mlsanityData to 7 Oct 2026

Universal Lower Bounds and Optimal Rates: Achieving Minimax Clustering Error in Sub-Exponential Mixture Models

arXiv 2402.15432 ↗First posted 23 Feb 2024math.ST cs.LG stat.MLcited by 3

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Abstract (arXiv)

Clustering is a pivotal challenge in unsupervised machine learning and is often investigated through the lens of mixture models. The optimal error rate for recovering cluster labels in Gaussian and sub-Gaussian mixture models involves ad hoc signal-to-noise ratios. Simple iterative algorithms, such as Lloyd's algorithm, attain this optimal error rate. In this paper, we first establish a universal lower bound for the error rate in clustering any mixture model, expressed through a Chernoff divergence, a more versatile measure of model information than signal-to-noise ratios. We then demonstrate that iterative algorithms attain this lower bound in mixture models with sub-exponential tails, notably emphasizing location-scale mixtures featuring Laplace-distributed errors. Additionally, for datasets better modelled by Poisson or Negative Binomial mixtures, we study mixture models whose distributions belong to an exponential family. In such mixtures, we establish that Bregman hard clustering, a variant of Lloyd's algorithm employing a Bregman divergence, is rate optimal.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 14 of its 34 references that are in the map. First posted on arXiv 23 Feb 2024.

Builds on

14 of 34 references.

References · 14

  1. 2016Statistical and Computational Guarantees of Lloyd's Algorithm and its Variantscited by 52
  2. 2016Community Detection in Degree-Corrected Block Modelscited by 36
  3. 2019Iterative Algorithm for Discrete Structure Recoverycited by 21
  4. 2020Cutoff for exact recovery of Gaussian mixture modelscited by 18
  5. 2019Asymptotic Bayes risk for Gaussian mixture in a semi-supervised settingcited by 15
  6. 2018Robust Bregman Clusteringcited by 7
  7. 2021Optimal Clustering in Anisotropic Gaussian Mixture Modelscited by 7
  8. 2023Exact Recovery and Bregman Hard Clustering of Node-Attributed Stochastic Block Modelcited by 7
  9. 2020Community recovery in non-binary and temporal stochastic block modelscited by 6
  10. 2021Minimax Supervised Clustering in the Anisotropic Gaussian Mixture Model: A new take on Robust Interpolationcited by 6
  11. 2023Can semi-supervised learning use all the data effectively? A lower bound perspectivecited by 5
  12. 2023Adversarially robust clustering with optimality guaranteescited by 4
2 more
  1. 2022A robust model-based clustering based on the geometric median and the Median Covariation Matrixcited by 2
  2. 2023Consistency of Lloyd's Algorithm Under Perturbationscited by 2

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Citing papers per year: 2025: 3.

3 papers.

  1. 2025Near-Optimal Clustering in Mixture of Markov Chainscited by 1
  2. 2025Optimal Graph Clustering without Edge Density Signalscited by 0
  3. 2025Minimizing Human Intervention in Online Classificationcited by 0