# Optimal spectral initializers impact on phase retrieval phase transitions -- an RDT view

arXiv: https://arxiv.org/abs/2506.18279 · HTML: https://mlsanity.com/paper/2506.18279/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 115 references that are in the map.
- First posted on arXiv 23 Jun 2025.

## Builds on (most cited first, 30 of 40)

- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/1708.03395/) (2017)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution](https://mlsanity.com/paper/1711.10467/) (2017)
- [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)
- [Subdominant Dense Clusters Allow for Simple Learning and High Computational Performance in Neural Networks with Discrete Synapses](https://mlsanity.com/paper/1509.05753/) (2015)
- [Fundamental Limits of Weak Recovery with Applications to Phase Retrieval](https://mlsanity.com/paper/1708.05932/) (2017)
- [Phase retrieval in high dimensions: Statistical and computational phase transitions](https://mlsanity.com/paper/2006.05228/) (2020)
- [Inverting Deep Generative models, One layer at a time](https://mlsanity.com/paper/1906.07437/) (2019)
- [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)
- [Phase Transitions of Spectral Initialization for High-Dimensional Nonconvex Estimation](https://mlsanity.com/paper/1702.06435/) (2017)
- [PhaseMax: Convex Phase Retrieval via Basis Pursuit](https://mlsanity.com/paper/1610.07531/) (2016)
- [Regularly random duality](https://mlsanity.com/paper/1303.7295/) (2013)
- [Another look at the Gardner problem](https://mlsanity.com/paper/1306.3979/) (2013)
- [Limits of local algorithms over sparse random graphs](https://mlsanity.com/paper/1304.1831/) (2013)
- [Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow](https://mlsanity.com/paper/1506.03382/) (2015)
- [Local entropy as a measure for sampling solutions in Constraint Satisfaction Problems](https://mlsanity.com/paper/1511.05634/) (2015)
- [Optimization of the Sherrington-Kirkpatrick Hamiltonian](https://mlsanity.com/paper/1812.10897/) (2018)
- [Exact asymptotics for phase retrieval and compressed sensing with random generative priors](https://mlsanity.com/paper/1912.02008/) (2019)
- [Fully lifted random duality theory](https://mlsanity.com/paper/2312.00070/) (2023)
- [Phase retrieval with random Gaussian sensing vectors by alternating projections](https://mlsanity.com/paper/1609.03088/) (2016)
- [Phase Retrieval via Randomized Kaczmarz: Theoretical Guarantees](https://mlsanity.com/paper/1706.09993/) (2017)
- [Structured signal recovery from quadratic measurements: Breaking sample complexity barriers via nonconvex optimization](https://mlsanity.com/paper/1702.06175/) (2017)
- [Clustering of solutions in the symmetric binary perceptron](https://mlsanity.com/paper/1911.06756/) (2019)
- [Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation](https://mlsanity.com/paper/1610.04210/) (2016)

## Abstract (arXiv)

We analyze the relation between spectral initializers and theoretical limits of descending phase retrieval algorithms (dPR). In companion paper [104], for any sample complexity ratio, α, parametric manifold, mathcal PM(α), is recognized as a critically important structure that generically determines dPRs abilities to solve phase retrieval (PR). Moreover, overlap between the algorithmic solution and the true signal is positioned as a key mathcal PM's component. We here consider the so-called overlap optimal spectral initializers (OptSpins) as dPR's starting points and develop a generic Random duality theory (RDT) based program to statistically characterize them. In particular, we determine the functional structure of OptSpins and evaluate the starting overlaps that they provide for the dPRs. Since mathcal PM's so-called flat regions are highly susceptible to local jitteriness and as such are key obstacles on dPR's path towards PR's global optimum, a precise characterization of the starting overlap allows to determine if such regions can be successfully circumvented. Through the presented theoretical analysis we observe two key points in that regard: textbf(i) dPR's theoretical phase transition (critical α above which they solve PR) might be difficult to practically achieve as the mathcal PM's flat regions are large causing the associated OptSpins to fall exactly within them; and textbf(ii) Opting for so-called ``safer compression'' and slightly increasing α (by say 15%) shrinks flat regions and allows OptSpins to fall outside them and dPRs to ultimately solve PR. Numerical simulations are conducted as well and shown to be in an excellent agreement with theoretical predictions.