Randomized Tensor Robust Principal Component Analysis

Removing noise from hyperspectral images can be very beneficial for improving classification accuracy. Recently, tensor robust principal component analysis (TRPCA) has been successfully employed to reduce noise in hyperspectral images. In TRPCA, a minimization involving a tensor nuclear norm and a ℓ1-norm is employed to separate the low-rank hyperspectral image from the sparse noise. Tensor nuclear norm minimization is solved by iteratively performing tensor singular value thresholding (T-SVT). However, TRPCA possesses high computational complexity primarily due to the implementation of the T-SVT operator. The conventional approach for T-SVT is first to perform full tensor singular value decomposition (T-SVD), and then to shrink the singular values of the frontal slices in the frequency domain. However, when the solution of tensor nuclear norm minimization is a low-rank tensor, a good strategy is to incrementally find the singular values until they fall below the threshold. In this letter, we propose a randomized blocked algorithm for computing tensor singular value thresholding, and we leverage the compression attained by the fast Fourier transform (FFT) to accelerate TRPCA. Numerical experiments indicate that our method is significantly faster performing than TRPCA via the full T-SVD while maintaining classification accuracy.