Low-Rank Phase Retrieval with Structured Tensor Models

We study the low-rank phase retrieval problem, where the objective is to recover a sequence of signals (typically images) given the magnitude of linear measurements of those signals. Existing solutions involve recovering a matrix constructed by vectorizing and stacking each image. These algorithms model this matrix to be low-rank and leverage the low-rank property to decrease the sample complexity required for accurate recovery. However, when the number of available measurements is more limited, these low-rank matrix models can often fail. We propose an algorithm called Tucker-Structured Phase Retrieval (TSPR) that models the sequence of images as a tensor rather than a matrix that we factorize using the Tucker decomposition. This factorization reduces the number of parameters that need to be estimated, allowing for a more accurate reconstruction in the under-sampled regime. Interestingly, we observe that this structure also has improved performance in the over-determined setting when the Tucker ranks are chosen appropriately. We demonstrate the effectiveness of our approach on real video datasets under several different measurement models.