Multi-Swap $k$-Means++

• URL: http://arxiv.org/abs/2309.16384v2
• Date: Fri, 25 Oct 2024 18:14:44 GMT
• Title: Multi-Swap $k$-Means++
• Authors: Lorenzo Beretta, Vincent Cohen-Addad, Silvio Lattanzi, Nikos Parotsidis,
• Abstract summary: The $k$-means++ algorithm of Arthur and Vassilvitskii (SODA 2007) is often the practitioners' choice algorithm for optimizing the popular $k$-means clustering objective. Lattanzi and Sohler (ICML) proposed augmenting $k$-means++ with $O(k log log k)$ local search steps to yield a $c$-approximation to the $k$-means clustering problem.
• Score: 30.967186562175893
• License:
• Abstract: The $k$-means++ algorithm of Arthur and Vassilvitskii (SODA 2007) is often the practitioners' choice algorithm for optimizing the popular $k$-means clustering objective and is known to give an $O(\log k)$-approximation in expectation. To obtain higher quality solutions, Lattanzi and Sohler (ICML 2019) proposed augmenting $k$-means++ with $O(k \log \log k)$ local search steps obtained through the $k$-means++ sampling distribution to yield a $c$-approximation to the $k$-means clustering problem, where $c$ is a large absolute constant. Here we generalize and extend their local search algorithm by considering larger and more sophisticated local search neighborhoods hence allowing to swap multiple centers at the same time. Our algorithm achieves a $9 + \varepsilon$ approximation ratio, which is the best possible for local search. Importantly we show that our approach yields substantial practical improvements, we show significant quality improvements over the approach of Lattanzi and Sohler (ICML 2019) on several datasets.

Related papers

• Almost-linear Time Approximation Algorithm to Euclidean $k$-median and $k$-means [4.271492285528115]
  We focus on the Euclidean $k$-median and $k$-means problems, two of the standard ways to model the task of clustering. In this paper, we almost answer this question by presenting an almost linear-time algorithm to compute a constant-factor approximation.
  arXiv  Detail & Related papers  (2024-07-15T20:04:06Z)
• A Scalable Algorithm for Individually Fair K-means Clustering [77.93955971520549]
  We present a scalable algorithm for the individually fair ($p$, $k$)-clustering problem introduced by Jung et al. and Mahabadi et al. A clustering is then called individually fair if it has centers within distance $delta(x)$ of $x$ for each $xin P$. We show empirically that not only is our algorithm much faster than prior work, but it also produces lower-cost solutions.
  arXiv  Detail & Related papers  (2024-02-09T19:01:48Z)
• Simple, Scalable and Effective Clustering via One-Dimensional Projections [10.807367640692021]
  Clustering is a fundamental problem in unsupervised machine learning with many applications in data analysis. We introduce a simple randomized clustering algorithm that provably runs in expected time $O(mathrmnnz(X) + nlog n)$ for arbitrary $k$. We prove that our algorithm achieves approximation ratio $smashwidetildeO(k4)$ on any input dataset for the $k$-means objective.
  arXiv  Detail & Related papers  (2023-10-25T16:37:45Z)
• Do you know what q-means? [50.045011844765185]
  Clustering is one of the most important tools for analysis of large datasets. We present an improved version of the "$q$-means" algorithm for clustering. We also present a "dequantized" algorithm for $varepsilon which runs in $Obig(frack2varepsilon2(sqrtkd + log(Nd))big.
  arXiv  Detail & Related papers  (2023-08-18T17:52:12Z)
• Global $k$-means$++$: an effective relaxation of the global $k$-means clustering algorithm [0.20305676256390928]
  The $k$-means algorithm is a prevalent clustering method due to its simplicity, effectiveness, and speed. We propose the emphglobal $k$-meanstexttt++ clustering algorithm, which is an effective way of acquiring quality clustering solutions.
  arXiv  Detail & Related papers  (2022-11-22T13:42:53Z)
• A Nearly Tight Analysis of Greedy k-means++ [1.452875650827562]
  The $k$-means++ algorithm is known to return a $Theta(log k)$ approximate solution in expectation. We present nearly matching lower and upper bounds for the greedy $k$-means++ algorithm.
  arXiv  Detail & Related papers  (2022-07-16T13:49:07Z)
• Scalable Differentially Private Clustering via Hierarchically Separated Trees [82.69664595378869]
  We show that our method computes a solution with cost at most $O(d3/2log n)cdot OPT + O(k d2 log2 n / epsilon2)$, where $epsilon$ is the privacy guarantee. Although the worst-case guarantee is worse than that of state of the art private clustering methods, the algorithm we propose is practical.
  arXiv  Detail & Related papers  (2022-06-17T09:24:41Z)
• TURF: A Two-factor, Universal, Robust, Fast Distribution Learning Algorithm [64.13217062232874]
  One of its most powerful and successful modalities approximates every distribution to an $ell$ distance essentially at most a constant times larger than its closest $t$-piece degree-$d_$. We provide a method that estimates this number near-optimally, hence helps approach the best possible approximation.
  arXiv  Detail & Related papers  (2022-02-15T03:49:28Z)
• Fuzzy Clustering with Similarity Queries [56.96625809888241]
  The fuzzy or soft objective is a popular generalization of the well-known $k$-means problem. We show that by making few queries, the problem becomes easier to solve.
  arXiv  Detail & Related papers  (2021-06-04T02:32:26Z)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.