Weakly supervised learning on pre-image problem in kernel methods

This paper presents a novel alternative approach, namely weakly supervised learning (WSL), to learn the pre-image of a feature vector in the feature space induced by a kernel. It is known that the exact pre-image may typically seldom exist, since the input space and the feature space are not isomorphic in general, and an approximate solution is required in past. The proposed WSL, however, would find an appropriate rather than only a purely approximate solution. WSL is able to involve some weakly supervised prior knowledge into the study of pre-image. The prior knowledge is weak and no class label of the sample is required, providing only information of positive class and negative class which should properly depend on applications. The proposed algorithm is demonstrated on kernel principal component analysis (KPCA) with application to illumination normalization and image denoising on faces. Evaluations of the performance of the proposed algorithm show notable improvement as comparing with some well-known existing approaches.