具有多模态扩展的低秩矩阵混合隶属度模型
Mixed Membership Model of Low-rank Matrices with Multimodal Extension
AI总结：
针对矩阵值数据提出带多模态扩展的低秩混合隶属度模型，兼具可解释群体原型与连续受试者隶属度，在模拟及人类连接组计划数据中表现优异，达极小极大最优重构率。
AI中文摘要：
矩阵值观测数据出现在多重网络、神经成像等领域，这些领域中群体层面的模式通常是低秩的，且受试者可能同时表现出多个潜在模式。现有的张量主成分分析（tensor PCA）方法能提供连续的受试者得分，但其载荷矩阵作为群体原型可能难以解释；而低秩聚类虽能得到可解释的原型，但采用硬标签。我们针对矩阵值数据提出了一种低秩混合隶属度模型，其中每个受试者矩阵的期望值是潜在低秩基矩阵的凸组合。该模型既能得到可解释的群体层面极端轮廓，又能得到连续的受试者层面隶属度。我们的多模态扩展在模态间共享隶属度，同时使用模态特定的基矩阵，且在单一模态不足时可恢复模型的可识别性。我们在纯受试者条件下建立了可识别性，提出了带谱初始化和低秩优化的约束最小二乘估计器及可扩展算法，并推导了非渐近误差界。该估计器在对数因子范围内达到极小极大最优重构率，在几何条件下基矩阵和隶属度的收敛率相互分离。模拟结果验证了理论收敛率并显示出优异性能。在人类连接组计划（Human Connectome Project）功能连接数据的分析中，所提方法识别出可解释的脑连接轮廓，其估计的隶属度与认知表型密切相关。
英文摘要：
Matrix-valued observations arise in multiplex networks, neuroimaging, and other domains where population-level patterns are often low-rank and subjects may express several latent patterns simultaneously. Existing tensor PCA methods provide continuous subject scores but their loading matrices can be difficult to interpret as population prototypes, while low-rank clustering yields interpretable prototypes with hard labels. We introduce a low-rank mixed membership model for matrix-valued data in which the expected value of each subject's matrix is a convex combination of latent low-rank basis matrices. The model yields both interpretable population-level extreme profiles and continuous subject-level memberships. Our multimodal extension shares memberships across modalities with modality-specific basis matrices and can restore identifiability when one modality is insufficient. We establish identifiability under a pure-subject condition, propose a constrained least-squares estimator and scalable algorithm with spectral initialization and low-rank refinement, and derive nonasymptotic error bounds. The estimator achieves a minimax-optimal reconstruction rate up to a logarithmic factor, with separate basis and membership convergence rates under a geometric condition. Simulations corroborate the theoretical rates and show strong performance. In an analysis of Human Connectome Project functional connectivity data, the proposed method identifies interpretable brain connectivity profiles whose estimated memberships are strongly associated with cognitive phenotypes.
