We consider binary classification restricted to a class of continuous piecewise linear functions whose decision boundaries are (possibly nonconvex) starshaped polyhedral sets, supported on a fixed polyhedral simplicial fan. We investigate the expressivity of these function classes and describe the combinatorial and geometric structure of the loss landscape, most prominently the sublevel sets, for two loss-functions: the 0/1-loss (discrete loss) and a log-likelihood loss function. In particular, we give explicit bounds on the VC dimension of this model, and concretely describe the sublevel sets of the discrete loss as chambers in a hyperplane arrangement. For the log-likelihood loss, we give sufficient conditions for the optimum to be unique, and describe the geometry of the optimum when varying the rate parameter of the underlying exponential probability distribution.

翻译：本文研究限定于一类连续分段线性函数的二分类问题，其决策边界为（可能非凸的）星形多面体集，并支撑于固定的多面体单纯形扇上。我们探究了这些函数类的表达能力，并针对两种损失函数——0/1损失（离散损失）和对数似然损失函数，描述了损失景观的组合与几何结构，特别是其子水平集。具体而言，我们给出了该模型VC维度的显式界，并将离散损失的子水平集具体描述为超平面排列中的腔室。对于对数似然损失，我们给出了最优解唯一的充分条件，并描述了当基础指数概率分布的速率参数变化时最优解的几何特性。