Adaptive and fuzzy locality discriminant analysis for dimensionality reduction

Linear discriminant analysis (LDA) uses labeled samples for acquiring a discriminant projection direction, by which data of different categories are separated into distinct groups in a lower-dimensional subspace. In response to the issue that LDA lacks robustness to non-Gaussian data with rich local information, improvements on LDA explore the subspace manifold structure by building a fully connected similarity graph. However, these methods are vulnerable to the interference of noisy and redundant information. In this paper, we propose a new local LDA method, named adaptive and fuzzy locality discriminant analysis (AFLDA), which aims at extracting precise and compact features. Firstly, an adaptive and fuzzy k-means strategy is adopted, where the membership between data and corresponding subclass centers for each class is learned to establish a hybrid bipartite graph and capture local information. Secondly, we design a discrete and probability constraint imposed on the membership matrix to explore the intricate structure of multimodal data. Moreover, the subblock partition for each class makes data accommodate to multimodal subclasses. Thirdly, the maximum total scatter regularization term is introduced, which amply disperses the data to enhance the recognition of local structure and avoid trivial solutions. Finally, to eliminate the interference of noisy and redundant information, AFLDA learns an optimized subspace, where cluster centers and the membership matrix are updated alternately. Promising results in experiments illustrate the efficacy of the model.