揭示了传统相位恢复与梯度优化的数学等价性，打通了经典方法与可微物理模型的桥梁。
On the conditional equivalence of phase retrieval algorithms
- 将GS算法的幅度替换视为幅值平方损失的单位梯度下降
- 证明相位恢复在高斯幅度噪声下等价于负对数似然最大化
- 提供局部贝叶斯更新视角，指导迭代松弛策略设计
相位恢复——从强度测量中恢复复值场——通常通过Gerchberg-Saxton（GS）算法变体求解，该方法被理解为在测量平面间交替投影。然而，现代计算成像越来越多依赖基于梯度的优化和自动微分。本文表明，这两种方法在数学上完全等价：GS算法中的幅度替换步骤恰好是幅值最小二乘损失的单位梯度下降。这一等价性使得经典相位恢复方法能无缝集成到可微物理流水线中。我们进一步识别出两种互补的概率解释：全局上，幅值损失即高斯幅度噪声下的负对数似然；局部上，每一步投影可视为以传播场为先验的贝叶斯更新。局部视角为迭代相位恢复中的松弛策略提供了定性指导。
原文摘要 · Abstract (English)
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.
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