Supervised Non-negative Matrix Factorization Induced by Huber Loss

Abstract

Non-negative matrix factorization (NMF) has the ability for non-negative feature extraction and is successfully exploited for parts-based image representation. Most NMF-based algorithms utilize loss function with $l_2$-norm or Kullback-Leibler divergence to evaluate the quality of factorization. However, these measurements are sensitive to noise and outliers. Also, NMF is an unsupervised learning method and thus cannot acquire the desired performance in classification tasks. To address the problems of the NMF algorithm, this paper proposes a supervised non-negative matrix factorization (HSNMF) approach using Huber loss function, which is more robust to noise and outliers than $l_2$-norm. To enhance the discriminative power of NMF, we establish the objective function by incorporating two quantities including intra-class and inter-class information into the Huber loss function. The updating rules of HSNMF are derived using KKT conditions. The proposed HSNMF is shown to be convergent via the auxiliary function technique. Experimental results on face recognition demonstrate the robustness and superior performance of our algorithm when compared with the state-of-the-art algorithms.