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

$k$ 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.Comment: 32 pages, 12 figures, 7 table

spa: Semi-Supervised Semi-Parametric Graph-Based Estimation in R

In this paper, we present an R package that combines feature-based (X) data and graph-based (G) data for prediction of the response Y . In this particular case, Y is observed for a subset of the observations (labeled) and missing for the remainder (unlabeled). We examine an approach for fitting Y = Xò + f(G) where ò is a coefficient vector and f is a function over the vertices of the graph. The procedure is semi-supervised in nature (trained on the labeled and unlabeled sets), requiring iterative algorithms for fitting this estimate. The package provides several key functions for fitting and evaluating an estimator of this type. The package is illustrated on a text analysis data set, where the observations are text documents (papers), the response is the category of paper (either applied or theoretical statistics), the X information is the name of the journal in which the paper resides, and the graph is a co-citation network, with each vertex an observation and each edge the number of times that the two papers cite a common paper. An application involving classification of protein location using a protein interaction graph and an application involving classification on a manifold with part of the feature data converted to a graph are also presented.