发表机构
Computing Department; Federal University of São Carlos（计算机系; 圣卡洛斯联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出曲率感知半径收缩（CARSANN）框架，利用局部流形曲率调整邻域空间范围，在70余个OpenML数据集上较标准k-NN实现显著且具统计意义的分类性能提升。
AI 中文摘要
最近邻分类从根本上依赖于局部性的定义，但传统的k近邻（k-NN）在整个特征空间中施加相同的邻域基数，该假设对于在底层流形上局部几何差异显著的数据可能并不适用。我们提出用于自适应最近邻分类的曲率感知半径收缩（CARSANN），这是一种几何驱动的框架，可根据局部几何复杂度调整每个邻域的空间范围。CARSANN首先使用TwoNN估计本征维数，并通过主成分分析构建本征表示；随后使用基于形状算子的公式估计局部平均曲率，并以此控制邻域尺度：高度弯曲的区域会受到更强的半径收缩，而近似平坦的区域则保留更宽的空间范围。与仅修改邻域数量或局部度量的方法不同，CARSANN明确调整局部证据的空间范围。在超过70个真实世界OpenML数据集上的实验表明，CARSANN始终优于标准k-NN，且与自适应最近邻方法具有竞争力。在使用相同基础邻域大小的受控对比中，CARSANN在45个数据集中的40个上实现了更高的平衡准确率，将平均平衡准确率从0.6506提升至0.7528。该优势在与固定k=5的k-NN对比中依然存在，Friedman检验和Nemenyi检验证实这些改进具有统计显著性。这些结果表明，局部流形曲率可作为调整邻域支撑的有效几何控制变量，为基于基数的最近邻自适应提供了互补范式。
英文摘要
Nearest neighbor classification relies fundamentally on how locality is defined, yet conventional $k$-NN imposes the same neighborhood cardinality throughout the feature space. This assumption can be inadequate for data whose local geometry varies substantially across the underlying manifold. We introduce Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification (CARSANN), a geometry-driven framework that adapts the spatial support of each neighborhood according to local geometric complexity. CARSANN first estimates intrinsic dimensionality using TwoNN and constructs an intrinsic representation through principal component analysis. Local mean curvature is then estimated using a shape-operator-based formulation and controls neighborhood scale: highly curved regions receive stronger radius shrinkage, whereas approximately flat regions retain broader spatial support. Unlike methods that modify only the number of neighbors or the local metric, CARSANN explicitly adapts the spatial extent of local evidence. Experiments on more than 70 real-world OpenML datasets show that CARSANN consistently improves upon standard $k$-NN and is competitive with adaptive nearest-neighbor methods. In a controlled comparison using the same base neighborhood size, CARSANN achieves higher balanced accuracy on 40 of 45 datasets, increasing mean balanced accuracy from 0.6506 to 0.7528. The advantage also persists against $k$-NN with fixed $k=5$. Friedman and Nemenyi tests confirm that the improvements are statistically significant. These results indicate that local manifold curvature can serve as an effective geometric control variable for adapting neighborhood support, providing a complementary paradigm to cardinality-based nearest-neighbor adaptation.
Comments28 pages, 2 figures and 3 tables
