Sequential Spectral Clustering of Data Sequences

We study the problem of nonparametric clustering of data sequences, where each data sequence comprises i.i.d. samples generated from an unknown distribution. The true clusters are the clusters obtained using the Spectral clustering algorithm (SPEC) on the pairwise distance between the true distributions corresponding to the data sequences. Since the true distributions are unknown, the objective is to estimate the clusters by observing the minimum number of samples from the data sequences for a given error probability. To solve this problem, we propose the Sequential Spectral clustering algorithm (SEQ-SPEC), and show that it stops in finite time almost surely and is exponentially consistent. We also propose a computationally more efficient algorithm called the Incremental Approximate Sequential Spectral clustering algorithm (IA-SEQ-SPEC). Through simulations, we show that both our proposed algorithms perform better than the fixed sample size SPEC, the Sequential $K$-Medoids clustering algorithm (SEQ-KMED) and the Sequential Single Linkage clustering algorithm (SEQ-SLINK). The IA-SEQ-SPEC, while being computationally efficient, performs close to SEQ-SPEC on both synthetic and real-world datasets. To the best of our knowledge, this is the first work on spectral clustering of data sequences under a sequential framework.

Online Clustering with Bandit Information

We study the problem of online clustering within the multi-armed bandit framework under the fixed confidence setting. In this multi-armed bandit problem, we have $M$ arms, each providing i.i.d. samples that follow a multivariate Gaussian distribution with an {\em unknown} mean and a known unit covariance. The arms are grouped into $K$ clusters based on the distance between their means using the Single Linkage (SLINK) clustering algorithm on the means of the arms. Since the true means are unknown, the objective is to obtain the above clustering of the arms with the minimum number of samples drawn from the arms, subject to an upper bound on the error probability. We introduce a novel algorithm, Average Tracking Bandit Online Clustering (ATBOC), and prove that this algorithm is order optimal, meaning that the upper bound on its expected sample complexity for given error probability $\delta$ is within a factor of 2 of an instance-dependent lower bound as $\delta \rightarrow 0$. Furthermore, we propose a computationally more efficient algorithm, Lower and Upper Confidence Bound-based Bandit Online Clustering (LUCBBOC), inspired by the LUCB algorithm for best arm identification. Simulation results demonstrate that the performance of LUCBBOC is comparable to that of ATBOC. We numerically assess the effectiveness of the proposed algorithms through numerical experiments on both synthetic datasets and the real-world MovieLens dataset. To the best of our knowledge, this is the first work on bandit online clustering that allows arms with different means in a cluster and $K$ greater than 2.