双射不变量理论中的逼近定理
Approximation theorems in bilipschitz invariant theory
AI总结：
本文研究了双射不变量理论中低失真嵌入问题，证明了三种情况下的最小失真可通过最大滤波器组与线性变换的组合近似实现。
摘要：
双射不变量理论关注将轨道空间低失真嵌入欧几里得空间。截至目前，只有少数情况已知具有最小可能失真的嵌入，包括：(a)平面旋转，(b)实相位恢复，以及(c)有限反射群。本文证明，对于这三种情况，最小可能的失真几乎由最大滤波器组与线性变换的组合实现。我们的证明是一个两步过程：首先，我们证明只需展示某种Lipschitz函数空间的包含关系；其次，我们通过根本不同的方法证明该包含关系，适用于这三种情况。我们还表明，这些情况与某些相关函数空间的相互作用不同，这表明统一处理将具有挑战性。
英文摘要：
Bilipschitz invariant theory concerns low-distortion embeddings of orbit spaces into Euclidean space. To date, embeddings with the smallest-possible distortion are known for only a few cases, to include: (a) planar rotations, (b) real phase retrieval, and (c) finite reflection groups. Here, we prove that for all three of these cases, the smallest possible distortion is nearly achieved by a composition of a "max filter bank" with a linear transformation. Our proof amounts to a two-step process: first, we show it suffices to demonstrate a certain inclusion of Lipschitz function spaces, and second, we prove that inclusion, using fundamentally different approaches for the three cases. We also show that these cases interact differently with a few related function spaces, which suggests that a unified treatment would be nontrivial.
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