Adaptive local linear discriminant analysis

Dimensionality reduction plays a significant role in high-dimensional data processing, and Linear Discriminant Analysis (LDA) is a widely used supervised dimensionality reduction approach. However, a major drawback of LDA is that it is incapable of extracting the local structure information, which is crucial for handling multimodal data. In this article, we propose a novel supervised dimensionality reduction method named Adaptive Local Linear Discriminant Analysis (ALLDA), which adaptively learns a k-nearest neighbors graph from data themselves to extract the local connectivity of data. Furthermore, the original high-dimensional data usually contains noisy and redundant features, which has a negative impact on the evaluation of neighborships and degrades the subsequent classification performance. To address this issue, our method learns the similarity matrix and updates the subspace simultaneously so that the neighborships can be evaluated in the optimal subspaces where the noises have been removed. Through the optimal graph embedding, the underlying sub-manifolds of data in intra-class can be extracted precisely. Meanwhile, an efficient iterative optimization algorithm is proposed to solve the minimization problem. Promising experimental results on synthetic and real-world datasets are provided to evaluate the effectiveness of proposed method.