Adaptive $k$-nearest neighbor classifier based on the local estimation of the shape operator
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.
Submission Length: Regular submission (no more than 12 pages of main content)
Previous TMLR Submission Url: https://openreview.net/forum?id=ol9Pl2Ogtp
Changes Since Last Submission: We would like to thank the Editor and reviewers for their careful reading and constructive comments on our paper. In this revision, we have addressed all reviewers' comments, and an extensive point-by-point reply is provided in our "Reply Letter" (8 pages) attached to this submission as a Supplementary Material, for your appreciation. We greatly appreciate the encouragement to revise and resubmit our work through this new (re)submission.
Assigned Action Editor: ~Alessandro_Sperduti1
Submission Number: 3036
Loading