论文标题
分段线性通过凸功能的差异回归
Piecewise Linear Regression via a Difference of Convex Functions
论文作者
论文摘要
我们提出了一种新的分段线性回归方法，该方法利用将凸功能（DC函数）拟合到数据的差异。这些是函数$ f $，可以表示为差异$ ϕ_1 -DAS_2 $，用于选择凸功能$ ϕ_1，ϕ_2 $。该方法通过类似于Max-affine回归的方式来估算分段衬里凸函数，该方法的差异近似于数据。该函数的选择是由一个新的seminorm在DC函数上正规化的，该函数控制了估计的$ \ ell_ \ infty $ lipschitz常数。即使在高维度中，也可以通过二次编程有效地实现所得的方法，并显示出接近Minimax统计风险。我们从经验上验证了该方法，表明该方法实际上是可以实现的，并且具有与现有数据集中的现有回归/分类方法相当的性能。
We present a new piecewise linear regression methodology that utilizes fitting a difference of convex functions (DC functions) to the data. These are functions $f$ that may be represented as the difference $ϕ_1 - ϕ_2$ for a choice of convex functions $ϕ_1, ϕ_2$. The method proceeds by estimating piecewise-liner convex functions, in a manner similar to max-affine regression, whose difference approximates the data. The choice of the function is regularised by a new seminorm over the class of DC functions that controls the $\ell_\infty$ Lipschitz constant of the estimate. The resulting methodology can be efficiently implemented via Quadratic programming even in high dimensions, and is shown to have close to minimax statistical risk. We empirically validate the method, showing it to be practically implementable, and to have comparable performance to existing regression/classification methods on real-world datasets.
