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.
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.
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.
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.
arXiv:2607. 09490v1 Announce Type: cross Abstract: Terminal embeddings have emerged as a powerful tool for dimension reduction.
arXiv:2502. 00168v5 Announce Type: replace-cross Abstract: Supervised dimensionality reduction maps labeled data into a low-dimensional feature space while preserving class separation.
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.
arXiv:2606. 02765v1 Announce Type: cross Abstract: Model dimension ($d_{model}$) is a fundamental hyperparameter in transformer language models, yet its role in setting the geometric limits of feature representation remains under-explored.
arXiv:2601. 20844v3 Announce Type: replace-cross Abstract: This paper studies the Minimal Embeddable Dimension (MED): the least dimension in which there exists a configuration of $m$ object vectors so that every subset of size at most $k$ is exactly retrieved by score comparison.
The paper introduces a unified bundle adjustment framework that jointly optimizes camera parameters, sparse 3D points, and richer geometric features such as lines, coplanarity, and parallelism. It classifies features into scalable ones with direct 2D measurements and higher‑order groups that can be treated as camera‑like entities, allowing group constraints and cross‑feature relations to be expressed via 2D reprojection errors. This approach preserves the sparsity of classical point‑based BA, maintains numerical stability, and achieves runtime comparable to point‑only BA while producing richer 3D structures and improved accuracy.
This paper introduces the Sierpiński-Knopp (SK) Wasserstein distance, a fast metric between persistence diagrams. The SK-Wasserstein distance, denoted $d_{\mathrm{SK}}$, maps diagram points and their...
arXiv:2608. 01039v1 Announce Type: cross Abstract: Trajectory similarity learning is fundamental to efficient trajectory retrieval under complex distance measures.
arXiv:2607. 03112v1 Announce Type: cross Abstract: We revisit random projections for reducing the dimension of high-dimensional polygonal curves.