A max filtering local stability theorem with application to weighted phase retrieval and cryo-EM

TL;DR

The paper addresses embedding orbit spaces V/G into Euclidean space with bilipschitz fidelity by analyzing max filter banks, a family of convex G-invariant feature maps. It develops a desingularization-based local bilipschitz result on the regular set R(G) and couples it with a Voronoi-cell framework to obtain generic bilipschitz embeddings when the template count n exceeds a complexity threshold involving χ(G) and c. Applications include stable weighted phase retrieval (extending complex/quaternionic phase retrieval to weighted representations) and an improved approach to cryo-EM nearest-neighbor problems via bilipschitz embeddings of coefficient spaces under C^*-action. The work integrates geometric, semialgebraic, and Riemannian tools to characterize principal and regular orbits and to justify the practical relevance of these embeddings in high-dimensional, symmetry-aware data analysis.

Abstract

Given an inner product space $V$ and a group $G$ of linear isometries, max filtering offers a rich class of convex $G$-invariant maps. In this paper, we identify sufficient conditions under which these maps are locally bilipschitz on $R(G)$, the set of orbits with maximal dimension, with respect to the quotient metric on the orbit space $V/G$. Central to our proof is a desingularization theorem, which applies to open, dense neighborhoods around each orbit in $R(G)/G$ and may be of independent interest. As an application, we provide guarantees for stable weighted phase retrieval. That is, we construct componentwise convex bilipschitz embeddings of weighted complex (resp.\ quaternionic) projective spaces. These spaces arise as quotients of direct sums of nontrivial unitary irreducible complex (resp.\ quaternionic) representations of the group of unit complex numbers $S^1\cong \operatorname{SO}(2)$ (resp.\ unit quaternions $S^3\cong \operatorname{SU}(2)$). We also discuss the relevance of such embeddings to a nearest-neighbor problem in single-particle cryogenic electron microscopy (cryo-EM), a leading technique for resolving the spatial structure of biological molecules.

Figures (9)

• Figure 1: Illustration for \ref{['ex.3d voronoi SOReflection']}, showing instances of $x$, $y$, and $z$ along with their orbits and Voronoi cells, all of which are precisely described in the referenced example.
• Figure 2: Illustration for \ref{['prop.daduk proposition']}. (left) Here, $M = \{(t,t^2)\in\mathbb R^2: t\in\mathbb R\}$ and $U = \{(x_1,x_2)\in\mathbb R^2:x_1 > 0\}$. The line $L = \{(0,x_2)\in\mathbb R^2:x_2 > \frac{1}{2}\}$ is the set of points with multiple nearest neighbors to $M$, and its closure is $\overline L = L\cup \{(0,\frac{1}{2})\}$. For each $z\in U \cup L$ and $x \in U\cap \arg\min_{p\in M}\|p-z\|$, the neighborhood $U$ satisfies assertions (b) and (c) in \ref{['prop.daduk proposition']}. For $z_0\in R:= \{(0,x_2)\in\mathbb R^2:x_2 < \frac{1}{2}\}$, it holds that $\arg\min_{p\in M}\|p-z\| = \{(0,0)\}$, and any neighborhood $U$ of $R$ with $U\cap \overline L = \varnothing$ satisfies those assertions with respect to each $z_0\in R$. (right) Here, $M = \{(\cos(\theta),\sin(\theta))\in\mathbb R^2: \theta\in (-\frac{\pi}{3},\frac{\pi}{3})\}$ and $U = \operatorname{int}(\operatorname{cone}(M))$. For each $z\in U\cup \{(0,0)\}$ and $x \in \arg\min_{p\in M}\|p-z\|$, the neighborhood $U$ satisfies assertions (b) and (c) in \ref{['prop.daduk proposition']}. For nonzero $z\in U^c$, the set $\arg\min_{p\in M}\|p-z\|$ is empty.
• Figure 3: Illustration for Case 4 in \ref{['ex.circle UXVX']} when $w_1=1$ and $w_2 = 2$. The graph with the highest $y$-intercept of $3$ corresponds to the function $y = 4\cos(x)-\cos(2x)$. It has a unique global maximum over $[-\pi,\pi]$, attained as a flat local maximum at $x=0$. The other graphs, with $y$-intercepts $k-1$, correspond to the functions $y = k\cos(x)-\cos(2x)$ for $k\in \{1,2,3,3.5\}$. Each attains its global maximum over $[-\pi,\pi]$ at $x=\pm \cos^{-1}\left(k/4\right)\neq 0$. While the values of $k$ here do not exactly correspond to $4-\frac{1}{n}$, the behavior of the global maxima remains the same for that sequence. (We thank Aleksei Kulikov for bringing this example to our attention.)
• Figure 4: This figure is an aiding illustration for \ref{['app.cut point symmetry']}. It is an adaptation of Figure 13.2.1 in DoCarmo:92.
• Figure 5: This figure is an aiding illustration for the proof of \ref{['claim.construction of S']}. For brevity, we define $\log(U) := \exp|_{T_W}^{-1}(U)$. The dashed lines connecting the left and right halves of the figure are solely for visualizing the transition by magnification from $G$ to its Lie algebra.

Theorems & Definitions (97)

• Example 1: Example 1 in CahillIM:24
• Definition 2
• Definition 3
• Theorem 4
• Theorem 5
• Corollary 6
• Remark 7
• Definition 8
• Definition 9
• Theorem 10