Complete theoretical guarantees for tensor phase retrieval

Develop a rigorous, provably accurate, and computationally efficient theoretical framework for tensor phase retrieval of low Tucker-rank tensors under Gaussian sensing, providing recovery algorithms and performance guarantees that do not rely on specialized initializations or restrictive assumptions.

Background

Phase retrieval seeks to recover a signal from magnitude-only measurements, and extending it to tensors with low Tucker rank introduces additional algebraic and computational challenges. The paper emphasizes that, compared to vectors and matrices, tensor estimation results lag behind; although this work establishes local convergence under RAIC given a good initialization, a comprehensive theory that guarantees recovery without such prerequisites has not been established.

The authors position their contributions relative to the literature by noting that general theoretical results for tensor phase retrieval had not been available, motivating the need for a complete treatment beyond local analyses.