# Low-Rank Phase Retrieval with Structured Tensor Models

arXiv: https://arxiv.org/abs/2202.08260 · HTML: https://mlsanity.com/paper/2202.08260/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 21 references that are in the map.
- First posted on arXiv 15 Feb 2022.

## Builds on (most cited first, 9 of 9)

- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Phase Retrieval using Alternating Minimization](https://mlsanity.com/paper/1306.0160/) (2013)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [Tucker Tensor Regression and Neuroimaging Analysis](https://mlsanity.com/paper/1304.5637/) (2013)
- [ISLET: Fast and Optimal Low-rank Tensor Regression via Importance Sketching](https://mlsanity.com/paper/1911.03804/) (2019)
- [Learning Mixtures of Separable Dictionaries for Tensor Data: Analysis and Algorithms](https://mlsanity.com/paper/1903.09284/) (2019)
- [Toward Long Distance, Sub-diffraction Imaging Using Coherent Camera Arrays](https://mlsanity.com/paper/1510.08470/) (2015)
- [Low-Rank Phase Retrieval via Variational Bayesian Learning](https://mlsanity.com/paper/1811.01574/) (2018)

## Abstract (arXiv)

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.