δ-Norm-Based Robust Regression With Applications to Image Analysis

Up to now, various matrix norms (e.g., $ {l_{1}}$ -norm, $ {l_{2}}$ -norm, $ {l_{2,1}}$ -norm, etc.) have been widely leveraged to form the loss function of different regression models, and have played an important role in image analysis. However, the previous regression models adopting the existing norms are sensitive to outliers and, thus, often bring about unsatisfactory results on the heavily corrupted images. This is because their adopted norms for measuring the data residual can hardly suppress the negative influence of noisy data, which will probably mislead the regression process. To address this issue, this paper proposes a novel $ {\delta }$ (delta)-norm to count the nonzero blocks around an element in a vector or matrix, which weakens the impacts of outliers and also takes the structure property of examples into account. After that, we present the $ {\delta }$ -norm-based robust regression (DRR) in which the data examples are mapped to the kernel space and measured by the proposed $ {\delta }$ -norm. By exploring an explicit kernel function, we show that DRR has a closed-form solution, which suggests that DRR can be efficiently solved. To further handle the influences from mixed noise, DRR is extended to a multiscale version. The experimental results on image classification and background modeling datasets validate the superiority of the proposed approach to the existing state-of-the-art robust regression models.