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.

翻译：双李普希兹不变性理论研究轨道空间到欧几里得空间低失真嵌入问题。迄今为止，仅对少数情形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，对于这三种情形，最小可能失真均可通过"最大滤波器组"与线性变换的复合近似实现。我们的证明分两步：首先，证明只需验证李普希兹函数空间的某种包含关系；其次，对三种情形采用根本不同的方法证明该包含关系。我们还表明，这些情形与若干相关函数空间的交互方式各异，这暗示统一处理将具有非平凡性。