We consider estimating a compact set from finite data by approximating the support function of that set via sublinear regression. Support functions uniquely characterize a compact set up to closure of convexification, and are sublinear (convex as well as positive homogeneous of degree one). Conversely, any sublinear function is the support function of a compact set. We leverage this property to transcribe the task of learning a compact set to that of learning its support function. We propose two algorithms to perform the sublinear regression, one via convex and another via nonconvex programming. The convex programming approach involves solving a quadratic program (QP). The nonconvex programming approach involves training a input sublinear neural network. We illustrate the proposed methods via numerical examples on learning the reach sets of controlled dynamics subject to set-valued input uncertainties from trajectory data.

翻译：我们考虑通过次线性回归近似有限数据的紧集支撑函数，从而估计该紧集。支撑函数唯一刻画了紧集（直至凸闭包），且具有次线性性质（既是凸函数，也是一阶正齐次函数）。反之，任何次线性函数均为某个紧集的支撑函数。我们利用这一性质将学习紧集的任务转化为学习其支撑函数。我们提出了两种次线性回归算法，分别基于凸规划和非凸规划。凸规划方法涉及求解二次规划问题，而非凸规划方法则涉及训练输入次线性神经网络。我们通过数值示例展示了所提方法在从轨迹数据中学习受集值输入不确定性影响的受控动力学可达集的应用。