Guaranteed Tensor Decomposition: A Moment Approach

We develop a theoretical and computational framework to perform guaranteed tensor decomposition, which also has the potential to accomplish other tensor tasks such as tensor completion and denoising. We formulate tensor decomposition as a problem of measure estimation from moments. By constructing a dual polynomial, we demonstrate that measure optimization returns the correct CP decomposition under an incoherence condition on the rank-one factors. To address the computational challenge, we present a hierarchy of semidefinite programs based on sums-of-squares relaxations of the measure optimization problem. By showing that the constructed dual polynomial is a sum-of-squares modulo the sphere, we prove that the smallest SDP in the relaxation hierarchy is exact and the decomposition can be extracted from the solution under the same incoherence condition. One implication is that the tensor nuclear norm can be computed exactly using the smallest SDP as long as the rank-one factors of the tensor are incoherent. Numerical experiments are conducted to test the performance of the moment approach.