arXiv:2608. 15313v1 Announce Type: cross Abstract: In this paper, we propose SHOPCA (Shape Operator-based Principal Component Analysis), a novel method for unsupervised metric learning and dimensionality reduction that incorporates differential geometric information into the covariance structure of classical PCA.
By Alexandre L. M. Levada
arXiv:2607. 03145v1 Announce Type: cross Abstract: The informativeness of a training set is as consequential as its size, yet most sampling strategies remain agnostic to the intrinsic geometry of the data distribution.
By Alexandre L. M. Levada
arXiv:2606. 06329v1 Announce Type: new Abstract: Estimating local mean curvature at each point of a high-dimensional dataset is a key ingredient of geometry-aware machine learning algorithms, such as the Mean Curvature Boundary Points (MCBP) method.
By Alexandre L. M. Levada
LocaLeJEPA extends Joint-Embedding Predictive Architectures (JEPA) by introducing a manifold-theoretic reformulation that aligns representation learning with non‑Euclidean geometric priors. It incorporates a locally Euclidean distribution‑matching regularizer, LocaLeReg, to stabilize training and scale to large models. Experiments on ImageNet‑1K show that LocaLeJEPA outperforms the original LeJEPA across both hyperbolic and Euclidean backbones, improving linear and k‑NN classification as well as downstream tasks.
By Kartik Tandon, Alessio Borgi, Pavel Snopov, In\'es Garc\'ia-Redondo, Davide D'Ascenzo, Sebastiano Cultrera di Montesano, Fabrizio Silvestri, Pietro Li\`o, Francesca Dominici, Randall Balestriero, Claudio Battiloro
The paper introduces Constant‑Curvature Sliced Gromov‑Wasserstein (CCSGW), a new divergence for aligning probability distributions on heterogeneous constant‑curvature spaces such as hyperbolic and spherical manifolds. It extends sliced Gromov‑Wasserstein by adding geodesic‑based one‑dimensional projections for spherical spaces, enabling efficient and principled comparison across manifolds with different curvatures while preserving intrinsic geometric relationships. The authors provide theoretical analysis showing that CCSGW controls intrinsic geometric discrepancy and demonstrate consistent performance gains when integrated into mixed‑curvature learning tasks like graph anomaly detection, node classification, and multimodal learning.
By Shanglin Li, Wenjing Lu, Muyang Li, Nicu Sebe, Ziheng Chen
The paper introduces View distance, a novel metric that projects high‑dimensional data onto all pairwise two‑dimensional planes and sums the Euclidean distances across these projections. It satisfies metric axioms, couples features, suppresses redundancy, and captures anisotropic geometry. To make it scalable, the authors propose a plane‑selection strategy using iterative Maximum Weight Matching, reducing complexity from ω(n²) to ω(k) and demonstrating competitive performance on twelve datasets.
By Yiqun Zhang, Hou-biao Li