We present a framework for performing efficient regression in general metric spaces. Roughly speaking, our regressor predicts the value at a new point by computing a Lipschitz extension -- the smoothest function consistent with the observed data -- while performing an optimized structural risk minimization to avoid overfitting. The offline (learning) and online (inference) stages can be solved by convex programming, but this naive approach has runtime complexity O(n^3), which is prohibitive for large datasets. We design instead an algorithm that is fast when the the doubling dimension, which measures the "intrinsic" dimensionality of the metric space, is low. We use the doubling dimension multiple times; first, on the statistical front, to bound fat-shattering dimension of the class of Lipschitz functions (and obtain risk bounds); and second, on the computational front, to quickly compute a hypothesis function and a prediction based on Lipschitz extension. Our resulting regressor is both asymptotically strongly consistent and comes with finite-sample risk bounds, while making minimal structural and noise assumptions.

提出了一种基于Lipschitz extension的回归框架，通过结构风险最小来避免过拟合，使用凸优化求解离线学习及在线预测，同时解决了大数据集的处理问题。

通过近似利普希茨扩展在度量空间中高效完成回归