New Method Improves Dimensionality Reduction by Incorporating Symmetries.

Spectral embedding methods are widely used for reducing the dimensionality of high-dimensional datasets and for clustering, particularly when data possesses inherent low-dimensional structures. However, many real-world datasets exhibit symmetries, such as rotational invariance, which standard spectral embedding techniques typically ignore. These methods treat symmetry-related data points as distinct, potentially obscuring the true underlying geometry. This research proposes a new approach called Group Invariant Spectral Embedding, which directly integrates these symmetries into the affinity kernels used for constructing graph Laplacians. The focus is on Riemannian data manifolds with symmetries defined by a compact Lie group. The analysis demonstrates that graph Laplacians built from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space of the manifold by the group. A key implication of this is improved convergence rates, as the effective dimension of the problem is reduced by the dimension of the symmetry group. Experimental validation on datasets with SO(2) or SO(3) symmetry confirms that this G-invariant spectral embedding successfully recovers the intrinsic geometry of the data, a task where standard methods often fail.