An Unbalanced Optimal Transport-Based Approach for Robust Dictionary Learning

Dictionary learning (DL) is a pivotal task in machine learning and signal processing, involving extracting representative features from a given dataset. However, conventional DL models are known to be highly sensitive to outliers. To circumvent this issue, we introduce a new and robust DL model based on unbalanced optimal transport (UOT). Compared to DL models based on conventional robust distances and the Wasserstein distance, our model not only captures and leverages the structural information within the data but also demonstrates strong resilience to outliers. By employing the structure of the proposed robust DL model, we develop a novel hybrid block coordinate descent (BCD) algorithm. The proposed algorithm maintains computational tractability by exploiting special block structures of the subproblems. In addition, we establish the convergence of our algorithm without the Lipschitz smooth condition. Through extensive experimentation, we validate our theoretical results and demonstrate the effectiveness of the proposed method on synthetic data, MNIST data, Olivetti faces dataset, and hyperspectral images (HSIs) datasets.