高维线性混合模型的可扩展学习算法
Scalable Algorithms for Learning High-Dimensional Linear Mixed Models
摘要：
线性混合模型（Linear mixed models，简称LMM）广泛用于线性回归中观测值的相依性建模，在许多应用领域中得到了广泛的应用。在大数据上，LIFE的参数估计可能在计算上是禁止的。最先进的学习算法需要计算复杂度，其至少线性地依赖于协变量的维度$p$，并且通常使用不提供理论保证的算法。我们提出了可扩展的算法，学习高维线性次线性计算复杂度依赖于$p$。我们的方法的关键是新的双估计，只使用核函数的数据，和快速计算技术的基础上的子采样随机Hadamard变换。我们为我们的学习算法提供了理论保证，证明了参数估计的鲁棒性。最后，我们补充的理论与实验的大合成和真实数据。
英文摘要：
Linear mixed models (LMMs) are used extensively to model dependecies of observations in linear regression and are used extensively in many application areas. Parameter estimation for LMMs can be computationally prohibitive on big data. State-of-the-art learning algorithms require computational complexity which depends at least linearly on the dimension $p$ of the covariates, and often use heuristics that do not offer theoretical guarantees. We present scalable algorithms for learning high-dimensional LMMs with sublinear computational complexity dependence on $p$. Key to our approach are novel dual estimators which use only kernel functions of the data, and fast computational techniques based on the subsampled randomized Hadamard transform. We provide theoretical guarantees for our learning algorithms, demonstrating the robustness of parameter estimation. Finally, we complement the theory with experiments on large synthetic and real data.
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