Authors:
(1) Andrew Draganov, Aarhus University and All authors contributed equally to this research;
(2) David Saulpic, Université Paris Cité & CNRS;
(3) Chris Schwiegelshohn, Aarhus University. Table of Links Abstract and 1 Introduction 2 Preliminaries and Related Work 2.1 On Sampling Strategies 2.2 Other Coreset Strategies 2.3 Coresets for Database Applications 2.4 Quadtree Embeddings 3 Fast-Coresets 4 Reducing the Impact of the Spread 4.1 Computing a crude upper-bound 4.2 From Approximate Solution to Reduced Spread 5 Fast Compression in Practice 5.1 Goal and Scope of the Empirical Analysis 5.2 Experimental Setup 5.3 Evaluating Sampling Strategies 5.4 Streaming Setting and 5.5 Takeaways 6 Conclusion 7 Acknowledgements 8 Proofs, Pseudo-Code, and Extensions and 8.1 Proof of Corollary 3.2 8.2 Reduction of k-means to k-median 8.3 Estimation of the Optimal Cost in a Tree 8.4 Extensions to Algorithm 1 References 6 Conclusion In this work, we discussed the theoretical and practical limits of compression algorithms for center-based clustering. We proposed the first nearly-linear time coreset algorithm for k-median and k-means. Moreover, the algorithm can be parameterized to achieve an asymptotically optimal coreset size. Subsequently, we conducted a thorough experimental analysis comparing this algorithm with fast sampling heuristics. In doing so, we find that although the Fast-Coreset algorithm achieves the best compression guarantees among its competitors, naive uniform sampling is already a sufficient compression for downstream clustering tasks in well-behaved datasets. Furthermore, we find that intermediate heuristics interpolating between uniform sampling and coresets play an important role in balancing efficiency and accuracy. Although this closes the door on the highly-studied problem of optimally small and fast coresets for k-median and k-means, open questions of wider scope still remain. For example, when does sensitivity sampling guarantee accurate compression with optimal space in linear time and can these conditions be formalized? Furthermore, sensitivity sampling is incompatible with paradigms such as fair-clustering [8, 15, 21, 43, 56] and it is unclear whether one can expect that a linear-time method can optimally compress a dataset while adhering to the fairness constraints. 7 Acknowledgements Andrew Draganov and Chris Schwiegelshohn are partially supported by the Independent Research Fund Denmark (DFF) under a Sapere Aude Research Leader grant No 1051-00106B. David Sauplic has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101034413. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Andrew Draganov, Aarhus University and All authors contributed equally to this research; (2) David Saulpic, Université Paris Cité & CNRS; (3) Chris Schwiegelshohn, Aarhus University. Authors: Authors: (1) Andrew Draganov, Aarhus University and All authors contributed equally to this research; (2) David Saulpic, Université Paris Cité & CNRS; (3) Chris Schwiegelshohn, Aarhus University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminaries and Related Work 2.1 On Sampling Strategies 2.1 On Sampling Strategies 2.2 Other Coreset Strategies 2.2 Other Coreset Strategies 2.3 Coresets for Database Applications 2.3 Coresets for Database Applications 2.4 Quadtree Embeddings 2.4 Quadtree Embeddings 3 Fast-Coresets 3 Fast-Coresets 4 Reducing the Impact of the Spread 4 Reducing the Impact of the Spread 4.1 Computing a crude upper-bound 4.1 Computing a crude upper-bound 4.2 From Approximate Solution to Reduced Spread 4.2 From Approximate Solution to Reduced Spread 5 Fast Compression in Practice 5 Fast Compression in Practice 5.1 Goal and Scope of the Empirical Analysis 5.1 Goal and Scope of the Empirical Analysis 5.2 Experimental Setup 5.2 Experimental Setup 5.3 Evaluating Sampling Strategies 5.3 Evaluating Sampling Strategies 5.4 Streaming Setting and 5.5 Takeaways 5.4 Streaming Setting and 5.5 Takeaways 6 Conclusion 6 Conclusion 7 Acknowledgements 7 Acknowledgements 8 Proofs, Pseudo-Code, and Extensions and 8.1 Proof of Corollary 3.2 8 Proofs, Pseudo-Code, and Extensions and 8.1 Proof of Corollary 3.2 8.2 Reduction of k-means to k-median 8.2 Reduction of k-means to k-median 8.3 Estimation of the Optimal Cost in a Tree 8.3 Estimation of the Optimal Cost in a Tree 8.4 Extensions to Algorithm 1 8.4 Extensions to Algorithm 1 References References 6 Conclusion In this work, we discussed the theoretical and practical limits of compression algorithms for center-based clustering. We proposed the first nearly-linear time coreset algorithm for k-median and k-means. Moreover, the algorithm can be parameterized to achieve an asymptotically optimal coreset size. Subsequently, we conducted a thorough experimental analysis comparing this algorithm with fast sampling heuristics. In doing so, we find that although the Fast-Coreset algorithm achieves the best compression guarantees among its competitors, naive uniform sampling is already a sufficient compression for downstream clustering tasks in well-behaved datasets. Furthermore, we find that intermediate heuristics interpolating between uniform sampling and coresets play an important role in balancing efficiency and accuracy. Although this closes the door on the highly-studied problem of optimally small and fast coresets for k-median and k-means, open questions of wider scope still remain. For example, when does sensitivity sampling guarantee accurate compression with optimal space in linear time and can these conditions be formalized? Furthermore, sensitivity sampling is incompatible with paradigms such as fair-clustering [8, 15, 21, 43, 56] and it is unclear whether one can expect that a linear-time method can optimally compress a dataset while adhering to the fairness constraints. 7 Acknowledgements Andrew Draganov and Chris Schwiegelshohn are partially supported by the Independent Research Fund Denmark (DFF) under a Sapere Aude Research Leader grant No 1051-00106B. David Sauplic has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101034413. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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