Mathematical Foundations of Intelligence: An "Erlangen Programme" for AI

在1872年，菲利克斯克莱因出版了他现在著名的埃尔兰根计划，他在其中处理几何作为研究的不变量，正式使用群论。这种全新的方法可以将世纪出现的不同类型的非欧几里德几何联系在一起，并对几何和数学产生了深刻的方法论和文化影响。新领域的数学，如外部演算，代数拓扑，理论的纤维束和层，和范畴理论出现作为一个延续克莱因的蓝图。埃尔兰根计划也是20世纪上半叶物理学发展的基础，诺特定理和规范不变性的概念成功地为电磁、弱和强相互作用提供了统一框架，并在20世纪70年代的标准模型中达到顶峰。现在是人工智能“埃尔兰根计划”的时候了，基于严格的数学原理，可以更好地理解现有的人工智能方法以及新一代的方法，这些方法保证了表达和泛化能力，更好的可解释性，可扩展性以及数据和计算效率。正如克莱因的埃尔兰根计划的思想渗透到其他学科，并在数学，物理学等领域产生了新的理论一样，我们将在我们的人工智能研究计划中从这些类比中汲取灵感。通过诉诸强大的工具，从数学和算法领域有时被认为是“异国情调”的应用领域，新的理论见解和计算模型可以得出。 我们的“埃尔兰根人工智能计划”将研究现代人工智能/机器学习系统的四个基本问题，努力提供严格的数学答案。如何发现数据中隐藏的结构，并用几何和拓扑语言表达，以便ML模型利用？我们是否可以使用几何和拓扑工具来分析ML模型，以了解它们何时以及如何工作和失败？我们如何保证学习从这些结构中受益，并利用这些见解开发更好、更高效、更安全的新模型？最后，我们如何在未来的人工智能系统中使用这些模型来做出可能影响数十亿人的决策？该中心位于牛津大学，地理覆盖范围广泛，将汇集数学，算法和计算领域的领先专家，支持AI/ML系统及其在科学和工业环境中的应用。一些中心参与者有以前成功合作的记录，而其他合作是新的。拟议中心的研究计划旨在打破不同领域之间的障碍，并将一个多样化和地理分布的英国领先专家群体聚集在一起，以实现强大的交叉受精。在AI/ML领域，我们的工作将有助于从目前尚未开发的数学领域开发工具。相反，我们的计划将有助于吸引理论家对新问题和应用的注意。

In 1872, Felix Klein published his now famous Erlangen Programme, in which he treated geometry as the study of invariants, formalised using group theory. This radically new approach allowed tying together different types of non-Euclidean geometries that had emerged in the nineteenth century and has had a profound methodological and cultural impact on geometry in particular and mathematics in general. New fields of mathematics such as exterior calculus, algebraic topology, the theory of fibre bundles and sheaves, and category theory emerged as a continuation of Klein's blueprint. The Erlangen Programme was also fundamental for the development of physics in the first half of the twentieth century, with Noether's theorem and the notion of gauge invariance successfully providing a unification framework for electromagnetic, weak, and strong interactions, culminating in the Standard Model in the 1970s.Now is the time for an "Erlangen Programme" for AI, based on rigorous mathematical principles that would bring better understanding of existing AI methods as well as a new generation of methods that have guaranteed expressive and generalisation power, better interpretability, scalability, and data- and computational-efficiency. Just as the ideas of Klein's Erlangen Programme spilled into other disciplines and produced new theories in mathematics, physics, and beyond, we will draw inspiration from these analogies in our AI research programme. By resorting to powerful tools from the mathematical and algorithmic fields sometimes considered "exotic" in applied domains, new theoretical insights and computational models can be derived. Our "Erlangen Programme of AI" will study four fundamental questions that underlie modern AI/ML systems, striving to provide rigorous mathematical answers. How can hidden structures in data be discovered and expressed in the language of geometry and topology in order to be exploited by ML models? Can we use geometric and topological tools to characterise ML models in order to understand when and how they work and fail? How can we guarantee learning to benefit from these structures, and use these insights to develop better, more efficient, and safer new models? Finally, how can we use such models in future AI systems that make decisions potentially affecting billions of people? With a centre at Oxford, and broad geographic coverage of the UK, the Hub will bring together leading experts in mathematical, algorithmic, and computational fields underpinning AI/ML systems as well as their applications in scientific and industrial settings. Some of the Hub participants have a track record of previous successful work together, while other collaborations are new. The research programme in the proposed Hub is intended to break barriers between different fields and bring a diverse and geographically-distributed cohort of leading UK experts rarely seen together with the purpose of strong cross-fertilisation. In the fields of AI/ML, our work will contribute to the exploitation of tools from currently underexplored mathematical fields. Conversely, our programme will help attract the attention of theoreticians to new problems and applications.