流形间拓扑富映射的深可逆逼近
Deep Invertible Approximation of Topologically Rich Maps between Manifolds
摘要：
我们怎样才能设计出神经网络，使其能够对拓扑上有趣的流形之间的映射进行稳定的通用逼近？答案是坐标投影。基于拓扑数据分析（TDA）的神经网络使用诸如持久同源性之类的工具来学习数据的拓扑签名并稳定训练，但可能不是通用近似器或具有稳定逆。其他架构普遍近似子流形上的数据分布，但仅当后者由单个图表给出时，使它们无法学习改变拓扑的地图。通过利用局部bilipschitz映射，覆盖空间和局部同胚之间的拓扑相似性，并通过使用机器学习中的通用近似参数，我们发现一种形式为$\mathcal{T} \circ p \circ \mathcal{E}$的新型网络，其中$\mathcal{E}$是内射网络，$p$是固定坐标投影，$\mathcal{T}$是双射网络，是嵌入在$\mathbb{R}^n$中的紧致光滑子流形之间的局部自同态的泛逼近器。我们强调的情况下，目标地图改变拓扑结构。此外，我们发现，通过约束投影$p$，我们的网络的多值反演可以计算而不牺牲普遍性。作为一个应用程序，我们表明，学习一个群不变函数与未知的组行动自然减少到学习局部有限群的同构的问题。我们的理论使我们能够恢复轨道的群体行动。我们还概述了我们的架构，以解决分子成像的对称性的可能的扩展。最后，我们的分析通知生成问题的拓扑表达的起始空间的选择。
英文摘要：
How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use tools such as persistent homology to learn topological signatures of data and stabilize training but may not be universal approximators or have stable inverses. Other architectures universally approximate data distributions on submanifolds but only when the latter are given by a single chart, making them unable to learn maps that change topology. By exploiting the topological parallels between locally bilipschitz maps, covering spaces, and local homeomorphisms, and by using universal approximation arguments from machine learning, we find that a novel network of the form $\mathcal{T} \circ p \circ \mathcal{E}$, where $\mathcal{E}$ is an injective network, $p$ a fixed coordinate projection, and $\mathcal{T}$ a bijective network, is a universal approximator of local diffeomorphisms between compact smooth submanifolds embedded in $\mathbb{R}^n$. We emphasize the case when the target map changes topology. Further, we find that by constraining the projection $p$, multivalued inversions of our networks can be computed without sacrificing universality. As an application, we show that learning a group invariant function with unknown group action naturally reduces to the question of learning local diffeomorphisms for finite groups. Our theory permits us to recover orbits of the group action. We also outline possible extensions of our architecture to address molecular imaging of molecules with symmetries. Finally, our analysis informs the choice of topologically expressive starting spaces in generative problems.
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