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G-invariant diffusion maps

Eitan Rosen, Xiuyuan Cheng, Yoel Shkolnisky

TL;DR

The paper develops diffusion-map embeddings that respect group symmetries of data lying on a $G$-invariant manifold. By leveraging the $G$-invariant graph Laplacian and its harmonic analysis via irreducible unitary representations, it constructs both $G$-equivariant and $G$-invariant diffusion maps, with corresponding diffusion distances tied to random-walk densities on group-orbit spaces. The approach yields practical embeddings that cluster and align data while accounting for symmetry, and it is demonstrated on synthetic $SO(2)$-actions and a random tomography task. The results enable robust analysis of high-dimensional, symmetry-rich data and have potential applications in imaging and cryo-electron microscopy where symmetry-induced nuisances or ambiguities arise.

Abstract

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which is closed under the action of a continuous matrix group. An example of such a data set is images whose planar rotations are arbitrary. The G-invariant graph Laplacian, introduced in Part I of this work, admits eigenfunctions in the form of tensor products between the elements of the irreducible unitary representations of the group and eigenvectors of certain matrices. We employ these eigenfunctions to derive diffusion maps that intrinsically account for the group action on the data. In particular, we construct both equivariant and invariant embeddings, which can be used to cluster and align the data points. We demonstrate the utility of our construction in the problem of random computerized tomography.

G-invariant diffusion maps

TL;DR

The paper develops diffusion-map embeddings that respect group symmetries of data lying on a -invariant manifold. By leveraging the -invariant graph Laplacian and its harmonic analysis via irreducible unitary representations, it constructs both -equivariant and -invariant diffusion maps, with corresponding diffusion distances tied to random-walk densities on group-orbit spaces. The approach yields practical embeddings that cluster and align data while accounting for symmetry, and it is demonstrated on synthetic -actions and a random tomography task. The results enable robust analysis of high-dimensional, symmetry-rich data and have potential applications in imaging and cryo-electron microscopy where symmetry-induced nuisances or ambiguities arise.

Abstract

The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which is closed under the action of a continuous matrix group. An example of such a data set is images whose planar rotations are arbitrary. The G-invariant graph Laplacian, introduced in Part I of this work, admits eigenfunctions in the form of tensor products between the elements of the irreducible unitary representations of the group and eigenvectors of certain matrices. We employ these eigenfunctions to derive diffusion maps that intrinsically account for the group action on the data. In particular, we construct both equivariant and invariant embeddings, which can be used to cluster and align the data points. We demonstrate the utility of our construction in the problem of random computerized tomography.
Paper Structure (21 sections, 14 theorems, 161 equations, 9 figures, 3 algorithms)

This paper contains 21 sections, 14 theorems, 161 equations, 9 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. The $G$-invariant graph Laplacian
  3. $G$-equivariant diffusion maps
  4. $G$-invariant diffusion maps
  5. Numerical experiments
  6. Basic examples
  7. Random computerized tomography with random angles and shifts
  8. Summary and future work
  9. Compact matrix Lie groups
  10. Matrix Lie groups and actions
  11. Harmonic analysis over matrix Lie groups
  12. Proof of Lemma \ref{['eqvDmaps:eqvEigenVecLemma']}
  13. Proofs of results from Section \ref{['sec:GEqvDmaps']}
  14. Proof of Theorem \ref{['eqvDmaps:diffDistEigProp']}
  15. Proof of Proposition \ref{['eqvDmaps:truncEmbedEqvProp']}
  16. ...and 6 more sections

Key Result

Theorem 2

(GGL) For each $\ell\in \mathcal{I} _G$, let be the $Nd_\ell\times Nd_\ell$ block-diagonal matrix whose $i^{\text{th}}$ block of size $d_\ell\times d_\ell$ on the diagonal is given by the product of the scalar $D_{ii}$ from GinvDef:DiiDef with the $d_\ell\times d_\ell$ identity matrix. Then, the normalized $G$-invariant graph Laplacian $\tilde

Figures (9)

  • Figure 1: Figure (a) shows the $SO(2)$-equivariant diffusion distances \ref{['eqvDmaps:eqvDist']} of the points in the simulated data set $X$ from $x_i\in X$ (marked by a blue dot), depicted as a heat map superimposed on the torus $\mathbb{T}^2\subset \mathbb{R}^3$. Figure (b) shows the distances \ref{['eqvDmaps:eqvDist']} of the points in $X$ from $x_{N+1}$ (marked by the green dot). The images indicate that rotating $x_i$ by $180^\circ$ rotates the heat distribution by the same angle.
  • Figure 2: Figure (a) displays the $SO(2)$-invariant diffusion distance of the points in the simulated data set $X$ from $x_i\in X$ (marked by the blue point). Figure (b) shows the same simulation repeated with points sampled from the 2-sphere. In both cases, the distances are constant over the orbits induced by the action of $SO(2)$ by rotations about the $z$-axis, namely, the horizontal circles.
  • Figure 3: Heat distributions resulting from repeating the simulation depicted in Figure \ref{['fig:eqvTori']} above with data sampled from the $2$-sphere. In this case, the green point is a rotation of $x_i$ (marked by the blue point) by $60^\circ$ about the $z$-axis. The distribution of the diffusion distances of the points in the simulated data set from the rotation of $x_i$ by $60^\circ$ in Figure (b) is obtained by rotating the distribution of the distances from $x_i$ in Figure (a) by $60^\circ$, showing a similar qualitative picture to that of the simulation with the torus.
  • Figure 4: Shepp-Logan phantom reconstructed from 256 shifted random projections at various levels of noise, after centering them by using our method.
  • Figure 5: Angles $\tilde{\varphi}_{(i)}$ obtained by ordering the true projection angles $\varphi_i$ according to the order of the sorted angles $\tilde{\varphi}_i$ of \ref{['numerSec:ordAng']}, plotted against the angles $\varphi_{(i)}$ obtained by sorting $\varphi_i$. The angles $\tilde{\varphi_i}$ were estimated by passing to Algorithm \ref{['numerSec:ordAlg']} the class-averages of the $32$ nearest neighbors of each $x_i\in X$, determined by the $S^1$-invariant diffusion maps.
  • ...and 4 more figures

Theorems & Definitions (42)

  • Definition 1: The $G$-invariant graph Laplacian
  • Theorem 2
  • Lemma 3
  • proof
  • Definition 4
  • Definition 5
  • Theorem 6
  • proof
  • Proposition 7
  • proof
  • ...and 32 more