A Quantum Tensor Network-Based Viewpoint for Modeling and Analysis of Time Series Data
Pragatheeswaran Vipulananthan, Kamal Premaratne, Dilip Sarkar, Manohar N. Murthi
TL;DR
The paper presents a quantum tensor network-based framework for time series that furnishes interpretable uncertainty quantification by mapping the kernel mean embedding ($KME$) into a 1-D spin-chain Hamiltonian within a reproducing kernel Hilbert space ($RKHS$) and analyzing its eigen-modes via Schrödinger dynamics with perturbation theory. A finite target space of Hermitian spin operators is constructed, and a quantum correlation matrix ($M_{oldsymbol{1}}$) guides selection of Hamiltonians that admit the $KME$ as an eigen-mode; perturbation-based UQ yields first-order corrections to uncertainty measures. Empirical validation on synthetic change point data and a VidTIMIT-based TS clustering task demonstrates competitive performance against baselines and yields interpretable uncertainty signals aligned with regime changes. The framework advances interpretability in time-series analysis by tying data structure to physically meaningful eigen-modes and deterministic UQ, with potential for robust decision-making in real-world TS applications.
Abstract
Accurate uncertainty quantification is a critical challenge in machine learning. While neural networks are highly versatile and capable of learning complex patterns, they often lack interpretability due to their ``black box'' nature. On the other hand, probabilistic ``white box'' models, though interpretable, often suffer from a significant performance gap when compared to neural networks. To address this, we propose a novel quantum physics-based ``white box'' method that offers both accurate uncertainty quantification and enhanced interpretability. By mapping the kernel mean embedding (KME) of a time series data vector to a reproducing kernel Hilbert space (RKHS), we construct a tensor network-inspired 1D spin chain Hamiltonian, with the KME as one of its eigen-functions or eigen-modes. We then solve the associated Schr{ö}dinger equation and apply perturbation theory to quantify uncertainty, thereby improving the interpretability of tasks performed with the quantum tensor network-based model. We demonstrate the effectiveness of this methodology, compared to state-of-the-art ``white box" models, in change point detection and time series clustering, providing insights into the uncertainties associated with decision-making throughout the process.