Towards a bilipschitz invariant theory

TL;DR

This work develops a framework for embedding orbit spaces $V/G$ arising from group actions on real Hilbert spaces into Hilbert spaces via bilipschitz maps. It introduces the metric quotient $V//G$, analyzes how bilipschitz invariants can be extended from the sphere to the whole space through homogeneous extensions, and proves fundamental limits: bilipschitz invariants cannot be differentiable at points with nontrivial stabilizers, while certain free actions admit bilipschitz polynomial invariants via extension. The paper also derives distortion-minimization results using semidefinite programming, and thoroughly treats quotients by permutations and translations, establishing both positive embedding results (e.g., a 1-dimensional permutation quotient) and no-go results (e.g., higher-dimensional permutation quotients). Overall, the results illuminate when orbit-space data can be processed with Euclidean tools and quantify the best possible distortions for several natural group actions, with implications for invariant feature construction and orbit recovery tasks. $V$, $G$, and quotient constructions are central to the theory, and the developed techniques offer practical guidance for designing bilipschitz invariants in data-analysis pipelines.

Abstract

Consider the quotient of a Hilbert space by a subgroup of its automorphisms. We study whether this orbit space can be embedded into a Hilbert space by a bilipschitz map, and we identify constraints on such embeddings.

Figures (3)

• Figure 1: Illustration of Example \ref{['ex.finite dimensional real phase retrieval']}. Here, we take $V:=\mathbb{R}^2$ and $G:=\{\pm\operatorname{id}\}\leq O(2)$. The horizontal and vertical axes represent input and output distances, respectively. (left) Draw a million pairs of vectors $x,y\in\mathbb{R}^2$ with standard gaussian distribution and plot the output distance $\|xx^\top-yy^\top\|_F$ versus the input distance $d([x],[y])$. One can show that if the input distance is $a>0$, then the output distance can take any value in $[a^2/\sqrt{2},\infty)$. The lower bound is depicted in red. Notably, the map $[x]\mapsto xx^\top$ is neither Lipschitz nor lower Lipschitz. (middle) Draw a million pairs of vectors $x,y\in\mathbb{R}^2$ uniformly from the unit circle and plot the output distance $\|xx^\top-yy^\top\|_F$ versus the input distance $d([x],[y])$. One can show that that if the input distance is $a>0$, then the output distance resides in the interval $[a,\sqrt{2}a]$. These bounds are depicted in red. Notably, the map $[x]\mapsto xx^\top$ is bilipschitz when restricted to $S(\mathbb{R}^2)/G$. (right) Draw a million pairs of vectors $x,y\in\mathbb{R}^2$ with standard gaussian distribution and plot the output distance $\|\frac{1}{\|x\|}xx^\top-\frac{1}{\|y\|}yy^\top\|_F$ versus the input distance $d([x],[y])$. One can show that that if the input distance is $a>0$, then the output distance resides in the interval $[a,\sqrt{2}a]$. These bounds are depicted in red. Notably, the map $[x]\mapsto \frac{1}{\|x\|}xx^\top$ is bilipschitz. More generally, Theorem \ref{['thm.bilipschitz homogeneous extension']} gives that the homogeneous extension of a bilipschitz map is bilipschitz.
• Figure 2: Illustration of homogeneous extension. See Definition \ref{['def.homogeneous extension']}.
• Figure 3: Illustration of Example \ref{['ex.max filtering is not differentiable']}. Max filtering invariants are piecewise linear. In this example, the orbits of the templates $z_1,z_2,z_3\in\mathbb{R}^2$ determine points of non-differentiability (namely, the boundaries of their Voronoi cells) in the resulting max filtering invariant. On the left, the group $G\leq O(2)$ consists of rotations by multiples of $2\pi/3$, while on the right, $G$ is the dihedral group of order $6$. Considering Theorem \ref{['thm.bilipschitz requires nondifferentiability']}(b), the non-differentiability of max filtering is an artifact of its bilipschitzness.

Theorems & Definitions (72)

• Example 1: Nearest neighbor search
• Example 2: Clustering
• Example 3: Visualization
• Example 4
• Example 5
• Example 6
• Lemma 7
• proof : Proof of Lemma \ref{['lem.metric quotient is categorical']}
• Lemma 8
• proof : Proof of Lemma \ref{['lem.metric quotient orbit closure']}