Connected decision boundaries are useful in several tasks like image segmentation, clustering, alpha-shape or defining a region in nD-space. However, the machine learning literature lacks methods for generating connected decision boundaries using neural networks. Thresholding an invex function, a generalization of a convex function, generates such decision boundaries. This paper presents two methods for constructing invex functions using neural networks. The first approach is based on constraining a neural network with Gradient Clipped-Gradient Penality (GCGP), where we clip and penalise the gradients. In contrast, the second one is based on the relationship of the invex function to the composition of invertible and convex functions. We employ connectedness as a basic interpretation method and create connected region-based classifiers. We show that multiple connected set based classifiers can approximate any classification function. In the experiments section, we use our methods for classification tasks using an ensemble of 1-vs-all models as well as using a single multiclass model on larger-scale datasets. The experiments show that connected set-based classifiers do not pose any disadvantage over ordinary neural network classifiers, but rather, enhance their interpretability. We also did an extensive study on the properties of invex function and connected sets for interpretability and network morphism with experiments on simulated and real-world data sets. Our study suggests that invex function is fundamental to understanding and applying locality and connectedness of input space which is useful for various downstream tasks.

翻译：连接决策边界在图像分割、聚类、阿尔法形状或定义n维空间区域等任务中非常有用。然而，机器学习领域缺乏使用神经网络生成连接决策边界的方法。对invex函数（凸函数的推广）设置阈值可生成此类决策边界。本文提出了两种使用神经网络构建invex函数的方法。第一种方法基于梯度裁剪-梯度惩罚（GCGP）约束神经网络，其中我们对梯度进行裁剪和惩罚。第二种方法则基于invex函数与可逆凸函数复合之间的关系。我们采用连通性作为基本解释方法，并创建基于连通区域的分类器。研究表明，多个基于连通集的分类器可以逼近任意分类函数。在实验部分，我们使用一对一模型集成以及单一多类模型在大规模数据集上执行分类任务。实验表明，基于连通集的分类器相对于普通神经网络分类器并无劣势，反而增强了可解释性。我们还通过模拟和真实数据集的实验，对invex函数和连通集在可解释性和网络同态方面的性质进行了广泛研究。研究表明，invex函数对于理解和应用输入空间的局部性与连通性具有基础性作用，可广泛应用于各类下游任务。