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 small-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 toy 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：连通决策边界在图像分割、聚类、α-形状定义或n维空间区域划分等任务中具有重要价值。然而，机器学习领域目前缺乏利用神经网络生成连通决策边界的方法。通过对可逆函数（凸函数的一种推广形式）进行阈值处理，可以生成此类连通决策边界。本文提出两种利用神经网络构建可逆函数的方法：第一种方法基于梯度裁剪-梯度惩罚（GCGP）的神经网络约束机制，通过对梯度进行裁剪和惩罚来实现；第二种方法则基于可逆函数与可逆凸函数复合的数学关系。我们将连通性作为基础解释方法，构建基于连通区域的分类器。理论证明表明，多个基于连通集的分类器能够逼近任意分类函数。在实验部分，我们采用两种方案进行分类任务验证：一是基于1-vs-all模型集成策略，二是直接在小型数据集上构建单一多分类模型。实验结果表明，基于连通集的分类器不仅未弱化传统神经网络分类器的性能，反而显著提升了模型的可解释性。此外，我们通过玩具数据集和真实数据集的系统性实验，深入探究了可逆函数与连通集在可解释性和网络形态学方面的特性。研究表明，可逆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意义，这为各类下游任务提供了重要理论支撑。