SURE Based Truncated Tensor Nuclear Norm Regularization for Low Rank Tensor Completion

Low rank tensor completion aims to recover the underlying low rank tensor obtained from its partial observations, this has a wide range of applications in Signal Processing and Machine Learning. A number of recent low rank tensor methods have successfully utilised the tensor singular value decomposition method with tensor nuclear norm minimisation via tensor singular value thresholding. This approach while proving to be effective has the potential issue that it may over or under shrink the singular values which will effect the overall performance. A truncated nuclear norm based method has been introduced which explicitly exploits the low rank assumption within the optimization in combination with tensor singular value thresholding. In this work the truncated nuclear norm approach is extended to incorporate a data driven approach based on Stein's unbiased risk estimation method which efficiently thresholds the singular values. Experimental results in a colour image denoising problem demonstrate the efficiency and accuracy of the method.