A Quantum-inspired Algorithm for General Minimum Conical Hull Problems

Maximum a posteriori (MAP) estimation is a central problem in machine and statistical learning [5, 22]. The general MAP problem has been proven to be NP hard [33]. Despite the hardness in the general case, there are two fundamental learning models, the matrix factorization and the latent variable model, that enable MAP problem to be solved in polynomial runtime under certain constraints [24, 25, 29, 30, 32]. The algorithms that have been developed for these learning models have been used extensively in machine learning with competitive performance, particularly on tasks such as subspace clustering, topic modeling, collaborative filtering, structure prediction, feature engineering, motion segmentation, sequential data analysis, and recommender systems [15, 25, 30]. A recent study demonstrates that MAP problems addressed by matrix factorization and the latent variable models can be reduced to the general minimum conical hull problem [41]. In particular, the general minimum conical hull problem transforms problems resolved by these two learning models into a geometric problem, whose goal is to identify a set of extreme data points with the smallest cardinality in dataset Y such that every data point in dataset X can be expressed as a conical combination of the identified extreme data points. Unlike the matrix factorization and the latent variable models that their optimizations generally suffer from the local minima, a unique global solution is guaranteed for the general minimum conical hull problem [41].