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Analysis and Computation of Geodesic Distances on Reductive Homogeneous Spaces

Remco Duits, Gijs Bellaard, Barbara Tumpach

TL;DR

The paper analyzes left-invariant metrics on the reductive homogeneous space $SE(3)/SO(2)$ and the problem of selecting a section from $G/H$ to lift computations to $G$ in geometric ML frameworks like PDE-G-CNNs and G-CNNs. It compares the minimal-distance fiber section $\sigma_d$ with computationally efficient sections $\sigma$ and $\sigma_\rho$, proving that $\sigma$ does not in general yield distance-minimizing fibers, though differences are small under logarithmic-norm approximations. A key result is that geodesics realizing fiber-minimization are horizontal with constant momentum and zero acceleration along the fiber, and these insights extend to reductive spaces with legal metrics. The work also shows that the smooth section $\sigma$ is locally close to $\sigma_d$ and $\sigma_\rho$, coinciding on $SO(3)/SO(2)\cong S^2$ in many cases, offering a computationally cheap yet faithful alternative for roto-translation invariant image analysis and related geometric applications on $SE(3)$.

Abstract

Many geometric machine learning and image analysis applications, require a left-invariant metric on the 5D homogeneous space of 3D positions and orientations SE(3)/SO(2). This is done in Equivariant Neural Networks (G-CNNs), or in PDE-Based Group Convolutional Neural Networks (PDE-G-CNNs), where the Riemannian metric enters in multilayer perceptrons, message passing, and max-pooling over Riemannian balls. In PDE-G-CNNs it is proposed to take the minimum left-invariant Riemannian distance over the fiber in SE(3)/SO(2), whereas in G-CNNs and in many geometric image processing methods an efficient SO(2)-conjugation invariant section is advocated. The conjecture rises whether that computationally much more efficient section indeed always selects distance minimizers over the fibers. We show that this conjecture does NOT hold in general, and in the logarithmic norm approximation setting used in practice we analyze the small (and sometimes vanishing) differences. We first prove that the minimal distance section is reached by minimal horizontal geodesics with constant momentum and zero acceleration along the fibers, and we generalize this result to (reductive) homogeneous spaces with legal metrics and commutative structure groups.

Analysis and Computation of Geodesic Distances on Reductive Homogeneous Spaces

TL;DR

The paper analyzes left-invariant metrics on the reductive homogeneous space and the problem of selecting a section from to lift computations to in geometric ML frameworks like PDE-G-CNNs and G-CNNs. It compares the minimal-distance fiber section with computationally efficient sections and , proving that does not in general yield distance-minimizing fibers, though differences are small under logarithmic-norm approximations. A key result is that geodesics realizing fiber-minimization are horizontal with constant momentum and zero acceleration along the fiber, and these insights extend to reductive spaces with legal metrics. The work also shows that the smooth section is locally close to and , coinciding on in many cases, offering a computationally cheap yet faithful alternative for roto-translation invariant image analysis and related geometric applications on .

Abstract

Many geometric machine learning and image analysis applications, require a left-invariant metric on the 5D homogeneous space of 3D positions and orientations SE(3)/SO(2). This is done in Equivariant Neural Networks (G-CNNs), or in PDE-Based Group Convolutional Neural Networks (PDE-G-CNNs), where the Riemannian metric enters in multilayer perceptrons, message passing, and max-pooling over Riemannian balls. In PDE-G-CNNs it is proposed to take the minimum left-invariant Riemannian distance over the fiber in SE(3)/SO(2), whereas in G-CNNs and in many geometric image processing methods an efficient SO(2)-conjugation invariant section is advocated. The conjecture rises whether that computationally much more efficient section indeed always selects distance minimizers over the fibers. We show that this conjecture does NOT hold in general, and in the logarithmic norm approximation setting used in practice we analyze the small (and sometimes vanishing) differences. We first prove that the minimal distance section is reached by minimal horizontal geodesics with constant momentum and zero acceleration along the fibers, and we generalize this result to (reductive) homogeneous spaces with legal metrics and commutative structure groups.

Paper Structure

This paper contains 9 sections, 7 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction and Background
  2. Research context
  3. Notation
  4. Sections $\sigma$, $\sigma_d$ and $\sigma_\rho$ of $\pi: \mathop{\mathrm{SE}}\nolimits(3) \rightarrow \mathop{\mathrm{SE}}\nolimits(3)/\mathop{\mathrm{SO}}\nolimits(2)$
  5. Minimal Distance Elements in the Fibers $[g]$
  6. Section $\sigma([g])$ is locally close to $\sigma_{\rho}([g])$ and $\sigma_d([g])$
  7. Conclusion
  8. Acknowledgments.
  9. Disclosure of Interests.

Key Result

theorem thmcountertheorem

Set $\ul{a}=(0,0,1)$, and $G=\mathop{\mathrm{SE}}\nolimits(3)$, $H=\textrm{Stab}_{\mathop{\mathrm{SE}}\nolimits(3)}(\ul{0},\ul{a})\equiv \mathop{\mathrm{SO}}\nolimits(2)$. Let $\mathcal{G}$ be a legal metric on $G$ w.r.t. $H$, with either $g_{11}$ positive (Riemannian case) or infinite (Sub-Riemanni

Figures (2)

  • Figure 1: Left to right: 1) start-frame $e$ and (far) end-frame $p$, 2) the exact SR geodesic DuitsJDCS being the minimizing curve in (\ref{['statement']}) with $g_{11}=g_{44}=1, g_{22}=g_{33}=\infty$ connecting $e$ and $p$, 3) the minimal horizontal exponential curve (ending with same orientation in blue). Above we visualize points $g=(\mathbf{x},R)$ in $\mathop{\mathrm{SE}}\nolimits(3)$ with a local frame in $T_{\ul{x}}(\mathbb{R}^3)$.
  • Figure 2: Illustration of Thm.\ref{['thm:sections']} and symmetry of Lemma \ref{['lem:ref']}. Left: Exp curves in $\mathop{\mathrm{SE}}\nolimits(3)$ that map unit element $[e]$ to a local orientation $[g_1]$ are plotted by their spatial projections along with a rotation frame. Right: section $\sigma$ can deviate from section $\sigma_{\rho}$ in $[g_1]$: $\textrm{Error}_{\mathcal{G}}(g_1)= 0.1$. Settings top: $[g_1]=\{g_1 e^{x A_6}\;|\; x \in (-\pi,\pi]\}$, $g_1=\exp(2 A_3 + \tfrac{7 \pi}{16} A_4 + \tfrac{7 \pi}{16} A_5)$, co-planar, $\mathcal{G}_e = \textrm{diag}(1, 1, 1, 1, 1, 0)$. Here the discrepancy of the log distance between taking section $g_1=\sigma([g_1]) \Leftrightarrow x=\alpha+\gamma=0$, and the actual minimizer $\sigma_{\rho}([g_1])$ over the fiber is visible. Settings bottom: For $g_2=\sigma([g_2])=\exp(\frac{1}{4}(A_3+A_2 + \frac{\pi}{14}A_5))$ (close to $e$) and $\mathcal{G}_{e} = \textrm{diag}(1, 1, 1, 0.01, 0.01, 0.05)$ the error vanishes.

Theorems & Definitions (22)

  • definition thmcounterdefinition: Section $\sigma$
  • definition thmcounterdefinition: Section $\sigma_d$
  • definition thmcounterdefinition: Section $\sigma_\rho$
  • definition thmcounterdefinition: Legal Metric
  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • proof
  • remark thmcounterremark
  • corollary thmcountercorollary
  • definition thmcounterdefinition: Reductive Homogeneous Space
  • ...and 12 more