Fast Learnings of Coupled Nonnegative Tensor Decomposition Using Optimal Gradient and Low-rank Approximation

Tensor decomposition is a fundamental technique widely applied in signal processing, machine learning, and various other fields. However, traditional tensor decomposition methods encounter limitations when jointly analyzing multi-block tensors, as they often struggle to effectively explore shared information among tensors. In this study, we first introduce a novel coupled nonnegative CANDECOMP/PARAFAC decomposition algorithm optimized by the alternating proximal gradient method (CoNCPD-APG). This algorithm is specially designed to address the challenges of jointly decomposing different tensors that are partially or fully linked, while simultaneously extracting common components, individual components and, core tensors. Recognizing the computational challenges inherent in optimizing nonnegative constraints over high-dimensional tensor data, we further propose the lraCoNCPD-APG algorithm. By integrating low-rank approximation with the proposed CoNCPD-APG method, the proposed algorithm can significantly decrease the computational burden without compromising decomposition quality, particularly for multi-block large-scale tensors. Simulation experiments conducted on synthetic data, real-world face image data, and two kinds of electroencephalography (EEG) data demonstrate the practicality and superiority of the proposed algorithms for coupled nonnegative tensor decomposition problems. Our results underscore the efficacy of our methods in uncovering meaningful patterns and structures from complex multi-block tensor data, thereby offering valuable insights for future applications.