Robust nonparametric nearest neighbor random process clustering

TL;DR

This paper introduces two robust clustering algorithms for noisy, finite-length observations of stationary ergodic random processes, demonstrating their effectiveness even with significant noise, missing data, and overlapping spectral features.

Contribution

The paper proposes and analyzes two novel spectral clustering algorithms for process data, providing theoretical guarantees and demonstrating superior performance in challenging scenarios.

Findings

Both algorithms succeed with high probability under noise and missing data.

The algorithms outperform existing methods in human motion sequence clustering.

Theoretical analysis quantifies the effects of overlap, noise, and missing data on clustering performance.

Abstract

We consider the problem of clustering noisy finite-length observations of stationary ergodic random processes according to their generative models without prior knowledge of the model statistics and the number of generative models. Two algorithms, both using the $L^1$-distance between estimated power spectral densities (PSDs) as a measure of dissimilarity, are analyzed. The first one, termed nearest neighbor process clustering (NNPC), relies on partitioning the nearest neighbor graph of the observations via spectral clustering. The second algorithm, simply referred to as $k$-means (KM), consists of a single $k$-means iteration with farthest point initialization and was considered before in the literature, albeit with a different dissimilarity measure. We prove that both algorithms succeed with high probability in the presence of noise and missing entries, and even when the generative process PSDs overlap significantly, all provided that the observation length is sufficiently large. Our results quantify the tradeoff between the overlap of the generative process PSDs, the observation length, the fraction of missing entries, and the noise variance. Finally, we provide extensive numerical results for synthetic and real data and find that NNPC outperforms state-of-the-art algorithms in human motion sequence clustering.