分区最小二乘法
Partitioned Least Squares
AI总结：
本文提出分区最小二乘模型，将输入特征分组并要求组内变量贡献相似，通过非凸问题的精确与近似求解方法评估组及变量重要性，并证明NP完全性。
摘要：
在本文中，我们提出了一种线性最小二乘模型的变体，该变体允许实践者将输入特征划分为若干组变量，并要求这些组内的变量对最终结果做出相似的贡献。输出结果使实践者能够评估每个组以及组内每个变量的重要性。我们正式证明了新公式不是凸的，并提供了两种处理该问题的替代方法：一种基于交替最小二乘方法的非精确方法；另一种基于使用指数数量的子问题重新表述该问题的精确方法，其最小值保证为最优解。我们正式证明了精确方法的正确性，并比较了两种解决方案，表明在分区数量较少的情况下，精确解在所需时间的一小部分内提供了更好的结果。为了完整性，我们还提供了一种替代的分支定界算法，当分区数量过多时，该算法可代替精确方法使用，并提供了本文引入的优化问题的NP完全性证明。
英文摘要：
In this paper we propose a variant of the linear least squares model allowing practitioners to partition the input features into groups of variables that they require to contribute similarly to the final result. The output allows practitioners to assess the importance of each group and of each variable in the group. We formally show that the new formulation is not convex and provide two alternative methods to deal with the problem: one non-exact method based on an alternating least squares approach; and one exact method based on a reformulation of the problem using an exponential number of sub-problems whose minimum is guaranteed to be the optimal solution. We formally show the correctness of the exact method and also compare the two solutions showing that the exact solution provides better results in a fraction of the time required by the alternating least squares solution (assuming that the number of partitions is small). For the sake of completeness, we also provide an alternative branch and bound algorithm that can be used in place of the exact method when the number of partitions is too large, and a proof of NP-completeness of the optimization problem introduced in this paper.
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