CLUSTERING IN BLOCK MARKOV CHAINS

• 成果类型: Article
• 署名作者: Sanders, Jaron; Proutiere, Alexandre; Yun, Se-Young
• 署名单位: Royal Institute of Technology; Delft University of Technology; Korea Advanced Institute of Science & Technology (KAIST)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1939
• 发表日期: 2020
• 页码: 3488-3512
• 关键词: semicircle law; spectrum

摘要

This paper considers cluster detection in Block Markov Chains (BMCs). These Markov chains are characterized by a block structure in their transition matrix. More precisely, the n possible states are divided into a finite number of K groups or clusters, such that states in the same cluster exhibit the same transition rates to other states. One observes a trajectory of the Markov chain, and the objective is to recover, from this observation only, the (initially unknown) clusters. In this paper, we devise a clustering procedure that accurately, efficiently and provably detects the clusters. We first derive a fundamental information-theoretical lower bound on the detection error rate satisfied under any clustering algorithm. This bound identifies the parameters of the BMC, and trajectory lengths, for which it is possible to accurately detect the clusters. We next develop two clustering algorithms that can together accurately recover the cluster structure from the shortest possible trajectories, whenever the parameters allow detection. These algorithms thus reach the fundamental detectability limit, and are optimal in that sense.