Group Invariant Spectral Embedding

2026-07-09

AI Summary

This work addresses the limitation of standard spectral embedding methods, which neglect invariance under symmetry groups such as rotations and thus fail to accurately recover the intrinsic geometry of symmetric data. The authors propose incorporating compact Lie group symmetries directly into the affinity kernel to construct a group-invariant graph Laplacian. Under the assumption of an underlying Riemannian manifold, they establish—for the first time—the convergence of graph Laplacians derived from three classes of invariant kernels to second-order differential operators on the corresponding quotient space. Notably, this convergence occurs at an accelerated rate due to the reduced effective dimensionality induced by the group action. Experiments demonstrate that, on data with SO(2) or SO(3) symmetries, the proposed method successfully recovers the intrinsic geometry, whereas conventional spectral embeddings remain inconsistent even in the infinite-sample limit.

Abstract

Spectral embedding methods are widely used for dimensionality reduction and clustering of high-dimensional datasets with intrinsic low-dimensional structures. Although many datasets of practical interest exhibit invariance under symmetries such as rotations, standard spectral embedding methods do not account for this, treating symmetry-related data points as unrelated. Our approach to this problem is to incorporate the symmetries directly into the affinity kernels used for spectral embedding. We analyze the case of a Riemannian data manifold $M$ with symmetries given by a compact Lie group~$G$ and prove that, under suitable conditions, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space $M/G$. Our analysis implies improved convergence rates, as the effective dimension drops according to the dimension of the group. We validate our approach on datasets with $\mathrm{SO}(2)$ or $\mathrm{SO}(3)$ symmetry, and show that $G$-invariant spectral embedding recovers the intrinsic geometry of the data, in contrast to standard spectral embedding, which fails to do so even in the limit of infinite data.

Problem

• spectral embedding
• group invariance
• symmetry
• dimensionality reduction
• graph Laplacian

Innovation

• group-invariant spectral embedding
• symmetry-aware kernels
• graph Laplacian convergence
• quotient manifold
• dimensionality reduction