Fast adaptive tubal rank-revealing algorithm for t-product based tensor approximation

Color images and video sequences can be modeled as three-way tensors, which admit low tubal-rank approximations via convex surrogate minimization. This optimization problem is efficiently addressed by tensor singular value thresholding (t-SVT). To mitigate the computational burden of tensor singular value decomposition (t-SVD) in each iteration, this paper introduces an adaptive randomized algorithm for tubal rank revelation in data tensors \(\mathcal{A}\). Our method selectively captures the principal information from frontal slices in the Fourier domain using a predefined threshold, obviating the need for priori tubal-rank and Fourier-domain singular values estimations while providing an explicit tensor approximation. Leveraging optimality results from matrix randomized SVD, we establish theoretical guarantees demonstrating that the proposed algorithm computes low tubal-rank approximations within constants dependent on data dimensions and the Fourier-domain singular value gap. Empirical evaluations validate its efficacy in image processing and background modeling tasks.