The paper investigates how to recover low‑dimensional latent geometry from a random geometric graph built from noisy, high‑dimensional data. It analyzes a spectral embedding algorithm on the Signal+Noise Graph Model, where vertices are perturbed points and edges are added when their inner product exceeds a threshold. In the regime where both the number of points and ambient dimension grow, the authors prove that, under a spectral gap condition, the leading eigenvectors and eigenvalues of the adjacency matrix can approximately recover the original point cloud up to an orthogonal transformation, and demonstrate the method on nested spheres and high‑dimensional sinusoid curves.
arXiv:2511. 11927v2 Announce Type: replace-cross Abstract: Principal Component Analysis (PCA) is a standard tool for extracting a low-rank signal from noisy observations.
The paper presents provable guarantees for a spectral method that recovers binary node labels on signed graphs with edge‑flip noise. It provides graph‑structure‑agnostic bounds on approximate inference accuracy and maximum angle deviation, using matrix concentration and eigenvector perturbation techniques. The results connect to the Cheeger constant and are validated with synthetic experiments, marking the first theoretical analysis of this spectral approach.
arXiv:2609.23668v1 Announce Type: cross Abstract: Inferring network topology from noisy node observations is a central problem in graph signal processing. In this paper, we consider Laplacian-constra...
arXiv:2510. 02308v2 Announce Type: replace Abstract: Estimating the tangent spaces of a data manifold is a fundamental problem in geometric data analysis.
arXiv:2608. 11321v1 Announce Type: cross Abstract: We study spectral clustering in the presence of a confounding latent geometry.
arXiv:2606. 29675v1 Announce Type: new Abstract: Bogomolny, Bohigas and Schmit (BBS) found that the spectrum of the pairwise distance matrix on N points sampled from a smooth d-dimensional manifold encodes a signature of the underlying geometry.
arXiv:2606. 11263v1 Announce Type: cross Abstract: Spectral methods rely fundamentally on the stability of principal eigenspaces under random perturbations.
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.
The paper introduces Spectral Connectivity-Regularized Graph Learning (SCoGL), a method for learning sparse graphs from limited data by incorporating Laplacian spectral priors that promote global connectivity. SCoGL extends the graphical lasso objective with a connectivity prior derived from Laplacian eigenvalues and uses projected gradient descent with Armijo backtracking for optimization. Experiments demonstrate that SCoGL improves graph recovery and enhances downstream tasks such as graph signal denoising when observations are scarce.
arXiv:2412. 16457v3 Announce Type: replace-cross Abstract: In this paper, we focus on the matching recovery problem between a pair of correlated Gaussian Wigner matrices with a latent vertex correspondence.
arXiv:2607. 14304v1 Announce Type: cross Abstract: We study sparse random geometric graphs generated by connecting pairs of high-dimensional vectors whose inner product exceeds a threshold.
arXiv:2609.37649v1 Announce Type: cross Abstract: Many datasets carry an intrinsic directionality: citations point backward in time, cells differentiate along lineages, and traffic follows preferred...