广义最小二乘矩阵分解
A Generalized Least Squares Matrix Decomposition
摘要：
许多大规模高维数据集中的变量都是结构化的，例如来自成像和时间序列中的规则网格测量，或来自气候研究中的时空测量。经典的多变量技术忽略了这些结构关系，往往导致性能不佳。我们提出了一个广义的奇异值分解（SVD）和主成分分析（PCA），是适合于大量的数据集与结构化变量或已知的双向依赖关系。通过找到最佳的低秩近似的数据相对于一个转置的二次范数，我们的分解，称为广义最小二乘矩阵分解（GMD），直接占结构关系。由于高维环境中的许多变量通常是不相关的或有噪声的，我们还通过添加双向惩罚来正则化矩阵分解，以鼓励稀疏性或平滑性。我们开发了快速计算算法，使用我们的方法进行广义PCA（GPCA），稀疏GPCA和功能GPCA的海量数据集。通过模拟和全脑功能磁共振成像的例子，我们证明了我们的方法的效用降维，信号恢复，和高维结构化数据的特征选择。
英文摘要：
Variables in many massive high-dimensional data sets are structured, arising for example from measurements on a regular grid as in imaging and time series or from spatial-temporal measurements as in climate studies. Classical multivariate techniques ignore these structural relationships often resulting in poor performance. We propose a generalization of the singular value decomposition (SVD) and principal components analysis (PCA) that is appropriate for massive data sets with structured variables or known two-way dependencies. By finding the best low rank approximation of the data with respect to a transposable quadratic norm, our decomposition, entitled the Generalized least squares Matrix Decomposition (GMD), directly accounts for structural relationships. As many variables in high-dimensional settings are often irrelevant or noisy, we also regularize our matrix decomposition by adding two-way penalties to encourage sparsity or smoothness. We develop fast computational algorithms using our methods to perform generalized PCA (GPCA), sparse GPCA, and functional GPCA on massive data sets. Through simulations and a whole brain functional MRI example we demonstrate the utility of our methodology for dimension reduction, signal recovery, and feature selection with high-dimensional structured data.
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