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.

翻译：张量分解是广泛应用于信号处理、机器学习及其他众多领域的基础技术。然而，传统张量分解方法在联合分析多块张量时存在局限，往往难以有效挖掘张量间的共享信息。本研究首先提出一种由交替近端梯度法优化的新型耦合非负CANDECOMP/PARAFAC分解算法（CoNCPD-APG）。该算法专门设计用于解决部分或完全关联的不同张量的联合分解问题，同时提取公共分量、独立分量与核心张量。针对高维张量数据上优化非负约束固有的计算挑战，我们进一步提出lraCoNCPD-APG算法。通过将低秩近似与所提出的CoNCPD-APG方法相结合，该算法能在不损失分解质量的前提下显著降低计算负担，尤其适用于多块大规模张量。在合成数据、真实人脸图像数据及两类脑电图（EEG）数据上进行的仿真实验，验证了所提算法在耦合非负张量分解问题中的实用性与优越性。我们的结果凸显了该方法从复杂多块张量数据中发掘有意义模式与结构的有效性，从而为未来应用提供了重要参考。