# Adaptive k-nearest neighbor classifier based on the local estimation of the shape operator

arXiv: https://arxiv.org/abs/2409.05084 · HTML: https://mlsanity.com/paper/2409.05084/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 37 references that are in the map.
- First posted on arXiv 8 Sep 2024.

## Builds on (most cited first, 4 of 4)

- [Blessing of dimensionality: mathematical foundations of the statistical physics of data](https://mlsanity.com/paper/1801.03421/) (2018)
- [Distance and Similarity Measures Effect on the Performance of K-Nearest Neighbor Classifier -- A Review](https://mlsanity.com/paper/1708.04321/) (2017)
- [Adaptive Estimation for Approximate k-Nearest-Neighbor Computations](https://mlsanity.com/paper/1902.09465/) (2019)
- [PL-kNN: A Parameterless Nearest Neighbors Classifier](https://mlsanity.com/paper/2209.12647/) (2022)

## Built on by (most cited first, 2 of 2)

- [Locally Adaptive One-Class Classifier Fusion with Dynamic ℓp-Norm Constraints for Robust Anomaly Detection](https://mlsanity.com/paper/2411.06406/) (2024)
- [Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification](https://mlsanity.com/paper/2608.27634/) (2026)

## Abstract (arXiv)

The k-nearest neighbor (k-NN) algorithm is one of the most popular methods for nonparametric classification. However, a relevant limitation concerns the definition of the number of neighbors k. This parameter exerts a direct impact on several properties of the classifier, such as the bias-variance tradeoff, smoothness of decision boundaries, robustness to noise, and class imbalance handling. In the present paper, we introduce a new adaptive k-nearest neighbours (kK-NN) algorithm that explores the local curvature at a sample to adaptively defining the neighborhood size. The rationale is that points with low curvature could have larger neighborhoods (locally, the tangent space approximates well the underlying data shape), whereas points with high curvature could have smaller neighborhoods (locally, the tangent space is a loose approximation). We estimate the local Gaussian curvature by computing an approximation to the local shape operator in terms of the local covariance matrix as well as the local Hessian matrix. Results on many real-world datasets indicate that the new kK-NN algorithm yields superior balanced accuracy compared to the established k-NN method and also another adaptive k-NN algorithm. This is particularly evident when the number of samples in the training data is limited, suggesting that the kK-NN is capable of learning more discriminant functions with less data considering many relevant cases.