Group Invariant Spectral Embedding

• URL: http://arxiv.org/abs/2607.08987v1
• Date: Thu, 09 Jul 2026 23:18:39 GMT
• Title: Group Invariant Spectral Embedding
• Abstract summary: We show that $G$-invariant spectral embedding recovers the intrinsic geometry of the data, in contrast to standard spectral embedding.<n>We validate our approach on datasets with $mathrmSO(2)$ or $mathrmSO(3)$ symmetry.
• Score: 2.1538077160255313
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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