The results give an empirical separation, on a real hierarchical-classification problem, between two natural latent geometries for a class-structured regularizer.
Abstract
We study a tree-structured regularizer over class-prototype layouts in a hierarchical-classification model and ask whether the choice of latent manifold for the prototypes (Euclidean R^d vs. the Poincare ball B^d_c) affects how well that regularizer can be satisfied without distorting the data likelihood. The two manifolds differ only in their volume growth: hyperbolic space grows exponentially with radius and embeds trees with provably lower distortion than R^d of matched dimension, so the structured regularizer should be cheaper to satisfy on B^d_c. Across 150 seed-replicated regularized maximum-likelihood fits spanning embedding dimension, curvature, and regularizer strength on WikiArt (27 styles, 81,446 paintings, frozen CLIP ViT-B/16 features), we find a single robust effect: Poincare prototypes preserve the topology of the nearest-neighbor graph in latent space substantially better than matched Euclidean prototypes (sibling recall@5 +8.7 pp, cousin recall +15.2 pp; paired-t p<10^-4, sign agreement 0.94), and the gap holds across three reference-tree definitions (hand-built lineage, CLIP-derived, and DINOv2-derived). On classification, Euclidean prototypes are tied with logistic regression on raw encoder features, indicating no detectable contribution from the latent geometry; only the hyperbolic fit improves on a k-NN encoder baseline for local retrieval. Global tree-fidelity comparisons are unstable across reference trees and we do not claim a winner. The results give an empirical separation, on a real hierarchical-classification problem, between two natural latent geometries for a class-structured regularizer.
This approach demonstrates that models trained on small-scale random graphs learn to extract universal distance-preserving features, achieving robust generalization to large-scale, real-world networks that match or exceed the fidelity of classical, exact landmark-based embeddings.
Spectral clustering methods for network data are commonly based on a few matrix representations, such as the adjacency matrix and the symmetric Laplacian. We study a continuum of degree-normalized spectral embeddings that includes these commonly used choices as special cases. Under a random dot product graph model, we establish a row-wise central limit theorem for this family of embeddings. The result provides an explicit description of how degree normalization affects both population geometry and the local uncertainty of embedded nodes. We use the limiting distributions to compare different normalizations in two-community stochastic block models through a projected-Gaussian Bayes-error diagnostic. These comparisons show that no single normalization is uniformly preferred. Instead, the favored normalization depends on network density, community imbalance, and block-probability structure. Typically, stronger normalization is favored in lower-density or more imbalanced settings. These results provide a unified distributional understanding of when and why alternative normalizations may improve spectral clustering.
This paper proposes a feature-graph-guided adaptive Log-L2,1 sparse NMF with anchor dual graphs under a logarithmic framework that jointly integrate sample structure preservation, feature structure preservation, and feature-aware sparse learning within a unified graph-NMF model.
Quanrun Li, Tao Ma, Fangchen Xu et al.· Mathematics· 0 citations
The findings illustrate that geometric insights grounded in hyperbolic geometry can offer powerful tools for understanding, embedding, and visualizing complex graph structures.
†. SalouaNaama, Kave Salamatian, M. Crovella· 0 citations
A membership-restricted Shepard rho resolves single-point and minority-population questions that many-pair statistics cannot -- questions on which even DREAMS, a recent local-plus-global hybrid, fails silently.
Hyperbolic embedding methods for collaborative filtering constrain all representations to the hyperboloid manifold, imposing a single geometry regardless of data characteristics. We introduce LightConeFM, which removes this constraint and allows embeddings to freely occupy any causal region of Lorentz-Minkowski space-timelike, lightlike, or spacelike-using only standard gradient descent without Riemannian optimization. Experiments on four real-world datasets reveal two consistent findings: (1) unconstrained embeddings outperform their constrained counterparts on every dataset (up to +7.0% AUC), and (2) the learned causal zone distribution predicts where hyperbolic geometry provides benefit over Euclidean alternatives (Pearson $r=0.94$, Spearman $\rho=1.0)$ -datasets with predominantly timelike users exhibit the largest gains, while predominantly spacelike datasets are better served by Euclidean methods. On a job recommendation dataset, LightConeFM achieves +5.99% AUC improvement in cold-start settings, indicating particular value for sparse, hierarchical domains.
K. Uyar· 2026 6th International Confe...· 0 citations
