Related papers: Diffusion Maps for Group-Invariant Manifolds

We study $\Gamma$-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on unions of intersecting manifolds of potentially different dimensions. Our investigation is motivated by problems of machine learning,…

The spectral structure of the Laplacian-Beltrami operator (LBO) on manifolds has been widely used in many applications, include spectral clustering, dimensionality reduction, mesh smoothing, compression and editing, shape segmentation,…

Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on…

The Novikov-Shubin invariants for a non-compact Riemannian manifold M can be defined in terms of the large time decay of the heat operator of the Laplacian on square integrable p-forms on M. For the (2n+1)-dimensional Heisenberg group H,…

We introduce a novel co-learning paradigm for manifolds naturally equipped with a group action, motivated by recent developments on learning a manifold from attached fibre bundle structures. We utilize a representation theoretic mechanism…

In this note we prove the following theorem: Let $G$ be a compact Lie group acting on a compact symplectic manifold $M$ in a Hamiltonian fashion. If $L$ is an $l$-dimensional closed invariant submanifold of $M$, on which the $G$-action is…

The Laplace-Beltrami operator has established itself in the field of non-rigid shape analysis due to its many useful properties such as being invariant under isometric transformation, having a countable eigensystem forming an orthornormal…

Given a smooth free action of a compact connected Lie group $G$ on a smooth compact manifold $M$, we show that the space of $G$-invariant Riemannian metrics on $M$ whose automorphism group is precisely $G$ is open dense in the space of all…

This paper introduces certain elliptic Harnack inequalities for harmonic functions in the setting of the product space $M \times X$, where $M$ is a (weighted) Riemannian Manifold and $X$ is a countable graph. Since some standard arguments…

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn…

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and…

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the…

Let $(M,\omega)$ be a symplectic manifold compact or convex at infinity. Consider a closed Lagrangian submanifold $L$ such that $\omega |_{\pi_2(M,L)}=0$ and $\mu|_{\pi_2(M,L)}=0$, where $\mu$ is the Maslov index. Given any Lagrangian…

We introduce a data-adaptive inversion method that integrates classical or deep learning-based approaches with iterative graph Laplacian regularization, specifically targeting acoustic impedance inversion - a critical task in seismic…

We generalize the Li-Yang notion of self-similar $k$-graph $(G,\Lambda)$ and its $C^*$-algebra $\mathcal{O}_{G,\Lambda}$ to any finitely aligned $k$-graph $\Lambda$. We then introduce an inverse semigroup model for $\mathcal{O}_{G,\Lambda}$…

We apply concepts from manifold regularization to develop new regularization techniques for training locally stable deep neural networks. Our regularizers are based on a sparsification of the graph Laplacian which holds with high…

We introduce a general method for learning representations that are equivariant to symmetries of data. Our central idea is to decompose the latent space into an invariant factor and the symmetry group itself. The components semantically…

We construct center-stable and center-unstable manifolds, as well as stable and unstable manifolds, for the nonlinear Klein-Gordon equation with a focusing energy sub-critical nonlinearity, associated with a family of solitary waves which…

In this paper, we investigate the spectrum of a class of weighted Laplacians on Cayley graphs and determine under what conditions the corresponding eigenspaces are generically irreducible. Specifically, we analyze the spectrum on…

Let $M$ be a complete Riemannian manifold and $G$ a Lie subgroup of the isometry group of $M$ acting freely and properly on $M.$ We study the Dirichlet Problem \begin{align*} \operatorname{div}\left( \frac{a\left( \left\Vert \nabla…