Change Point Detection in Hadamard Spaces by Alternating Minimization

Time series analysis of non-Euclidean data is highly challenging and crucial for many real-world applications. We address the problem of detecting multiple changes in time series within these complex data spaces. Hadamard spaces, which encompass important data spaces like positive semidefinite matrices, certain Wasserstein spaces, and hyperbolic spaces, provide the right general framework to address this complexity. We propose a computationally efficient two-step iterative optimization algorithm called HOP (Hadamard Optimal Partitioning) that detects changes in the sequence of so-called Fréchet means. Under mild conditions, the proposed method consistently estimates the change point locations. HOP is highly versatile, accommodating structural assumptions such as cyclic patterns and epidemic settings, making it unique in the literature. We validate its performance in synthetic and real-world scenarios, including applications in human gait analysis using EMG data with low SNR and behavioral analysis of animal motion.