A structure-protecting kernelized semi-supervised space adjustment for classification

Abstract

We introduce a semi-supervised space adjustment framework in this paper. In the introduced framework, the dataset contains two subsets: (a) training data subset (space-one data (SOD)) and (b) testing data subset (space-two data (STD)). Our semi-supervised space adjustment framework learns under three assumptions: (I) it is assumed that all data points in the SOD are labeled, and only a minority of the data points in the STD are labeled (we call the labeled space-two data as LSTD), (II) the size of LSTD is very small comparing to the size of SOD, and (III) it is also assumed that the data of SOD and the data of STD have different distributions. We denote the unlabeled space-two data by ULSTD, which is equal to STD - LSTD. The aim is to map the training data, i.e., the data from the training labeled data subset and those from LSTD (note that all labeled data are considered to be training data, i.e., SOD ∪ LSTD) into a shared space (ShS). The mapped SOD, ULSTD, and LSTD into ShS are named MSOD, MULSTD, and MLSTD, respectively. The proposed method does the mentioned mapping in such a way that structures of the data points in SOD and MSOD, in STD and MSTD, in ULSTD and MULSTD, and in LSTD and MLSTD are the same. In the proposed method, the mapping is proposed to be done by a principal component analysis transformation on kernelized data. In the proposed method, it is tried to find a mapping that (a) can maintain the neighbors of data points after the mapping and (b) can take advantage of the class labels that are known in STD during transformation. After that, we represent and formulate the problem of finding the optimal mapping into a non-linear objective function. To solve it, we transform it into a semidefinite programming (SDP) problem. We solve the optimization problem with an SDP solver. The examinations indicate the superiority of the learners trained in the data mapped by the proposed approach to the learners trained in the data mapped by the state of the art methods.