张量数据的独立分量分析
Independent component analysis for tensor-valued data
摘要：
在预处理张量值数据（例如图像和视频）时，一个常见的过程是将观测向量化，并将所得向量应用于独立分量分析（ICA）的许多方法之一。然而，原始数据的张量结构在矢量化中丢失，作为更合适的替代方案，我们提出了矩阵和张量四阶盲识别（MFOBI和TFOBI）。在经典的四阶盲识别（FOBI）的这些张量扩展中，我们假设混合的Kronecker结构，并在所观察到的张量的每个方向上同时执行FOBI。我们讨论了MFOBI和TFOBI背后的理论和假设，并提供了两种不同的算法和相关的估计的解混矩阵以及它们的渐近性质。最后，通过仿真比较了该方法与经典的FOBI方法在矢量化数据上的性能，并给出了一个实际的数据聚类实例。
英文摘要：
In preprocessing tensor-valued data, e.g. images and videos, a common procedure is to vectorize the observations and subject the resulting vectors to one of the many methods used for independent component analysis (ICA). However, the tensor structure of the original data is lost in the vectorization and, as a more suitable alternative, we propose the matrix- and tensor fourth order blind identification (MFOBI and TFOBI). In these tensorial extensions of the classic fourth order blind identification (FOBI) we assume a Kronecker structure for the mixing and perform FOBI simultaneously on each direction of the observed tensors. We discuss the theory and assumptions behind MFOBI and TFOBI and provide two different algorithms and related estimates of the unmixing matrices along with their asymptotic properties. Finally, simulations are used to compare the method's performance with that of classical FOBI for vectorized data and we end with a real data clustering example.
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