Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Collaborative Research: RI: Medium: Bridging the Semantic-Metric Gap via Multinocular Image Integration, Learned models for molecular reconstruction in cryogenic electron microscopy, NSF-BSF: Computational Methods for Shape Space Analysis in Structural Biology, FDT-BioTech: A digital twin framework for developing and analyzing virtual patient cohorts to enable virtual clinical trials, Belmont Forum Collaborative Research: Negotiating Ocean Conflicts among RIvals for Sustainable and Equitable Solutions
OpenAlex ID: W7168214239
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