joint work with Brahim Tamadazte, Assoweh Mohamed Ibrahim
In this paper, we study the low-tubal-rank tensor completion problem, i.e., to recover a third-order tensor by observing a subset of elements selected uniformly at random. We propose a fast iterative algorithm, called Tubal-Alt-Min. The unknown low-tubal-rank tensor is parameterised as the product of two much smaller tensors which naturally encode the low-tubal-rank property. Tubal-Alt-Min alternates between estimating those two tensors using least squares minimisation. We establish a "local is global" property for the minimisers, which implies convergence of Tubal-Alt-Min to global solutions.
joint work with Jian-Feng Cai, Tianming Wang
Spectral compressed sensing is about reconstructing spectrally sparse data from incomplete information. Two difference classes of nonconvex optimization algorithms are introduced to tackle this problem. Theoretical recovery guarantees will be presented for the proposed algorithms under certain random models, showing that the sampling complexity is essentially proportional to the intrinsic dimension of the problems rather the ambient dimension. Empirical observations demonstrate the efficacy of the algorithms.