arXiv:2509. 24458v2 Announce Type: replace-cross Abstract: We study $\Gamma$-convergence of graph Dirichlet energies and spectral convergence of graph Laplacians on unions of intersecting manifolds of potentially different dimensions.
arXiv:2502.04312v3 Announce Type: replace Abstract: Contrastive learning leverages data augmentation to develop feature representation without relying on large labeled datasets. However, despite its...
arXiv:2607. 06644v1 Announce Type: cross Abstract: Determinantal point processes have recently emerged as a kernel-based alternative to standard independent sampling for constructing efficient minibatches, coresets, and other compact representations of large-scale datasets.
arXiv:2510. 25354v3 Announce Type: replace Abstract: Hypergraphs provide a natural framework for modeling multiway interactions.
arXiv:2607.06644v2 Announce Type: replace Abstract: Determinantal point processes (DPPs) have recently emerged as a kernel-based alternative to standard independent sampling for constructing efficien...
When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold $(\mathcal{M}, g)$ of $\mathbb{R}^d$, a key estimate for the geodesic distance $d_g$ is that there exists $K > 0$ such that $0 \leq d_g(p, q)^2 - \|p-q\|^2 \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$. We observe that more generally, when $\mathcal{M}$ is equipped with a smooth symmetric divergence $D$ satisfying a non-degeneracy condition and $g$ is given by $g_p := \frac{1}{2}\mathrm{Hess}_p(D(p, \cdot))$ for all $p \in \mathcal{M}$, there exists $K > 0$ such that $\left| D(p, q) - d_g(p, q)^2 \right| \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$.
arXiv:2510. 02308v2 Announce Type: replace Abstract: Estimating the tangent spaces of a data manifold is a fundamental problem in geometric data analysis.
arXiv:2607. 08987v1 Announce Type: new Abstract: Spectral embedding methods are widely used for dimensionality reduction and clustering of high-dimensional datasets with intrinsic low-dimensional structures.
arXiv:2607. 05892v1 Announce Type: cross Abstract: When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold $(\mathcal{M}, g)$ of $\mathbb{R}^d$, a key estimate for the geodesic distance $d_g$ is that there exists $K > 0$ such that $0 \leq d_g(p, q)^2 - \|p-q\|^2 \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$.
arXiv:2607. 19305v1 Announce Type: cross Abstract: Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations.
The paper introduces a geometry‑aware graph construction method that adaptively selects Gaussian kernel bandwidths per node to align the kernel’s spectral complexity with the intrinsic dimensionality of the underlying manifold. By matching the kernel’s effective rank to a local intrinsic dimension estimate derived from a minimum spanning tree, the method operates within a manifold‑consistent log‑log scaling regime. Experiments on CIFAR‑100 demonstrate that this adaptive bandwidth approach consistently improves leave‑one‑out classification and label propagation accuracy compared to fixed‑bandwidth and other adaptive techniques.
arXiv:2607. 06497v1 Announce Type: new Abstract: We introduce EntroPath, a manifold learning method that recovers geodesic geometry from data graphs through ensembles of diffusion paths.
The paper proposes using concepts from low‑dimensional topology—specifically Morse theory and cobordism—to enhance graph diffusion models, introducing the MG‑Diff pipeline. It provides theoretical guarantees that the Morse‑theoretic guidance remains stable under small perturbations when a positive decision‑gap exists. The authors demonstrate the approach on spatio‑temporal graph forecasting and graph regeneration, suggesting broader potential for topology in machine learning.