Recommender systems inspired by the structure of quantum theory
  Physicists use quantum models to describe the behavior of physical systems. Quantum models owe their success to their interpretability, to their relation to probabilistic models (quantization of classical models) and to their high predictive power. Beyond physics, these properties are valuable in general data science. This motivates the use of quantum models to analyze general nonphysical datasets. Here we provide both empirical and theoretical insights into the application of quantum models in data science. In the theoretical part of this paper, we firstly show that quantum models can be exponentially more efficient than probabilistic models because there exist datasets that admit low-dimensional quantum models and only exponentially high-dimensional probabilistic models. Secondly, we explain in what sense quantum models realize a useful relaxation of compressed probabilistic models. Thirdly, we show that sparse datasets admit low-dimensional quantum models and finally, we introduce a method to compute hierarchical orderings of properties of users (e.g., personality traits) and items (e.g., genres of movies). In the empirical part of the paper, we evaluate quantum models in item recommendation and observe that the predictive power of quantum-inspired recommender systems can compete with state-of-the-art recommender systems like SVD++ and PureSVD. Furthermore, we make use of the interpretability of quantum models by computing hierarchical orderings of properties of users and items. This work establishes a connection between data science (item recommendation), information theory (communication complexity), mathematical programming (positive semidefinite factorizations) and physics (quantum models). 
Cyril Stark
cs.LG cs.IT math.IT math.OC quant-ph stat.ML