near-convex objectives. The $\alpha\beta$3-SVD factorization and the corresponding total-sensitivity bounds support coresets for SVM, logistic regression, M-estimators, and $\alpha\beta$4-regression, while remaining compatible with streaming and merge-and-reduce constructions (Tukan et al., 2020). The paper’s interpretation is that stability comes from controlling sensitivities through structural properties of the loss family rather than through problem-specific combinatorics.

4. Optimization and learned stability in deep learning

In deep learning, stability is often operational rather than purely geometric. A systematic benchmark on CNNs and transformers evaluates coreset stability through predictable performance as the effective data per epoch varies, robustness across architectures and datasets, total-time efficiency, and the absence of catastrophic failures at small subset sizes (Gupta et al., 2023). Under that notion, random subset selection is often more robust and stable than more elaborate methods when total time is counted, uniform per-class sampling is described as not the appropriate choice for highly non-uniform class complexity, and pretrained transformers remain stable at very small coreset sizes on some natural-image targets while CNNs generalize better without the right pretraining or on non-natural data (Gupta et al., 2023).

A more formal notion appears in posterior-smoothed deep-learning coresets. A subset $\alpha\beta$5 is defined to be $\alpha\beta$6-stable if

$\alpha\beta$7

This stability condition implies Hessian alignment,

$\alpha\beta$8

and Newton-step similarity, while the corresponding convergence analysis yields rates of order $\alpha\beta$9 and $\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$0 in the two noise regimes analyzed there (Chang et al., 21 Nov 2025). The underlying claim is that stable coresets should align induced and full loss landscapes, not merely match gradients at one iterate.

A different route uses training trajectories directly. Correlation of Loss Differences defines

$\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$1

or, in the bias-aware form used in practice,

$\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$2

for samples in class $\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$3 (Nagaraj et al., 27 Aug 2025). The paper proves a convergence bound in which the error is upper-bounded by an alignment term and by the representativeness of the validation set, and reports that CLD transfers across architectures with less than $\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$4 degradation, remains stable when only early checkpoints are used, and exhibits inherent bias reduction via per-class validation alignment (Nagaraj et al., 27 Aug 2025).

Learned average-loss coresets generalize this distributional viewpoint. If the learned coreset satisfies empirical average-loss approximation and the weight-sum constraint

$\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$5

then with probability at least $\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$6,

$\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$7

The coreset guarantee is therefore tied to the query distribution $\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$8, not to a worst-case supremum over the query space (Maalouf et al., 2021). A plausible implication is that this family of methods trades adversarial robustness for stability with respect to the actual optimization or deployment distribution.

5. Recurring algorithmic motifs

Several technical primitives recur across stable-coreset constructions. Carathéodory reductions are among the oldest and most persistent. Accurate coresets for sums, means, covariance, PCA, and linear regression compress data by preserving exact low-order moments, often with bounded nonnegative weights and support size depending only on the ambient dimension (Jubran et al., 2019). In density estimation, Carathéodory is used in Fourier space to build weighted coreset KDEs that attain minimax rate with near-minimal coreset size (Turner et al., 2020).

Sensitivity sampling is the dominant probabilistic primitive. It is the mechanism behind worst-case and stability-optimal $\cost_2(y, P) \eqdef \frac{1}{|P|}\sum_{p \in P} \|y - p\|_2,$9-means coresets (Bansal et al., 2024), and it also underlies generic coreset constructions for near-convex losses via total-sensitivity bounds derived from $Q \subseteq P$0-SVD factorizations (Tukan et al., 2020). The common interpretation is that stability comes from controlling the maximal influence of individual points across the relevant query family.

Random projection and dimensionality reduction are another recurring pattern. Johnson–Lindenstrauss type embeddings make it possible to replace ambient dimension dependence by intrinsic quantities such as $Q \subseteq P$1 in $Q \subseteq P$2-means streaming coresets (Bury et al., 2015), while terminal embeddings are used to remove explicit $Q \subseteq P$3-dependence in stable geometric-median coresets (Carmel et al., 29 Jun 2026).

Composability through merge-and-reduce or distributed aggregation is repeatedly identified as a stability mechanism over time. Gaussian-mixture coresets are explicitly described as efficiently constructible in distributed and streaming settings (Lucic et al., 2017), and accurate coresets admit streaming and dynamic maintenance through repeated reduction of local summaries (Jubran et al., 2019). This suggests that one axis of stability is temporal: the coreset remains valid under data arrival, partition, or local update.

Finally, several recent constructions replace geometric uniformity by optimization alignment. Posterior smoothing (Chang et al., 21 Nov 2025), validation-trajectory correlation (Nagaraj et al., 27 Aug 2025), and learned average-loss matching (Maalouf et al., 2021) all fit this template. Their shared premise is that a stable coreset should preserve the part of the loss landscape actually visited by training.

6. Limits, lower bounds, and open directions

The literature also delineates hard limits. For geometric median, even weak coresets require $Q \subseteq P$4 query complexity, and the stable-coreset upper bound is tight up to the logarithmic factor (Carmel et al., 29 Jun 2026). For approximation-preserving coresets in arbitrary metrics, any $Q \subseteq P$5-approximation preserving coreset with $Q \subseteq P$6 must have size

$Q \subseteq P$7

on some instances (Prabhu et al., 15 Jun 2026). For stable $Q \subseteq P$8-means subset coresets built from input points, $Q \subseteq P$9 is unavoidable even under cost stability (Bansal et al., 2024).

High dimension remains a fundamental obstruction in statistical settings. For nonparametric density estimation, the minimal coreset size for minimax-optimal risk grows to $(\varepsilon,\eta)$0 as $(\varepsilon,\eta)$1, showing that aggressive compression and statistical stability become incompatible in the high-dimensional regime considered there (Turner et al., 2020).

Several papers identify open methodological questions. In deep learning, the benchmark study on CNNs and transformers explicitly notes the absence of a formal stability theory for coreset methods, calls for adaptive per-class coreset construction, and raises questions about transformers on non-natural domains and about the balance between selection cost and training cost (Gupta et al., 2023). Posterior-smoothed stable coresets leave open the extension to more complex data types such as audio and video, and the use of more structured posteriors (Chang et al., 21 Nov 2025). Robust clustering under stochastic noise identifies streaming, heavy-tailed noise, and adversarial noise as open directions beyond the Bernstein-type setting analyzed there (Huang et al., 27 Oct 2025). Learned average-loss coresets, meanwhile, inherit a dependence on the chosen query distribution $(\varepsilon,\eta)$2, so their guarantees are only as good as the distributional proxy used in training (Maalouf et al., 2021).

Taken together, these developments suggest that stable coresets are best understood not as a single theorem schema, but as a spectrum of compression guarantees. At one end lie strong coresets that stabilize the entire query space; at the other lie learned and trajectory-based coresets that stabilize only the parts of the loss landscape or query distribution that matter in practice. Between them lie stable, approximation-preserving, and noise-aware notions that preserve relative comparisons, constrained optima, or true-risk behavior under explicitly modeled perturbations.