Subspace clustering is an important unsupervised clustering approach. It is based on the assumption that the high-dimensional data points are approximately distributed around several low-dimensional linear subspaces. The majority of the prominent subspace clustering algorithms rely on the representation of the data points as linear combinations of other data points, which is known as a self-expressive representation. To overcome the restrictive linearity assumption, numerous nonlinear approaches were proposed to extend successful subspace clustering approaches to data on a union of nonlinear manifolds. In this comparative study, we provide a comprehensive overview of nonlinear subspace clustering approaches proposed in the last decade. We introduce a new taxonomy to classify the state-of-the-art approaches into three categories, namely locality preserving, kernel based, and neural network based. The major representative algorithms within each category are extensively compared on carefully designed synthetic and real-world data sets. The detailed analysis of these approaches unfolds potential research directions and unsolved challenges in this field.

子空间聚类是一种重要的无监督聚类方法。它基于高维数据点近似分布在几个低维线性子空间周围的假设。大多数突出的子空间聚类算法依赖于将数据点表示为其他数据点的线性组合，这被称为自我表达表示。为了克服限制性线性假设，提出了许多非线性方法来将成功的子空间聚类方法扩展到非线性流形联合上的数据。在这项比较研究中，我们全面概述了过去十年中提出的非线性子空间聚类方法。我们引入了一种新的分类法，将最先进的方法分为三类，即基于局部性、基于核和基于神经网络。每个类别中的主要代表性算法都在精心设计的合成和现实世界数据集上进行了广泛的比较。对这些方法的详细分析揭示了该领域的潜在研究方向和未解决的挑战。