B样条回归节点选择的精确惩罚方法
Exact Penalty Method for Knot Selection of B-Spline Regression
AI总结：
本文提出一种基于非凸正则化最小二乘重构的B样条回归节点选择新方法，通过GIST算法求解并证明其满足基数约束，从而获得最多依赖指定数量节点的样条回归模型。
摘要：
本文提出了一种在估计B样条回归模型的同时选择节点的新方法。这种同时选择节点和模型的方法并非易事，但我们的策略通过在通常应用的最小二乘法上采用非凸正则化使其成为可能。更具体地说，受直接指定要使用的节点数量（上限）的约束的启发，我们提出了一种（无约束）正则化最小二乘重构，随后证明该重构与启发它的基数约束公式等价。对所得公式进一步修改，使我们能够针对一类非光滑非凸优化问题，采用一种称为GIST的近端梯度型算法。我们证明，在温和的技术假设下，该算法可达到问题的局部最小值。由于已证明该问题的任何局部最小值都满足基数约束，因此所提出的算法可用于获得最多仅依赖于指定数量节点的样条回归模型。数值实验展示了我们的方法在合成和真实数据集上的表现。
英文摘要：
This paper presents a new approach to selecting knots at the same time as estimating the B-spline regression model. Such simultaneous selection of knots and model is not trivial, but our strategy can make it possible by employing a nonconvex regularization on the least square method that is usually applied. More specifically, motivated by the constraint that directly designates (the upper bound of) the number of knots to be used, we present an (unconstrained) regularized least square reformulation, which is later shown to be equivalent to the motivating cardinality-constrained formulation. The obtained formulation is further modified so that we can employ a proximal gradient-type algorithm, known as GIST, for a class of non-smooth non-convex optimization problems. We show that under a mild technical assumption, the algorithm is shown to reach a local minimum of the problem. Since it is shown that any local minimum of the problem satisfies the cardinality constraint, the proposed algorithm can be used to obtain a spline regression model that depends only on a designated number of knots at most. Numerical experiments demonstrate how our approach performs on synthetic and real data sets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
