Unsupervised Learning of Nonlinear Mixtures: Identifiability and Algorithm

Linear mixture models (LMMs) have proven very useful in a plethora of applications, e.g., topic modeling, clustering, and speech / audio separation. As a critical aspect of the LMM, identifiability of the model parameters is well-studied, under frameworks such as independent component analysis and constrained matrix factorization. Nevertheless, when the linear mixtures are distorted by unknown nonlinear functions - which is well-motivated and more realistic in many cases - the associated identifiability issues are far less studied. This work focuses on parameter identification of a nonlinear mixture model that is motivated by a number of real-world applications, e.g., hyperspectral imaging and magnetic resonance imaging. A novel identification criterion is proposed and the associated identifiability issues are studied. A practical implementation based on a judiciously designed neural network is proposed to realize the criterion, and an effective learning algorithm is proposed. Numerical results on synthetic and real application data corroborate the effectiveness of the proposed method.