# On the conditional equivalence of phase retrieval algorithms

arXiv: https://arxiv.org/abs/2606.07257 · HTML: https://mlsanity.com/paper/2606.07257/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 25 references that are in the map.
- First posted on arXiv 5 Jun 2026.

## Builds on (most cited first, 5 of 5)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Adorym: A multi-platform generic x-ray image reconstruction framework based on automatic differentiation](https://mlsanity.com/paper/2012.12686/) (2020)
- [Using Automatic Differentiation as a General Framework for Ptychographic Reconstruction](https://mlsanity.com/paper/1902.03920/) (2019)
- [Sparse Reconstruction of Wavefronts using an Over-Complete Phase Dictionary](https://mlsanity.com/paper/2411.02985/) (2024)

## Abstract (arXiv)

Phase retrieval - recovering a complex-valued field from intensity measurements - is typically solved using variants of the Gerchberg-Saxton (GS) algorithm, understood as alternating projections between measurement planes. Meanwhile, modern computational imaging increasingly relies on gradient-based optimization and automatic differentiation. Here we show that these two approaches are mathematically identical: the GS magnitude replacement step is exactly a unit gradient descent step on an amplitude least-squares loss. This equivalence enables seamless integration of classical phase retrieval with differentiable physics pipelines. We further identify two complementary probabilistic interpretations of this equivalence: globally, the amplitude loss is the negative log-likelihood under Gaussian amplitude noise; locally, each projection step arises as a Bayesian update with the propagated field as prior. The local view provides qualitative guidance for relaxation in iterative phase retrieval.