Convex regression is the problem of fitting a convex function to a data set consisting of input-output pairs. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data, i.e. a function that is the maximum eigenvalue of an affine matrix expression of the input. This method represents a significant generalization of polyhedral (also called max-affine) regression, in which a polyhedral function (a maximum of a fixed number of affine functions) is fit to the data. We prove bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. We also analyze an alternating minimization algorithm for the non-convex optimization problem of fitting the best spectrahedral function to a given data set. We show that this algorithm converges geometrically with high probability to a small ball around the optimal parameter given a good initialization. Finally, we demonstrate the utility of our approach with experiments on synthetic data sets as well as real data arising in applications such as economics and engineering design.
翻译: convex 回归是将一个 convex 函数与由输入- 输出对配组成的数据集相配的问题。 我们对此问题提出了一个叫做光谱回归的新方法, 我们在这个方法中将光谱回归功能与数据相配, 即一个函数是输入方形矩阵表达的最大偏移值。 这个方法代表了多元回归( 也称为最大偏移) 的显著概括性, 即多元函数( 固定数的直角函数的最大数目) 适合数据。 我们证明光谱函数通过统计风险分析可以大致接近任意的convex 函数的界限。 我们还分析了一种交替最小化算法, 用于将最佳光谱仪功能与给定数据集相匹配的非convelx优化问题。 我们显示, 在良好的初始化情况下, 该算法在地理学上与一个小球的高度概率相交汇。 最后, 我们展示了我们对合成数据集的实验方法以及经济学和工程设计等应用中产生的真实数据的实用性。