Solving optimization tasks based on functions and losses with a topological flavor is a very active, growing field of research in data science and Topological Data Analysis, with applications in non-convex optimization, statistics and machine learning. However, the approaches proposed in the literature are usually anchored to a specific application and/or topological construction, and do not come with theoretical guarantees. To address this issue, we study the differentiability of a general map associated with the most common topological construction, that is, the persistence map. Building on real analytic geometry arguments, we propose a general framework that allows us to define and compute gradients for persistence-based functions in a very simple way. We also provide a simple, explicit and sufficient condition for convergence of stochastic subgradient methods for such functions. This result encompasses all the constructions and applications of topological optimization in the literature. Finally, we provide associated code, that is easy to handle and to mix with other non-topological methods and constraints, as well as some experiments showcasing the versatility of our approach.
翻译:以地形学的口味解决基于功能和损失的优化任务是一个非常活跃的、日益扩大的数据科学和地形数据分析研究领域,其应用不精细的优化、统计和机器学习。然而,文献中建议的方法通常以具体的应用和/或地形学结构为基础,没有理论保障。为解决这一问题,我们研究了与最常见的地形构造(即持久性地图)相关的一般地图的可差异性。在真实的分析几何学参数的基础上,我们提出了一个总框架,使我们能够以非常简单的方式界定和计算持久性基于功能的梯度。我们还为此类功能的随机分层方法的趋同提供了简单、明确和充分的条件。这一结果涵盖了文献中所有地形优化的构造和应用。最后,我们提供了相关的代码,易于处理和与其他非地形学方法和制约相结合,以及一些实验展示了我们方法的多功能。