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: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.
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.
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.
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:2606. 15760v1 Announce Type: new Abstract: A significant gap exists between theory and practice in deep learning.
The paper introduces a Nested Inductive Bias framework that uses a two‑stage diffeomorphic composition to pull back non‑Euclidean target geometries onto symmetric positive definite (SPD) manifolds. This approach allows the construction of curvature‑aligned Riemannian classifiers that respect both matrix constraints and the intrinsic relational geometry of data. Empirical results on kinematic, signal processing, and synthetic benchmarks show that class separability degrades when metric curvature does not match the data distribution, and the authors also propose the Rational Conformal Metric (RCM) for robust vectorized architectures.
arXiv:2606. 17022v1 Announce Type: cross Abstract: A central objective of machine learning is to identify structure and patterns in data.
arXiv:2606. 14334v1 Announce Type: new Abstract: High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension.
This paper proposes ``FlatVPR,'' a novel geometric rectification paradigm that effectively bridges the trade-off between map lightweightness and localization accuracy in visual place recognition (VPR) by enforcing a feature manifold structure where any descriptor between two adjacent anchors $\mathbf{z}_A$ and $\mathbf{z}_B$ can be accurately reconstructed via linear interpolation $\hat{\mathbf{z}}_{pseudo} = (1-t)\mathbf{z}_A + t\mathbf{z}_B$, where $t \in [0,1]$ denotes the relative position. While state-of-the-art foundation models such as DINOv2-ViT-S/14 provide robust semantic features, their latent manifolds exhibit prominent curvature, projecting uniform linear motion in physical space onto highly non-linear trajectories in the feature space, which hinders reliable reconstruction under sparse anchor conditions.
arXiv:2606. 01734v1 Announce Type: cross Abstract: This paper proposes ``FlatVPR,'' a novel geometric rectification paradigm that effectively bridges the trade-off between map lightweightness and localization accuracy in visual place recognition (VPR) by enforcing a feature manifold structure where any descriptor between two adjacent anchors $\mathbf{z}_A$ and $\mathbf{z}_B$ can be accurately reconstructed via linear interpolation $\hat{\mathbf{z}}_{pseudo} = (1-t)\mathbf{z}_A + t\mathbf{z}_B$, where $t \in [0,1]$ denotes the relative position.
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.
arXiv:2607. 17377v1 Announce Type: new Abstract: The paper presents a new parameter-efficient adaptation method called ChebyMA (Chebyshev Manifold Adaptation).