Provable initialization for tensor phase retrieval

Develop a provably accurate initialization method for Riemannian gradient descent in low Tucker-rank tensor phase retrieval under Gaussian sensing, ensuring the initialization satisfies the necessary proximity condition to the ground truth (for example, |X0 − X|F ≤ c Amin(X)) with high probability.

Background

The authors provide local convergence guarantees for tensor phase retrieval conditioned on an accurate initialization that lies within a specified radius of the ground truth, which is standard in Riemannian optimization for tensors. However, they explicitly state that they do not have a provably accurate initialization procedure and instead use a heuristic based on mixing the ground truth with a random tensor.

This highlights an unresolved component necessary for end-to-end guarantees: a statistically justified initializer that works from measurements alone and places the iterate within the basin of attraction required by the local analysis.