通过几何方法实现无辅助信息的非线性成分解耦，突破传统依赖假设的限制。
Diverse Influence Component Analysis: A Geometric Approach to Nonlinear Mixture Identifiability
- 利用混合函数雅可比矩阵的凸几何特性，设计体积最大化准则
- 在无需辅助信号或独立性假设下实现成分可识别性
- 适合需要强解耦表示的场景，如因果推断与表征学习
从未知的非线性混合中识别潜在成分是机器学习中的基础挑战，广泛应用于解耦表征学习和因果推断等任务。以往的非线性独立成分分析（nICA）研究表明，辅助信号（如弱监督）可支持条件独立潜变量的可识别性。近期方法尝试引入结构假设（如混合函数雅可比矩阵的稀疏性）以降低要求。本文提出多样影响成分分析（DICA），利用混合函数雅可比矩阵的凸几何特性，设计雅可比体积最大化（J-VolMax）准则，通过鼓励潜变量对观测变量的影响多样性来实现成分识别。在合理条件下，该方法无需依赖辅助信息、潜变量独立性或雅可比稀疏性假设即可实现可识别性。这些结果拓展了可识别性分析的边界，为现有方法提供了互补视角。
原文摘要 · Abstract (English)
Latent component identification from unknown nonlinear mixtures is a foundational challenge in machine learning, with applications in tasks such as disentangled representation learning and causal inference. Prior work in nonlinear independent component analysis (nICA) has shown that auxiliary signals -- such as weak supervision -- can support identifiability of conditionally independent latent components. More recent approaches explore structural assumptions, e.g., sparsity in the Jacobian of the mixing function, to relax such requirements. In this work, we introduce Diverse Influence Component Analysis (DICA), a framework that exploits the convex geometry of the mixing function's Jacobian. We propose a Jacobian Volume Maximization (J-VolMax) criterion, which enables latent component identification by encouraging diversity in their influence on the observed variables. Under reasonable conditions, this approach achieves identifiability without relying on auxiliary information, latent component independence, or Jacobian sparsity assumptions. These results extend the scope of identifiability analysis and offer a complementary perspective to existing methods.
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