Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification

Alexandre L. M. Levada

cs.LGAug 27, 2026

Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification

Authors: Alexandre L. M. Levada

Organizations: Computing Department Federal University of São Carlos 13565-905, São Carlos-SP, Brazil

Abstract

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.

Figures & tables

Figure 1: Geometric interpretation of the shape operator. As one moves along a tangent direction $\mathbf{v}$ , the shape operator measures the directional variation of the unit normal vector, providing a quantitative description of the local bending of the manifold.

Figure 2: Geometric motivation of the CARSANN radius-shrinkage mechanism. Regions of higher local curvature exhibit a faster departure from the tangent-space approximation and therefore receive a smaller effective neighborhood radius. Approximately flat regions can retain a larger spatial support.

Name	$n$	$d$	$c$	Name	$n$	$d$	$c$
11_Tumors	174	12,533	11	anneal	898	39	5
AP_Breast_Omentum	421	10,935	2	AP_Breast_Uterus	468	10,936	2
AP_Colon_Omentum	363	10,935	2	arcene	900	10,000	2
arsenic-female-bladder	478	9	2	arsenic-male-bladder	478	9	2
artificial-characters	6,000	7	10	CNS	60	7,129	2
cnae-9	1,080	857	9	coil-20	1,440	1,024	20

Table 1: Characteristics of the datasets used in the computational experiments:

n

denotes the number of samples,

d

denotes the number of features and

c

denotes the number of classes.

	Regular k-NN		DANN		Kk-NN		CARSANN
Datasets	BACC	F1	BACC	F1	BACC	F1	BACC	F1
11_Tumors	0.7815	0.7781	0.6404	0.6785	0.7852	0.8034	0.7872	0.7993
anneal	0.3669	0.7385	0.3922	0.7584	0.5881	0.7989	0.5812	0.7973
arsenic-male-bladder	0.7333	0.9664	0.7333	0.9664	0.8314	0.9770	0.8314	0.9770
artificial-characters	0.3171	0.3120	0.3144	0.3129	0.3894	0.3932	0.6382	0.6375
cnae-9	0.6827	0.7012	0.7221	0.7425	0.7187	0.7358	0.7315	0.7475

Table 2: Comparison of CARSANN with regular

k

-NN, DANN, and curvature-based

k

-NN (Kk-NN) across datasets. Balanced accuracy (BACC) and F1-score are reported for each classifier. The best result for each dataset and metric is highlighted in bold, while the final two rows report the corresponding mean and median values across all datasets.

	$k=k_{base}$				$k=5$
	k-NN (ambient)		CARSANN		k-NN (ambient)		CARSANN
Datasets	BACC	F1	BACC	F1	BACC	F1	BACC	F1
11_Tumors	0.7727	0.7721	0.7872	0.7993	0.6627	0.5788	0.7937	0.8099
anneal	0.5252	0.8325	0.5812	0.7973	0.4715	0.7836	0.5562	0.7892
AP_Breast_Omentum	0.5568	0.7513	0.8152	0.9129	0.7267	0.8579	0.8547	0.9258
AP_Breast_Uterus	0.8539	0.8997	0.8930	0.9342	0.8559	0.8889	0.9060	0.9313

Table 3: Comparison between CARSANN and standard

k

-NN in the ambient feature space under two neighborhood-size configurations. For

k=k_{\mathrm{base}}

, the same base neighborhood cardinality used by CARSANN is adopted by the standard

k

-NN classifier, providing a controlled comparison of the curvature-aware radius adaptation. For

k=5

, standard

k

-NN uses a fixed neighborhood size commonly adopted in practice. Results are reported in terms of balanced accuracy (BACC) and F1-score. The best result in each dataset and metric within each

k

configuration is highlighted in bold. The final two rows report the average and median performance across all datasets.

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Fixed Radius Near Neighbor (FRNN) search is an alternative to the widely used k Nearest Neighbors (kNN) search. Unlike kNN, FRNN determines a label or an estimate for a test sample based on all training samples within a predefined distance. While this approach is beneficial in certain scenarios, assuming a fixed maximum distance for all training samples can decrease the accuracy of the FRNN. Therefore, in this paper we propose the Adaptive Radius Near Neighbor (ARNN) and the Weighted ARNN (WARNN), which employ adaptive distances and in latter case weights. All three methods are compared to kNN and twelve of its variants for a regression problem, namely WiFi fingerprinting indoor positioning, using 22 different datasets to provide a comprehensive analysis. While the performances of the tested FRNN and ARNN versions were amongst the worse, three of the four best methods in the test were WARNN versions, indicating that using weights together with adaptive distances achieves performance comparable or even better than kNN variants.

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CuBAS: Information Geometric Curvature-Based Adaptive Sampling for Supervised Classification

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. We introduce CuBAS (Curvature-Based Adaptive Sampling), an information-geometric framework for adaptive data selection in supervised classification, grounded in the q-state Potts Markov random field (MRF) model. The central insight is that a labeled dataset can be viewed as a statistical manifold, on which local curvature, estimated via the ratio of second to first-order observed Fisher information, faithfully encodes the geometric complexity of the data distribution. We construct a k-nearest-neighbor graph over the labeled data and derive a closed-form curvature score at each vertex from the Potts sufficient statistics. This curvature signal partitions the graph into two complementary regimes: low-curvature regions, corresponding to smooth, homogeneous clusters, and high-curvature regions, concentrated around decision boundaries that are disproportionately informative for classification. By selecting nodes from both regimes, CuBAS constructs compact yet maximally informative training subsets. Empirical evaluation across more than 60 benchmark datasets demonstrates consistent and statistically significant improvements over random sampling and uncertainty-based baselines, across a wide range of labeling budgets and classifier architectures. CuBAS is computationally efficient (linear in the number of k-NN graph edges), theoretically grounded in the differential geometry of statistical manifolds, and interpretable in terms of the local shape operator of the data manifold.

Alexandre L. M. Levada

Federal University of São Carlos 13565-905, São Carlos-SP, Brazil

Information Geometry Adaptive Sampling Classification

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Abstract

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