A Graph-Based Semi-Supervised k Nearest-Neighbor Method for Nonlinear Manifold Distributed Data Classification

TL;DR

This paper introduces a graph-based semi-supervised kNN algorithm that effectively classifies nonlinear manifold distributed data by using a constrained Tired Random Walk for similarity measurement, outperforming traditional kNN methods.

Contribution

The paper proposes a novel graph-based kNN method utilizing constrained Tired Random Walks to improve classification on nonlinear manifold data, including an online version for sequential samples.

Findings

Effective classification of nonlinear manifold data.

Improved accuracy over traditional kNN algorithms.

Demonstrated robustness on synthetic and real-world datasets.

Abstract

$k$ Nearest Neighbors ($k$NN) is one of the most widely used supervised learning algorithms to classify Gaussian distributed data, but it does not achieve good results when it is applied to nonlinear manifold distributed data, especially when a very limited amount of labeled samples are available. In this paper, we propose a new graph-based $k$NN algorithm which can effectively handle both Gaussian distributed data and nonlinear manifold distributed data. To achieve this goal, we first propose a constrained Tired Random Walk (TRW) by constructing an $R$-level nearest-neighbor strengthened tree over the graph, and then compute a TRW matrix for similarity measurement purposes. After this, the nearest neighbors are identified according to the TRW matrix and the class label of a query point is determined by the sum of all the TRW weights of its nearest neighbors. To deal with online situations, we also propose a new algorithm to handle sequential samples based a local neighborhood reconstruction. Comparison experiments are conducted on both synthetic data sets and real-world data sets to demonstrate the validity of the proposed new $k$NN algorithm and its improvements to other version of $k$NN algorithms. Given the widespread appearance of manifold structures in real-world problems and the popularity of the traditional $k$NN algorithm, the proposed manifold version $k$NN shows promising potential for classifying manifold-distributed data.