Group Invariant Spectral Embedding
2026-07-09 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a way to improve how computers simplify complex high-dimensional data when the data has symmetrical patterns, like rotations. They modify standard methods to include these symmetries directly, making the analysis more accurate. Their math shows that this approach converges better and respects the true shape of the data when these symmetries are present. They tested it on data with rotational symmetries and found their method captures the data's true structure while traditional methods do not.
Spectral embeddingDimensionality reductionSymmetryLie groupRiemannian manifoldGraph LaplacianInvariant kernelsSO(2)SO(3)Quotient space
Authors
Yeari Vigder, Paulina Hoyos, David Thong, Joakim andén, Joe Kileel, Amit Moscovich
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.