Integrating invariance into data representations is a principled design in intelligent systems and web applications. Representations play a fundamental role, where systems and applications are both built on meaningful representations of digital inputs (rather than the raw data). In fact, the proper design/learning of such representations relies on priors w.r.t. the task of interest. Here, the concept of symmetry from the Erlangen Program may be the most fruitful prior -- informally, a symmetry of a system is a transformation that leaves a certain property of the system invariant. Symmetry priors are ubiquitous, e.g., translation as a symmetry of the object classification, where object category is invariant under translation. The quest for invariance is as old as pattern recognition and data mining itself. Invariant design has been the cornerstone of various representations in the era before deep learning, such as the SIFT. As we enter the early era of deep learning, the invariance principle is largely ignored and replaced by a data-driven paradigm, such as the CNN. However, this neglect did not last long before they encountered bottlenecks regarding robustness, interpretability, efficiency, and so on. The invariance principle has returned in the era of rethinking deep learning, forming a new field known as Geometric Deep Learning (GDL). In this tutorial, we will give a historical perspective of the invariance in data representations. More importantly, we will identify those research dilemmas, promising works, future directions, and web applications.

翻译：将不变性融入数据表示是智能系统和网络应用中的一项原则性设计。表示在其中发挥着基础性作用——系统和应用都构建在数字输入的有意义表示之上（而非原始数据）。事实上，此类表示的合理设计/学习依赖于针对特定任务的先验知识。在此，埃尔朗根纲领中的对称性概念可能是最具启发性的先验——非严格而言，系统的对称性是指保持系统特定属性不变的变换。对称性先验无处不在，例如平移作为物体分类的对称性，即物体类别在平移变换下保持不变。对不变性的探索与模式识别和数据挖掘本身同样古老。在深度学习时代之前，不变性设计已成为各种表示的基石，例如SIFT特征。当我们进入深度学习早期阶段时，不变性原理在很大程度上被忽视，并被数据驱动范式（如CNN）所取代。然而这种忽视并未持续太久，研究者很快在鲁棒性、可解释性、效率等方面遇到瓶颈。在重新审视深度学习的时代，不变性原理已强势回归，并形成了被称为几何深度学习的新领域。本教程将从历史视角梳理数据表示中的不变性研究脉络，更重要的是，我们将系统剖析该领域的研究困境、潜力成果、未来方向及网络应用前景。