Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing techniques back-propagate gradients through the persistence diagram, which is a summary of the topological features of a function. Their downside is that they provide information only at the critical points of the function. We propose a method that instead builds on persistence-sensitive simplification and translates the required changes to the persistence diagram into changes on large subsets of the domain, including both critical and regular points. This approach enables a faster and more precise topological regularization, the benefits of which we illustrate with experimental evidence.

中文翻译：

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通过持久性敏感优化进行拓扑正则化

优化是机器学习和统计中的关键工具，依赖正则化来减少过度拟合。传统的正则化方法控制解的范数以保证其平滑。最近，拓扑方法已经出现，作为一种对解决方案提供更精确和更具表现力的控制的方法，依靠持久的同源性来量化和降低其粗糙度。所有这些现有技术都通过持久图反向传播梯度，持久图是函数拓扑特征的总结。它们的缺点是仅在功能的关键点提供信息。我们提出了一种方法，该方法建立在持久性敏感的简化基础上，将对持久性图所需的更改转换为对域的大型子集（包括关键点和常规点）的更改。这种方法可以实现更快、更精确的拓扑正则化，我们用实验证据说明其好处。