This paper considers coresets for the robust $k$-medians problem with $m$ outliers, and new constructions in various metric spaces are obtained. Specifically, for metric spaces with a bounded VC or doubling dimension $d$, the coreset size is $O(m) + \tilde{O}(kd\varepsilon^{-2})$, which is optimal up to logarithmic factors. For Euclidean spaces, the coreset size is $O(m\varepsilon^{-1}) + \tilde{O}(\min\{k^{4/3}\varepsilon^{-2}, k\varepsilon^{-3}\})$, improving upon a recent result by Jiang and Lou (ICALP 2025). These results also extend to robust $(k,z)$-clustering, yielding, for VC and doubling dimension, a coreset size of $O(m) + \tilde{O}(kd\varepsilon^{-2z})$ with the optimal linear dependence on $m$. This extended result improves upon the earlier work of Huang et al. (SODA 2025). The techniques introduce novel dataset decompositions, enabling chaining arguments to be applied jointly across multiple components.

Robustness to outliers is important in machine learning. Many classical problems, including subspace embedding, clustering, and low-rank approximation, lack scalable, outlier-resilient algorithms. This paper considers machine learning problems of the form $\min_{x\in \mathbb{R}^d} F(x)$, where $F(x)=\sum_{i=1}^n F_i(x)$, and their robust counterparts $\min_{x\in\mathbb{R}^d} F^{(m)}(x)$, where $F^{(m)}(x)$ denotes the sum of all but the $m$ largest $F_i(x)$ values. We develop a general framework for constructing $\epsilon$-coresets for such robust problems, where an $\epsilon$-coreset is a weighted subset of functions $\{F_1(x),…,F_n(x)\}$ that provides a $(1+\epsilon)$-approximation to $F(x)$. Specifically, if the original problem $F$ has total sensitivity $T$ and admits a vanilla $\epsilon$-coreset of size $S$, our algorithm constructs an $\epsilon$-coreset of size $\tilde{O}(\frac{mT}{\epsilon})+S$ for the robust objective $F^{(m)}$. This coreset size can be shown to be near-tight for $\ell_2$ subspace embedding. Our coreset algorithm has scalable running time and leads to new or improved algorithms for the robust optimization problems. Empirical evaluations demonstrate that our coresets outperform uniform sampling on real-world data sets.

Designing small-sized coresets, which approximately preserve the costs of the solutions for large datasets, has been an important research direction for the past decade. We consider coreset construction for a variety of general constrained clustering problems. We introduce a general class of assignment constraints, including capacity constraints on cluster centers, and assignment structure constraints for data points (modeled by a convex body B ). We give coresets for clustering problems with such general assignment constraints that significantly generalize and improve known results. Notable implications include the first ε-coreset for capacitated and fair k-MEDIAN with m outliers in Euclidean spaces whose size is Õ (m + k2ε-4), generalizing and improving upon the prior bounds in [BCJ+ 22, HJLW23] (for capacitated k-MEDIAN, the coreset size bound obtained in [BCJ+22] is Õ (k3ε-6), and for k-MEDIAN with m outliers, the coreset size bound obtained in [HJLW23] is Õ (m + k3ε-5)), and the first ε-coreset of size poly(kε-1) for fault-tolerant clustering for various types of metric spaces.

Constructing small-sized coresets for various clustering problems in different metric spaces has attracted significant attention for the past decade. A central problem in the coreset literature is to understand what is the best possible coreset size for (k,z)-clustering in Euclidean space. While there has been significant progress in the problem, there is still a gap between the state-of-the-art upper and lower bounds. For instance, the best known upper bound for k-means (z=2) is min{O(k3/2 ε−2),O(k ε−4)} [Cohen-Addad, Larsen, Saulpic, Schwiegelshohn, Sheikh-Omar, NeurIPS’22], while the best known lower bound is Ω(kε−2) [Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC’22]. In this paper, we make significant progress on both upper and lower bounds. For a large range of parameters (i.e., ε, k), we have a complete understanding of the optimal coreset size. In particular, we obtain the following results: (1) We present a new coreset lower bound Ω(k ε−z−2) for Euclidean (k,z)-clustering when ε ≥ Ω(k−1/(z+2)). In view of the prior upper bound Õz(k ε−z−2) [Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC’22], the bound is optimal. The new lower bound is surprising since Ω(kε−2) [Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC’22] is “conjectured” to be the correct bound in some recent works (see e.g., [Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC’22; Cohen-Addad, Larsen, Saulpic, Schwiegelshohn, Sheikh-Omar, NeurIPS’22]). Our new lower bound instance is a delicate construction with multiple clusters of points, which is a significant departure from the previous construction in [Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC’22] that contains a single cluster of points. The new lower bound also implies improved lower bounds for (k,z)-clustering in doubling metrics. (2) For the upper bound, we provide efficient coreset construction algorithms for (k,z)-clustering with improved or optimal coreset sizes in several metric spaces. In particular, we provide an Õz(k2z+2/z+2 ε−2)-sized coreset, with a unfied analysis, for (k,z)-clustering for all z≥ 1 in Euclidean space. This upper bound improves upon the Õz(k2ε−2) upper bound by [Cohen-Addad, Larsen, Saulpic, Schwiegelshohn. STOC’22] (when k≤ ε−1), and matches the recent independent results [Cohen-Addad, Larsen, Saulpic, Schwiegelshohn, Sheikh-Omar, NeurIPS’22] for k-median and k-means (z=1,2) and extends them to all z≥ 1.

We consider the problem of constructing small coresets for $k$-Median in Euclidean spaces. Given a large set of data points $P\subset \mathbb{R}^d$, a coreset is a much smaller set $S\subset \mathbb{R}^d$, so that the $k$-Median costs of any $k$ centers w.r.t. $P$ and $S$ are close. Existing literature mainly focuses on the high-dimension case and there has been great success in obtaining dimension-independent bounds, whereas the case for small $d$ is largely unexplored. Considering many applications of Euclidean clustering algorithms are in small dimensions and the lack of systematic studies in the current literature, this paper investigates coresets for $k$-Median in small dimensions. For small $d$, a natural question is whether existing near-optimal dimension-independent bounds can be significantly improved. We provide affirmative answers to this question for a range of parameters. Moreover, new lower bound results are also proved, which are the highest for small $d$. In particular, we completely settle the coreset size bound for $1$-d $k$-Median (up to log factors). Interestingly, our results imply a strong separation between $1$-d $1$-Median and $1$-d $2$-Median. As far as we know, this is the first such separation between $k=1$ and $k=2$ in any dimension.

$(j,k)$-projective clustering is the natural generalization of the family of $k$-clustering and $j$-subspace clustering problems. Given a set of points $P$ in $\mathbb{R}^d$, the goal is to find $k$ flats of dimension $j$, i.e., affine subspaces, that best fit $P$ under a given distance measure. In this paper, we propose the first algorithm that returns an $L_\infty$ coreset of size polynomial in $d$. Moreover, we give the first strong coreset construction for general $M$-estimator regression. Specifically, we show that our construction provides efficient coreset constructions for Cauchy, Welsch, Huber, Geman-McClure, Tukey, $L_1-L_2$, and Fair regression, as well as general concave and power-bounded loss functions. Finally, we provide experimental results based on real-world datasets, showing the efficacy of our approach.

Motivated by practical generalizations of the classic k-median and k-means objectives, such as clustering with size constraints, fair clustering, and Wasserstein barycenter, we introduce a meta-theorem for designing coresets for constrained-clustering problems. The meta-theorem reduces the task of coreset construction to one on a bounded number of ring instances with a much-relaxed additive error. This reduction enables us to construct coresets using uniform sampling, in contrast to the widely-used importance sampling, and consequently we can easily handle constrained objectives. Notably and perhaps surprisingly, this simpler sampling scheme can yield coresets whose size is independent of n, the number of input points. Our technique yields smaller coresets, and sometimes the first coresets, for a large number of constrained clustering problems, including capacitated clustering, fair clustering, Euclidean Wasserstein barycenter, clustering in minor-excluded graph, and polygon clustering under Fréchet and Hausdorff distance. Finally, our technique yields also smaller coresets for 1-median in low-dimensional Euclidean spaces, specifically of size $\tilde{O}(\varepsilon^{-15})$ in $\mathbb{R}^{2}$ and $\tilde{O}(\varepsilon^{-16})$ in $\mathbb{R}^{3}$.

We provide the first coreset for clustering points in $\mathbb{R}^d$ that have multiple missing values (coordinates). Previous coreset constructions only allow one missing coordinate. The challenge in this setting is that objective functions, like \kMeans, are evaluated only on the set of available (non-missing) coordinates, which varies across points. Recall that an $\epsilon$-coreset of a large dataset is a small proxy, usually a reweighted subset of points, that $(1+\epsilon)$-approximates the clustering objective for every possible center set.Our coresets for $k$-Means and $k$-Median clustering have size $(jk)^{O(\min(j,k))} (\epsilon^{-1} d \log n)^2$, where $n$ is the number of data points, $d$ is the dimension and $j$ is the maximum number of missing coordinates for each data point. We further design an algorithm to construct these coresets in near-linear time, and consequently improve a recent quadratic-time PTAS for $k$-Means with missing values [Eiben et al., SODA 2021] to near-linear time.We validate our coreset construction, which is based on importance sampling and is easy to implement, on various real data sets. Our coreset exhibits a flexible tradeoff between coreset size and accuracy, and generally outperforms the uniform-sampling baseline. Furthermore, it significantly speeds up a Lloyd's-style heuristic for $k$-Means with missing values.

Coresets are modern data-reduction tools that are widely used in data analysis to improve efficiency in terms of running time, space and communication complexity. Our main result is a fast algorithm to construct a small coreset for k-Median in (the shortest-path metric of) an excluded-minor graph. Specifically we give the first coreset of size that depends only on k, ∊ and the excluded-minor size, and our running time is quasi-linear (in the size of the input graph). The main innovation in our new algorithm is that is iterative; it first reduces the n input points to roughly O(log n) reweighted points, then to O(log log n), and so forth until the size is independent of n. Each step in this iterative size reduction is based on the importance sampling framework of Feldman and Langberg (STOC 2011), with a crucial adaptation that reduces the number of distinct points, by employing a terminal embedding (where low distortion is guaranteed only for the distance from every terminal to all other points). Our terminal embedding is technically involved and relies on shortest-path separators, a standard tool in planar and excluded-minor graphs. Furthermore, our new algorithm is applicable also in Euclidean metrics, by simply using a recent terminal embedding result of Narayanan and Nelson (STOC 2019), which extends the Johnson-Lindenstrauss Lemma. We thus obtain an efficient coreset construction in high-dimensional Euclidean spaces, thereby matching and simplifying state-of-the-art results (Sohler and Woodruff, FOCS 2018; Huang and Vishnoi, STOC 2020). In addition, we also employ terminal embedding with additive distortion to obtain small coresets in graphs with bounded highway dimension, and use applications of our coresets to obtain improved approximation schemes, e.g., an improved PTAS for planar k-Median via a new centroid set.

We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems are essential to data analysis and used for example in road networks and data visualization. A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set, and it offers significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings. Our main result is a near-linear time construction of a coreset for k-Median in a general graph $G$, with size $O_{\epsilon, k}(\mathrm{tw}(G))$ where $\mathrm{tw}(G)$ is the treewidth of $G$, and we complement the construction with a nearly-tight size lower bound. The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of $O(\mathrm{tw}(G))$ on the shattering dimension under any point weights. We validate our coreset on real-world road networks, and our scalable algorithm constructs tiny coresets with high accuracy, which translates to a massive speedup of existing approximation algorithms such as local search for graph k-Median.

We design coresets for Ordered k-Median, a generalization of classical clustering problems such as k-Median and k-Center. Its objective function is defined via the Ordered Weighted Averaging (OWA) paradigm of Yager (1988), where data points are weighted according to a predefined weight vector, but in order of their contribution to the objective (distance from the centers). A powerful data-reduction technique, called a coreset, is to summarize a point set $X$ in $\mathbb{R}^d$ into a small (weighted) point set $X’$, such that for every set of $k$ potential centers, the objective value of the coreset $X’$ approximates that of $X$ within factor $1\pm \epsilon$. When there are multiple objectives (weights), the above standard coreset might have limited usefulness, whereas in a simultaneous coreset, the above approximation holds for all weights (in addition to all centers). Our main result is a construction of a simultaneous coreset of size $O_{\epsilon, d}(k^2 \log^2 |X|)$ for Ordered k-Median. We validate our algorithm on a real geographical data set, and we find our coreset leads to a massive speedup of clustering computations, while maintaining high accuracy for a range of weights.

We study the problem of constructing ε-coresets for the (k, z)-clustering problem in a doubling metric M(X, d). An ε-coreset is a weighted subset S ⊆ X with weight function w : S → ℝ≥0, such that for any k-subset C ∈ [X]k, it holds that Σx∈Sw(x) · dz(x, C) ∈ (1 ± ε) · Σx∈Xdz(x, C). We present an efficient algorithm that constructs an ε-coreset for the (k, z)-clustering problem in M(X, d), where the size of the coreset only depends on the parameters k, z, ε and the doubling dimension ddim(M). To the best of our knowledge, this is the first efficient c-coreset construction of size independent of |X| for general clustering problems in doubling metrics. To this end, we establish the first relation between the doubling dimension of M(X, d) and the shattering dimension (or VC-dimension) of the range space induced by the distance d. Such a relation is not known before, since one can easily construct instances in which neither one can be bounded by (some function of) the other. Surprisingly, we show that if we allow a small (1 ± ε)-distortion of the distance function d (the distorted distance is called the smoothed distance function), the shattering dimension can be upper bounded by O(ε-O(ddim(M))). For the purpose of coreset construction, the above bound does not suffice as it only works for unweighted spaces. Therefore, we introduce the notion of τ-error probabilistic shattering dimension, and prove a (drastically better) upper bound of O(ddim(M)·log(1/ε)+log log 1/τ) for the probabilistic shattering dimension for weighted doubling metrics. As it turns out, an upper bound for the probabilistic shattering dimension is enough for constructing a small coreset. We believe the new relation between doubling and shattering dimensions is of independent interest and may find other applications. Furthermore, we study robust coresets for (k, z)-clustering with outliers in a doubling metric. We show an improved connection between α-approximation and robust coresets. This also leads to improvement upon the previous best known bound of the size of robust coreset for Euclidean space [Feldman and Langberg, STOC 11]. The new bound entails a few new results in clustering and property testing. As another application, we show constant-sized (ε, k, z)centroid sets in doubling metrics can be constructed by extending our coreset construction. Prior to our result, constantsized centroid sets for general clustering problems were only known for Euclidean spaces. We can apply our centroid set to accelerate the local search algorithm (studied in [Friggstad et al., FOCS 2016]) for the (k, z)-clustering problem in doubling metrics.