# Total Least Squares Phase Retrieval

arXiv: https://arxiv.org/abs/2102.00927 · HTML: https://mlsanity.com/paper/2102.00927/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 42 references that are in the map.
- First posted on arXiv 1 Feb 2021.

## Builds on (most cited first, 16 of 16)

- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization](https://mlsanity.com/paper/1607.05447/) (2016)
- [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)
- [Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow](https://mlsanity.com/paper/1605.08285/) (2016)
- [Phase Retrieval from Coded Diffraction Patterns](https://mlsanity.com/paper/1310.3240/) (2013)
- [PhaseMax: Convex Phase Retrieval via Basis Pursuit](https://mlsanity.com/paper/1610.07531/) (2016)
- [Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow](https://mlsanity.com/paper/1506.03382/) (2015)
- [Random Projections through multiple optical scattering: Approximating kernels at the speed of light](https://mlsanity.com/paper/1510.06664/) (2015)
- [Sketchy Decisions: Convex Low-Rank Matrix Optimization with Optimal Storage](https://mlsanity.com/paper/1702.06838/) (2017)
- [Sparse Phase Retrieval via Truncated Amplitude Flow](https://mlsanity.com/paper/1611.07641/) (2016)
- [Reconstructing Point Sets from Distance Distributions](https://mlsanity.com/paper/1804.02465/) (2018)
- [Don't take it lightly: Phasing optical random projections with unknown operators](https://mlsanity.com/paper/1907.01703/) (2019)
- [Fast Optical System Identification by Numerical Interferometry](https://mlsanity.com/paper/1911.01006/) (2019)
- [Convex Total Least Squares](https://mlsanity.com/paper/1406.0189/) (2014)
- [Accelerating ptychographic reconstructions using spectral initializations](https://mlsanity.com/paper/2007.14139/) (2020)

## Built on by (most cited first, 2 of 2)

- [Glimpse: Generalized Locality for Scalable and Robust CT](https://mlsanity.com/paper/2401.00816/) (2024)
- [Differentiable Uncalibrated Imaging](https://mlsanity.com/paper/2211.10525/) (2022)

## Abstract (arXiv)

We address the phase retrieval problem with errors in the sensing vectors. A number of recent methods for phase retrieval are based on least squares (LS) formulations which assume errors in the quadratic measurements. We extend this approach to handle errors in the sensing vectors by adopting the total least squares (TLS) framework that is used in linear inverse problems with operator errors. We show how gradient descent and the specific geometry of the phase retrieval problem can be used to obtain a simple and efficient TLS solution. Additionally, we derive the gradients of the TLS and LS solutions with respect to the sensing vectors and measurements which enables us to calculate the solution errors. By analyzing these error expressions we determine conditions under which each method should outperform the other. We run simulations to demonstrate that our method can lead to more accurate solutions. We further demonstrate the effectiveness of our approach by performing phase retrieval experiments on real optical hardware which naturally contains both sensing vector and measurement errors.