Sparse principal component analysis (sPCA) enhances the interpretability of principal components (PCs) by imposing sparsity constraints on loading vectors (LVs). However, when used as a precursor to independent component analysis (ICA) for blind source separation (BSS), sPCA may underperform due to its focus on simplicity, potentially disregarding some statistical information essential for effective ICA. To overcome this limitation, a sophisticated approach is proposed that preserves the interpretability advantages of sPCA while significantly enhancing its source extraction capabilities. This consists of two tailored algorithms, dissociative PCA (DPCA1 and DPCA2), which employ adaptive and firm thresholding alongside gradient and coordinate descent approaches to optimize the proposed model dynamically. These algorithms integrate left and right singular vectors from singular value decomposition (SVD) through dissociation matrices (DMs) that replace traditional singular values, thus capturing latent interdependencies effectively to model complex source relationships. This leads to refined PCs and LVs that more accurately represent the underlying data structure. The proposed approach avoids focusing on individual eigenvectors, instead, it collaboratively combines multiple eigenvectors to disentangle interdependencies within each SVD variate. The superior performance of the proposed DPCA algorithms is demonstrated across four varied imaging applications including functional magnetic resonance imaging (fMRI) source retrieval, foreground-background separation, image reconstruction, and image inpainting. They outperformed traditional methods such as PCA+ICA, PPCA+ICA, SPCA+ICA, PMD, and GPower.

翻译：稀疏主成分分析（sPCA）通过对载荷向量（LVs）施加稀疏性约束，提升了主成分（PCs）的可解释性。然而，当将其作为独立成分分析（ICA）进行盲源分离（BSS）的前置步骤时，sPCA可能因过度关注简洁性而表现不佳，从而可能忽略对有效ICA至关重要的部分统计信息。为克服这一局限，本文提出一种先进方法，在保留sPCA可解释性优势的同时，显著增强了其源信号提取能力。该方法包含两种定制算法——解离主成分分析（DPCA1与DPCA2），它们采用自适应与硬阈值技术，结合梯度下降与坐标下降方法动态优化所提模型。这些算法通过解离矩阵（DMs）替代传统奇异值，整合奇异值分解（SVD）中的左右奇异向量，从而有效捕捉潜在相互依赖关系以建模复杂源信号关联。由此产生更精确表征底层数据结构的主成分与载荷向量。所提方法避免聚焦于单个特征向量，而是协同组合多个特征向量以解构每个SVD变量内的相互依赖关系。在功能磁共振成像（fMRI）源信号提取、前景-背景分离、图像重建与图像修复等四种不同成像应用中，所提DPCA算法均展现出优于传统方法（如PCA+ICA、PPCA+ICA、SPCA+ICA、PMD及GPower）的卓越性能。