Fast Discriminant Analysis with Adaptive Reconstruction Structure Preserving

Neighborhood reconstruction methods have been widely applied to feature engineering. Existing reconstruction-based discriminant analysis methods normally project high-dimensional data into a low-dimensional space while preserving the reconstruction relationships among samples. However, there are three limitations: 1) the reconstruction coefficients are learned based on the collaborative representation of all sample pairs, which requires the training time to be the cube of the number of samples; 2) these coefficients are learned in the original space, ignoring the interference of the noise and redundant features; and 3) there is a reconstruction relationship between heterogeneous samples; this will enlarge the similarity of heterogeneous samples in the subspace. In this article, we propose a fast and adaptive discriminant neighborhood projection model to tackle the above drawbacks. First, the local manifold structure is captured by bipartite graphs in which each sample is reconstructed by anchor points derived from the same class as that sample; this can avoid the reconstruction between heterogeneous samples. Second, the number of anchor points is far less than the number of samples; this strategy can reduce the time complexity substantially. Third, anchor points and reconstruction coefficients of bipartite graphs are updated adaptively in the process of dimensionality reduction, which can enhance the quality of bipartite graphs and extract discriminative features simultaneously. An iterative algorithm is designed to solve this model. Extensive results on toy data and benchmark datasets show the effectiveness and superiority of our model.