Using matrix-product states for time-series machine learning

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 wave functions while remaining tractable (polynomial scaling) 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, for the first time, we develop an MPS-based algorithm, MPSTime, for learning a joint probability distribution underlying a 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>χ</a:mi><a:mi>max</a:mi></a:msub></a:math> (with values for our applications ranging between <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>χ</b:mi><b:mi>max</b:mi></b:msub><b:mo>=</b:mo><b:mn>20</b:mn><b:mo>−</b:mo><b:mn>160</b:mn></b:mrow></b:math>), 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 that implements our approach, and can be used to reproduce all presented results. 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.