arXiv:2607. 01993v1 Announce Type: cross Abstract: The silhouette is one of the most widely used measures to assess the quality of a $k$-clustering of a dataset of $n$ elements.
By Ilie Sarpe, Federico Altieri, Andrea Pietracaprina, Geppino Pucci, Fabio Vandin
The paper introduces EGGroW, efficient algorithms for computing geodesic Gromov-Wasserstein distances using entropic Sinkhorn-like methods, GenusSink techniques, and random features. It addresses the cubic time complexity of traditional GWD calculations on dense intra-space distance matrices, enabling scalable comparisons of probabilistic distributions on general geodesic manifolds and graph shortest‑path distances. The authors demonstrate EGGroW’s effectiveness in downstream tasks such as 3D pose estimation and partial 3D template recovery, showing accurate results where Euclidean‑based methods fail while maintaining a light computational footprint.
By Krzysztof Marcin Choromanski, Derek Long, Ananya Parashar, Dwaipayan Saha
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
Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification (CARSANN) is a geometry-driven framework that adapts the spatial support of each neighborhood based on local geometric complexity. It estimates intrinsic dimensionality with TwoNN, builds an intrinsic representation via PCA, and uses a shape-operator-based estimate of local mean curvature to shrink the radius in highly curved regions while keeping a broader support in flatter areas. Experiments on over 70 OpenML datasets show that CARSANN consistently outperforms standard k‑NN and rivals other adaptive nearest‑neighbor methods, achieving a mean balanced accuracy increase from 0.6506 to 0.7528 and statistically significant improvements on most datasets.
By Alexandre L. M. Levada
The paper investigates the Platonic Representation Hypothesis, which posits that more capable models converge toward shared representations. By distinguishing relational structure (which samples are related) from metric geometry (quantitative relations like distances), the authors develop a controlled $2 imes2$ framework to evaluate both aspects at local and global scales. Their findings show that relational structure consistently converges across vision‑language and video‑text models, while metric geometry converges much more weakly, a pattern that persists even when using a Riemannian metric approximation.
By Junwon You, Mihyun Jang, Sangwoo Mo, Jae-Hun Jung
The paper introduces weighted t‑SNE, a variant of t‑SNE that incorporates feature importance scores from feature scorers into the 2‑D embedding of high‑dimensional data. It explains how to implement the method, proposes using the silhouette coefficient to evaluate the resulting visualizations, and demonstrates its use on synthetic and real‑world tabular datasets with nine different feature scorers. The approach allows practitioners to compare feature scorers visually and to interpret how machine‑learning models organize data based on learned feature importance.
By Bruno Iochins Grisci, Mario Inostroza-Ponta, M\'arcio Dorn