2010/04/29 by Daniil Ryabko, Ryabko, Daniil · 2 citations
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #Information Theory (cs.IT) #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #cs.IT #cs.LG #math.IT #stat.ML
paper · pdf · doi:10.48550/arxiv.1004.5194
in proceedings of ICML 2010. arXiv-admin note: for version 2 of this article please see: arXiv:1005.0826v1
arxiv created 2010/04/29 · openalex publication_date 2010/04/29 · arxiv updated 2010/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The problem of clustering is considered, for the case when each data point is a sample generated by a stationary ergodic process. We propose a very natural asymptotic notion of consistency, and show that simple consistent algorithms exist, under most general non-parametric assumptions. The notion of consistency is as follows: two samples should be put into the same cluster if and only if they were generated by the same distribution. With this notion of consistency, clustering generalizes such classical statistical problems as homogeneity testing and process classification. We show that, for the case of a known number of clusters, consistency can be achieved under the only assumption that the joint distribution of the data is stationary ergodic (no parametric or Markovian assumptions, no assumptions of independence, neither between nor within the samples). If the number of clusters is unknown, consistency can be achieved under appropriate assumptions on the mixing rates of the processes. (again, no parametric or independence assumptions). In both cases we give examples of simple (at most quadratic in each argument) algorithms which are consistent.