For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure. Our proofs and examples show that this geometric similarity between dataset and model is essential to trustworthy interpolation and perturbation, and we conjecture that this new concept will add value to the current debate regarding the unclear relationship between overfitting and generalization in applications of deep-learning. In typical DNN applications, an explicit geometric description of the model is impossible, but parallax can estimate topological features (components, cycles, voids, etc.) in the model by examining the effect on the Rips complex of geodesic distortions using the reference dataset. Thus, parallax indicates whether the model shares similar multiscale geometric features with the dataset. Parallax presents theoretically via topological data analysis [TDA] as a bi-filtered persistence module, and the key properties of this module are stable under perturbation of the reference dataset.

翻译：为提升人工智能系统的安全性与鲁棒性，我们引入拓扑视差作为理论与计算工具，用于比较训练模型与参考数据集，判断两者是否具有相似的多尺度几何结构。我们的证明与实例表明，数据集与模型间的几何相似性对可信插值与扰动至关重要，并推测这一新概念将为当前关于深度学习应用中过拟合与泛化之间模糊关系的讨论增添价值。在典型深度神经网络（DNN）应用中，对模型进行显式几何描述是不可能的，但视差可通过检查参考数据集对里普斯复形（Rips complex）的测地线扭曲效应，估计模型中的拓扑特征（如连通分量、环、空洞等）。因此，视差能够指示模型是否与数据集共享相似的多尺度几何特征。从理论层面，视差通过拓扑数据分析（TDA）呈现为双过滤持续模，且该模的关键性质在参考数据集扰动下保持稳定。