Exact Tensor Completion from Sparsely Corrupted Observations via Convex Optimization
This provides a theoretical foundation for tensor completion and robust PCA in applications like image and video processing, though it is incremental as it builds on existing tensor algebraic frameworks.
The paper tackles the problem of recovering a third-order tensor from a random subset of its corrupted entries by solving a convex optimization program, proving exact recovery with high probability under conditions of low tubal rank and sparse corruptions, and validating the approach with numerical experiments.
This paper conducts a rigorous analysis for provable estimation of multidimensional arrays, in particular third-order tensors, from a random subset of its corrupted entries. Our study rests heavily on a recently proposed tensor algebraic framework in which we can obtain tensor singular value decomposition (t-SVD) that is similar to the SVD for matrices, and define a new notion of tensor rank referred to as the tubal rank. We prove that by simply solving a convex program, which minimizes a weighted combination of tubal nuclear norm, a convex surrogate for the tubal rank, and the $\ell_1$-norm, one can recover an incoherent tensor exactly with overwhelming probability, provided that its tubal rank is not too large and that the corruptions are reasonably sparse. Interestingly, our result includes the recovery guarantees for the problems of tensor completion (TC) and tensor principal component analysis (TRPCA) under the same algebraic setup as special cases. An alternating direction method of multipliers (ADMM) algorithm is presented to solve this optimization problem. Numerical experiments verify our theory and real-world applications demonstrate the effectiveness of our algorithm.