Worst-Case Discriminative Feature Learning via Max-Min Ratio Analysis

We propose a novel discriminative feature learning method via Max-Min Ratio Analysis (MMRA) for exclusively dealing with the long-standing "worst-case class separation" problem. Existing technologies simply consider maximizing the minimal pairwise distance on all class pairs in the low-dimensional subspace, which is unable to separate overlapped classes entirely especially when the distribution of samples within same class is diverging. We propose a new criterion, i.e., Max-Min Ratio Analysis (MMRA) that focuses on maximizing the minimal ratio value of between-class and within-class scatter to extremely enlarge the separability on the overlapped pairwise classes. Furthermore, we develop two novel discriminative feature learning models for dimensionality reduction and metric learning based on our MMRA criterion. However, solving such a non-smooth non-convex max-min ratio problem is challenging. As an important theoretical contribution in this paper, we systematically derive an alternative iterative algorithm based on a general max-min ratio optimization framework to solve a general max-min ratio problem with rigorous proofs of convergence. More importantly, we also present another solver based on bisection search strategy to solve the SDP problem efficiently. To evaluate the effectiveness of proposed methods, we conduct extensive pattern classification and image retrieval experiments on several artificial datasets and real-world ScRNA-seq datasets, and experimental results demonstrate the effectiveness of proposed methods.