Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification
CARSANN introduces a geometry-driven framework that adapts neighborhood spatial support based on local manifold curvature, addressing a key limitation of standard k-NN which uses uniform neighborhood cardinality across the feature space The method estimates intrinsic dimensionality via TwoNN, constructs an intrinsic representation through PCA, and computes local mean curvature using a shape-operator-based formulation to control neighborhood radius shrinkage Experiments across 70+ OpenML datasets
Analysis
TL;DR
- CARSANN introduces a geometry-driven framework that adapts neighborhood spatial support based on local manifold curvature, addressing a key limitation of standard k-NN which uses uniform neighborhood cardinality across the feature space
- The method estimates intrinsic dimensionality via TwoNN, constructs an intrinsic representation through PCA, and computes local mean curvature using a shape-operator-based formulation to control neighborhood radius shrinkage
- Experiments across 70+ OpenML datasets demonstrate consistent improvements over standard k-NN, with balanced accuracy increasing from 0.6506 to 0.7528 on 45 controlled datasets (40 out of 45 showing gains)
- The approach provides a complementary paradigm to existing adaptive nearest-neighbor methods by explicitly adapting spatial extent rather than merely modifying neighbor count or local metric
- Statistical validation via Friedman and Nemenyi tests confirms the improvements are significant and not due to chance
Why It Matters
This work addresses a fundamental limitation in nearest-neighbor classification: the assumption that a fixed number of neighbors is equally appropriate across all regions of feature space, which breaks down when data lies on manifolds with varying geometric complexity. For AI practitioners working with real-world datasets that exhibit heterogeneous local structures, CARSANN offers a principled geometric adaptation that can improve classification performance without requiring architectural changes to existing ML pipelines. The curvature-based approach opens a new direction for adaptive neighborhood methods that could be extended beyond classification to regression and clustering tasks.
Technical Details
- Intrinsic dimensionality estimation: Uses TwoNN (Two-Nearest Neighbor) method to estimate the intrinsic dimensionality of local data distributions, enabling the model to understand the true complexity of the underlying manifold rather than relying on ambient feature space dimensions
- PCA-based intrinsic representation: Constructs a lower-dimensional intrinsic representation through principal component analysis, projecting data onto the most informative subspace before curvature estimation
- Shape-operator-based curvature estimation: Computes local mean curvature using a differential geometry formulation based on the shape operator (Weingarten map), which quantifies how the manifold bends at each point in feature space
- Curvature-controlled radius shrinkage: Applies stronger radius shrinkage in highly curved regions (where neighborhoods need to be smaller to capture local structure accurately) and retains broader spatial support in approximately flat regions (where larger neighborhoods are appropriate)
- Empirical validation: Tested on 70+ real-world OpenML datasets with controlled comparisons against standard k-NN and fixed k=5 baselines, using balanced accuracy as the primary metric with statistical significance confirmed through Friedman and Nemenyi non-parametric tests
Industry Insight
- The curvature-aware adaptation paradigm could become a valuable preprocessing or augmentation step for any pipeline relying on distance-based methods, particularly in domains like bioinformatics, medical diagnosis, and fraud detection where data manifolds are known to exhibit heterogeneous geometric properties
- Practitioners should consider evaluating CARSANN as a drop-in replacement for standard k-NN when working with high-dimensional or manifold-structured data, as the computational overhead of curvature estimation is modest relative to the potential accuracy gains
- This work suggests a broader research direction: incorporating differential geometric properties (curvature, torsion, geodesic distance) into classical machine learning algorithms could yield similar improvements across other distance-dependent methods such as DBSCAN, spectral clustering, and kernel-based approaches
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