JADE for Tensor-Valued Observations

Independent component analysis is a standard tool in modern data analysis and numerous different techniques for applying it exist. The standard methods however quickly lose their effectiveness when the data are made up of structures of higher order than vectors, namely, matrices or tensors (e.g., images or videos), being unable to handle the high amounts of noise. Recently, an extension of the classic fourth-order blind identification (FOBI) specially suited for tensor-valued observations was proposed and showed to outperform its vector version for tensor data. In this article, we extend another popular independent component analysis method, the joint approximate diagonalization of eigen-matrices (JADE), for tensor observations. In addition to the theoretical background, we also provide the asymptotic properties of the proposed estimator and use both simulations and real data to show its usefulness and superiority over its competitors. Supplementary material including the proofs of the theorems and the codes for running the simulations and the real data example are available online.

独立成分分析（Independent Component Analysis）是现代数据分析中的标准工具，目前已涌现出诸多不同的实现方法。然而，当数据由高于向量阶数的结构（即矩阵或张量（tensor），例如图像或视频）构成时，标准方法的性能会迅速衰减，且无法处理高强度噪声。近期，有研究提出了一款专为张量值观测设计的经典四阶盲辨识（fourth-order blind identification, FOBI）扩展方法，并证实其在张量数据场景下的表现优于其向量版本。本文中，我们针对张量观测场景，对另一款主流独立成分分析方法——特征矩阵联合近似对角化（joint approximate diagonalization of eigen-matrices, JADE）——进行了扩展。除理论背景铺垫外，本文还推导了所提估计量的渐近性质，并通过模拟实验与真实数据集验证了该方法的有效性与相较于同类竞争方法的优越性。包含定理证明、模拟实验及真实数据示例代码的补充材料均可在线获取。