Adaptive $k$-nearest neighbor classifier based on the local estimation of the shape operator (2409.05084v1)

Abstract: 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.

• The paper introduces kK-NN, a classifier that adaptively determines the number of neighbors based on local Gaussian curvature.
• It employs local covariance and Hessian matrices to estimate curvature, dynamically tailoring decision boundaries to data density.
• Experiments on 30 datasets demonstrate that kK-NN outperforms traditional k-NN methods, particularly with limited training data.

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

This paper, authored by Alexandre Luís Magalhães Levada, Frank Nielsen, and Michel Ferreira Cardia Haddad, presents a novel variant of the $k$-nearest neighbor ($k$-NN) classification algorithm, termed the $kK$-NN. This approach leverages local geometric properties to adaptively determine the number of neighbors, $k$, required for classification, which aims to address several inherent limitations of the traditional $k$-NN algorithm, including bias-variance tradeoff, decision boundary smoothness, robustness to noise, and class imbalance handling.

Methodology

The $kK$-NN algorithm adjusts the neighborhood size by utilizing the local Gaussian curvature. The core idea is that points with low curvature can have larger neighborhoods, approximating the local data shape well, whereas points with high curvature should have smaller neighborhoods due to the poor approximation by the tangent space. This is achieved through an innovative process involving the estimation of the local shape operator.

The shape operator's curvature is approximated via local covariance and Hessian matrices. Specifically, the local covariance matrix is used to estimate the metric tensor, while the Hessian matrix is used to compute the second fundamental form. By calculating the determinant of the product of these matrices, the algorithm approximates the local Gaussian curvature, thereby informing the adaptive neighborhood size for each data point.

The training phase of the $kK$-NN involves constructing a $k$-NN graph with $k = log_2~n$, where $n$ is the number of samples. After computing the curvature for all graph vertices and quantizing these curvatures into ten scores, the neighborhood size is adjusted by pruning edges in the $k$-NN graph based on these scores. For instance, a sample with a lower curvature retains more neighbors compared to a sample with higher curvature.

Results

The paper reports extensive computational experiments involving 30 real-world datasets. The authors identify that $kK$-NN consistently yields higher balanced accuracy than both the traditional $k$-NN and a rival adaptive $k$-NN algorithm. This superior performance is particularly evident in scenarios with limited training data. The introduction of local curvature information allows $kK$-NN to:

1. Avoid underfitting and overfitting by dynamically adjusting the neighborhood size.
2. Tailor decision boundaries in denser regions while smoothing them in sparser regions, thus adapting classification to the local feature space.
3. Isolate outliers, as high curvature points are often outliers and thus assigned fewer neighbors.

Implications and Discussion

The $kK$-NN algorithm's ability to dynamically adapt neighborhood sizes based on local geometric properties introduces a significant improvement in non-parametric classification methods. In practice, this method can better handle the complexities of real-world datasets, which often include noise and varying local densities. The $kK$-NN classifier's robustness to noise and outliers combined with its tailored decision boundaries can potentially find applications across various domains including computer vision, pattern recognition, and other fields where the $k$-NN algorithm is traditionally applied.

Future Directions

The authors suggest several avenues for future research:

1. Theoretical Investigations: Further exploration of the theoretical properties of curvature-adaptive classifiers, including convergence properties and theoretical performance bounds.
2. Image Processing: Application of $kK$-NN to image processing tasks, which can significantly benefit from local geometric adaptation.
3. Dimensionality Reduction and Metric Learning: Employing curvature-adaptive approaches in dimensionality reduction and metric learning tasks. This involves integrating shape operator-based adaptations as part of manifold learning techniques to facilitate improved clustering and classification in high-dimensional spaces.

The computational complexity of the proposed method, particularly the curvature estimation, remains a caveat. Given large datasets, preprocessing steps like Principal Component Analysis (PCA) might be required to manage the higher computational demand, ensuring the approach remains feasible in practice.

In summary, the $kK$-NN algorithm represents a methodologically sound and practically significant enhancement of the traditional $k$-NN approach. By embedding a curvature-based adaptive strategy, this approach effectively addresses several limitations associated with fixed parameter settings, yielding a classifier that is more flexible and better suited to handle the intricate structures typically present in real-world datasets.