Injectivity, stability, and positive definiteness of max filtering

TL;DR

This work analyzes max filtering as a tool for embedding orbit spaces $V/G$ into Euclidean space in a way that preserves the quotient metric up to a controllable distortion. It proves that 2d generic templates suffice for injectivity for any $G$ with closed orbits, and derives precise upper Lipschitz bounds via Voronoi decompositions, with tight results in the finite and antipodal cases. It introduces a sharp lower Lipschitz bound $\alpha(\{z_i\},G)$ and a practically useful Voronoi characteristic $\chi(G)$ to establish bilipschitz guarantees for generic templates and to bound distortion for random Gaussian templates, showing substantial improvements over prior bounds. Moreover, it characterizes when max filtering is a positive definite kernel, tying this property to polarity and finite Weyl groups, thereby linking invariant embeddings to geometric group actions. Together, these results clarify when max filtering provides stable, injective, and kernel-amenable embeddings suitable for transferring Euclidean ML methods to orbit spaces.

Abstract

Given a real inner product space V and a group G of linear isometries, max filtering offers a rich class of G-invariant maps. In this paper, we identify nearly sharp conditions under which these maps injectively embed the orbit space V/G into Euclidean space, and when G is finite, we estimate the map's distortion of the quotient metric. We also characterize when max filtering is a positive definite kernel.

Figures (5)

• Figure 1: (left) Illustration of Example \ref{['ex.voronoi 2d rot']} when $n=5$. The resulting Voronoi diagram $Q_x$ is sensitive to the argument of $x$. Note that $Q_x$ is the complement of the dashed lines. (right) Illustration of Example \ref{['ex.voronoi reflection']} when $G$ is generated by reflections across the dashed lines. The resulting Voronoi diagram $Q_x$ does not change when $x$ is perturbed to another point in $V_x$.
• Figure 2: Illustration for the proof of Theorem \ref{['thm.upper lipschitz bound finite']}. Here, $G\leq \operatorname O(2)$ consists of rotations by multiples of $\frac{2\pi}{5}$ radians. Points $x$ and $y$ are shown in green and blue colors respectively. Other points in their orbits are shown in the corresponding transparent color. Note that $d([x],[y])=\|x-y\|$. The templates $z_1,\dots, z_4$ are shown in transparent orange color along with their orbits. The union of the boundaries of their Voronoi diagrams (dashed orange lines) form the complement of $\cap_{i=1}^4 Q_{z_i}$. The line segment $[x,y]$ transverses through the connected components of $\cap_{i=1}^4 Q_{z_i}$, with $c_2$ and $c_3$ marking transition points.
• Figure 3: The group $G\leq \operatorname{O}(2)$ consists of rotations by multiples of $\frac{2\pi}{5}$ radians as in Example \ref{['ex.rotation Sxy 2d']} with $m=5$. (left) Illustration of Corollary \ref{['cor.choice function']}(a). The point $v_i(x)$ is the closest member of $z_i$'s (orange) orbit to $x$. This is the unique closest point since $x\in V_{v_i(x)}\subseteq Q_{z_i}$. (right) Illustration of Definition \ref{['def.S(x,y)']} and Corollary \ref{['cor.choice function']}. The orange dots in $V_x$ are the images $v_i(x)\in [z_i]$. Here, $S(x,y) = \{w_1,w_2\}$ are the elements of $[y]$ that have nontrivial open Voronoi overlap with $V_x$, i.e., $V_x\cap V_{w_j}\neq \varnothing$. For $f\in \mathcal{F}(x,y)$, one may choose $f(i) = w_j$ as long as $v_i(x)\in \overline {V_{w_j}}$.
• Figure 4: The group consists of rotations by multiples of $\frac{2\pi}{5}$ radians around the $z$-axis as in Example \ref{['ex.rotation Sxy 3d']} with $m=5$. Both pictures serve as an illustration for the discussion that follows Example \ref{['ex.rotation Sxy 3d']}. (left) The templates $z_1,z_2,z_3$ and the point $x$ lie on the upper half of the $z$-axis $W$ while $y\in W^c$. The Voronoi cell $V_x$ is given by the open upper half-space. While $S([x],[y]) = [y]$, there exists $f\in \mathcal{F}(x,y)$ such that $f(i)=y$ for each $i\in\{1,2,3\}$. (right) In this picture, $x,y\in W^c$ and $f(1) = w_2$ while $f(2)=f(3)=w_1$ so that $|\operatorname{im}(f)|=|S(x,y)|=2$, where $f$ is the singleton element of $\mathcal{F}(x,y)$.
• Figure 5: Illustration for the optimal partition in the proof of Lemma \ref{['lem.lower lipschitz optimal cases']}(a). Here, $G\leq \operatorname O(2)$ consists of rotations by multiples of $\frac{\pi}{2}$ radians and the figure zooms in on the Voronoi cell $V_x$ given by the open first quadrant of the plane. The orange points in the portion of $V_x$ below (resp. above) the diagonal comprise $\{v_i(x)\}_{i\in A}$ (resp. $\{v_i(x)\}_{i\in B}$). The vectors $z_A$ and $z_B$ are top eigenvectors corresponding to $\sum_{i\in A} v_i(x)v_i(x)^\top$ and $\sum_{i\in B} v_i(x)v_i(x)^\top$ respectively. Moreover, $x = z_A + z_B$ and $y = tRx$, where $R$ is the counterclockwise rotation by $\frac{\pi}{4}$ which forces $d([x],[y]) = \|x-y\|=\|x-R^{-2}y\|$, and $t>0$ is chosen in such a way that $x-R^{-2}y$ is orthogonal to $z_A$ and $x-y$ is orthogonal $z_B$; hence, each is parallel to the bottom eigenvectors of $\sum_{i\in A} v_i(x)v_i(x)^\top$ and $\sum_{i\in B} v_i(x)v_i(x)^\top$ respectively.

Theorems & Definitions (55)

• Definition 1
• Example 2: Example 1 in CahillIM:24
• Definition 3
• Example 4: Section 3.2.2 in CahillIMP:22
• Theorem 5
• Proposition 6: Lemma 10 in CahillIMP:22
• Definition 7: cf. DymG:22CahillIMP:22
• Proposition 8: Theorem 12 in CahillIMP:22
• Proposition 9
• proof