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.
Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised n…
Motivated by the numerical computation of the Mean Escape Time (MET) $τ:Ω\to\mathbb{R}$ of a stochastic process from a bounded domain $Ω\subseteq\mathbb{R}^d$, we study elliptic Dirichlet boundary value problems (BVPs) using boundary-enforced Physics-Informed Neural Networks (PIN…
We study aggregation of statistical evidence under unknown and potentially complex dependence using group-invariance. Building on permutation-based constructions that treat transformed datasets as exchangeable units, we aggregate evidence across statistics for each transformed da…
Sparse-support uncertainty is usually quantified by treating the dictionary as known, an assumption that can produce overconfident, label-dependent conclusions when the dictionary is learned from latent sparse mixtures. Near collisions of coherent atoms, a test signal may identif…
For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems ($4 \leq d \lesssim 10$), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-di…
We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space…