Modern technologies are producing a wealth of data with complex structures. For instance, in two-dimensional digital imaging, flow cytometry and electroencephalography, matrix-type covariates frequently arise when measurements are obtained for each combination of two underlying variables. To address scientific questions arising from those data, new regression methods that take matrices as covariates are needed, and sparsity or other forms of regularization are crucial owing to the ultrahigh dimensionality and complex structure of the matrix data. The popular lasso and related regularization methods hinge on the sparsity of the true signal in terms of the number of its non-zero coefficients. However, for the matrix data, the true signal is often of, or can be well approximated by, a low rank structure. As such, the sparsity is frequently in the form of low rank of the matrix parameters, which may seriously violate the assumption of the classical lasso. We propose a class of regularized matrix regression methods based on spectral regularization. A highly efficient and scalable estimation algorithm is developed, and a degrees-of-freedom formula is derived to facilitate model selection along the regularization path. Superior performance of the method proposed is demonstrated on both synthetic and real examples.

中文翻译：

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正则化矩阵回归。

现代技术正在产生大量结构复杂的数据。例如，在二维数字成像、流式细胞术和脑电图中，当为两个潜在变量的每个组合获得测量值时，经常会出现矩阵类型的协变量。为了解决这些数据产生的科学问题，需要以矩阵作为协变量的新回归方法，由于矩阵数据的超高维和复杂结构，稀疏性或其他形式的正则化至关重要。流行的套索和相关的正则化方法取决于真实信号在其非零系数数量方面的稀疏性。然而，对于矩阵数据，真实信号通常是低秩结构，或者可以很好地近似于低秩结构。像这样，稀疏性通常表现为矩阵参数的低秩，这可能严重违反经典套索的假设。我们提出了一类基于谱正则化的正则化矩阵回归方法。开发了一种高效且可扩展的估计算法，并导出了一个自由度公式，以方便沿正则化路径进行模型选择。所提出的方法的优越性能在合成和真实例子中都得到了证明。并推导出一个自由度公式，以方便沿正则化路径进行模型选择。所提出的方法的优越性能在合成和真实例子中都得到了证明。并推导出一个自由度公式，以方便沿正则化路径进行模型选择。所提出的方法的优越性能在合成和真实例子中都得到了证明。