In this work, we explore Partitioned Independent Component Analysis (PICA), an extension of the well-established Independent Component Analysis (ICA) framework. Traditionally, ICA focuses on extracting a vector of independent source signals from a linear combination of them defined by a mixing matrix. We aim to provide a comprehensive understanding of the identifiability of this mixing matrix in ICA. Significant to our investigation, recent developments by Mesters and Zwiernik relax these strict independence requirements, studying the identifiability of the mixing matrix from zero restrictions on cumulant tensors. In this paper, we assume alternative independence conditions, in particular, the PICA case, where only partitions of the sources are mutually independent. We study this case from an algebraic perspective, and our primary result generalizes previous results on the identifiability of the mixing matrix.

翻译：本文探讨了分部独立成分分析（PICA），这是经典独立成分分析（ICA）框架的扩展。传统上，ICA旨在从由混合矩阵定义的线性组合中提取独立的源信号向量。我们的目标是对ICA中该混合矩阵的可辨识性提供全面理解。重要地，Mesters和Zwiernik的最新研究放宽了这些严格的独立性要求，通过累积张量的零约束研究了混合矩阵的可辨识性。在本文中，我们假设了替代的独立性条件，特别是PICA情形，其中仅源信号的分区之间相互独立。我们从代数角度研究这一情形，我们的主要结果推广了先前关于混合矩阵可辨识性的结论。