Semidefinite programming relaxations for matrix completion, inverse scattering and blind deconvolution

The first instance, rank one matrix completion, was known to be solved by non-linear propagation algorithms without stability guarantees. The thesis closes the line of work on this problem by introducing a stable algorithm based on two levels of semidefinite programming relaxation. For this algorithm, recovery of the unknown matrix is first certified in the absence of noise, at the information limit, through the construction of a dual (sum of squares) polynomial. In passing, this dual polynomial also provides a rationale for the use of the trace norm in semidefinite programming. The dual polynomial is then used to derive a stability estimate for the noisy version of the problem.