# Using matrix-product states for time-series machine learning

arXiv: https://arxiv.org/abs/2412.15826 · HTML: https://mlsanity.com/paper/2412.15826/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 59 references that are in the map.
- First posted on arXiv 20 Dec 2024.

## Builds on (most cited first, 29 of 29)

- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Transformers in Time Series: A Survey](https://mlsanity.com/paper/2202.07125/) (2022)
- [InceptionTime: Finding AlexNet for Time Series Classification](https://mlsanity.com/paper/1909.04939/) (2019)
- [Real-valued (Medical) Time Series Generation with Recurrent Conditional GANs](https://mlsanity.com/paper/1706.02633/) (2017)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [HIVE-COTE 2.0: a new meta ensemble for time series classification](https://mlsanity.com/paper/2104.07551/) (2021)
- [Bake off redux: a review and experimental evaluation of recent time series classification algorithms](https://mlsanity.com/paper/2304.13029/) (2023)
- [A Survey on Diffusion Models for Time Series and Spatio-Temporal Data](https://mlsanity.com/paper/2404.18886/) (2024)
- [Deep Learning for Multivariate Time Series Imputation: A Survey](https://mlsanity.com/paper/2402.04059/) (2024)
- [aeon: a Python toolkit for learning from time series](https://mlsanity.com/paper/2406.14231/) (2024)
- [Supervised Learning with Quantum-Inspired Tensor Networks](https://mlsanity.com/paper/1605.05775/) (2016)
- [Time Series Data Imputation: A Survey on Deep Learning Approaches](https://mlsanity.com/paper/2011.11347/) (2020)
- [Quantum-inspired Machine Learning on high-energy physics data](https://mlsanity.com/paper/2004.13747/) (2020)
- [Highly comparative time-series analysis: The empirical structure of time series and their methods](https://mlsanity.com/paper/1304.1209/) (2013)
- [Deep Imputation of Missing Values in Time Series Health Data: A Review with Benchmarking](https://mlsanity.com/paper/2302.10902/) (2023)
- [Anomaly Detection with Tensor Networks](https://mlsanity.com/paper/2006.02516/) (2020)
- [Tensor Networks for Medical Image Classification](https://mlsanity.com/paper/2004.10076/) (2020)
- [Entanglement and Tensor Networks for Supervised Image Classification](https://mlsanity.com/paper/2007.06082/) (2020)
- [Entanglement-guided architectures of machine learning by quantum tensor network](https://mlsanity.com/paper/1803.09111/) (2018)
- [Matrix Product State Based Quantum Classifier](https://mlsanity.com/paper/1905.01426/) (2019)
- [A Multi-Scale Tensor Network Architecture for Classification and Regression](https://mlsanity.com/paper/2001.08286/) (2020)
- [Tensor Networks for Explainable Machine Learning in Cybersecurity](https://mlsanity.com/paper/2401.00867/) (2023)
- [A Practical Guide to Sample-based Statistical Distances for Evaluating Generative Models in Science](https://mlsanity.com/paper/2403.12636/) (2024)
- [A Review on Generative AI Models for Synthetic Medical Text, Time Series, and Longitudinal Data](https://mlsanity.com/paper/2411.12274/) (2024)
- [Generative Learning of Continuous Data by Tensor Networks](https://mlsanity.com/paper/2310.20498/) (2023)
- [A Matrix Product State Model for Simultaneous Classification and Generation](https://mlsanity.com/paper/2406.17441/) (2024)
- [Shortcut Matrix Product States and its applications](https://mlsanity.com/paper/1812.05248/) (2018)
- [Positive unlabeled learning with tensor networks](https://mlsanity.com/paper/2211.14085/) (2022)
- [Parameter inference from a non-stationary unknown process](https://mlsanity.com/paper/2407.08987/) (2024)

## Built on by (most cited first, 2 of 2)

- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [On the Approximation Complexity of Matrix Product Operator (MPO) Born Machines](https://mlsanity.com/paper/2605.11471/) (2026)

## Abstract (arXiv)

Matrix-product states (MPS) have proven to be a versatile ansatz for modeling quantum many-body physics. For many applications, and particularly in one-dimension, they capture relevant quantum correlations in many-body wavefunctions while remaining tractable to store and manipulate on a classical computer. This has motivated researchers to also apply the MPS ansatz to machine learning (ML) problems where capturing complex correlations in datasets is also a key requirement. Here, we develop and apply an MPS-based algorithm, MPSTime, for learning a joint probability distribution underlying an observed time-series dataset, and show how it can be used to tackle important time-series ML problems, including classification and imputation. MPSTime can efficiently learn complicated time-series probability distributions directly from data, requires only moderate maximum MPS bond dimension chirm max, with values for our applications ranging between chirm max = 20-160, and can be trained for both classification and imputation tasks under a single logarithmic loss function. Using synthetic and publicly available real-world datasets, spanning applications in medicine, energy, and astronomy, we demonstrate performance competitive with state-of-the-art ML approaches, but with the key advantage of encoding the full joint probability distribution learned from the data, which is useful for analyzing and interpreting its underlying structure. This manuscript is supplemented with the release of a publicly available code package MPSTime that implements our approach. The effectiveness of the MPS-based ansatz for capturing complex correlation structures in time-series data makes it a powerful foundation for tackling challenging time-series analysis problems across science, industry, and medicine.