论文标题
单调多指数模型的估计
Estimation of Monotone Multi-Index Models
论文作者
论文摘要
在具有$ K $索引向量的多指数模型中，输入变量通过使用索引向量的内部产品来转换。传输函数$ f：\ mathbb {r}^k \ to \ mathbb {r} $应用于这些内部产品以生成输出。因此，多指数模型是线性模型的概括。在本文中，我们考虑单调多指数模型。即，假定传递函数是坐标单调的。因此，单调多指数模型概括了线性回归和等渗回归，这是对坐标单调函数的估计。我们考虑非负索引向量的情况。我们提供了一种基于整数编程的算法，以估算单调多指数模型，并提供有关估计功能相对于地面真相的$ L_2 $损失的保证。
In a multi-index model with $k$ index vectors, the input variables are transformed by taking inner products with the index vectors. A transfer function $f: \mathbb{R}^k \to \mathbb{R}$ is applied to these inner products to generate the output. Thus, multi-index models are a generalization of linear models. In this paper, we consider monotone multi-index models. Namely, the transfer function is assumed to be coordinate-wise monotone. The monotone multi-index model therefore generalizes both linear regression and isotonic regression, which is the estimation of a coordinate-wise monotone function. We consider the case of nonnegative index vectors. We provide an algorithm based on integer programming for the estimation of monotone multi-index models, and provide guarantees on the $L_2$ loss of the estimated function relative to the ground truth.
