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Phase retrieval with semi-algebraic and ReLU neural network priors

Tamir Bendory, Nadav Dym, Dan Edidin, Arun Suresh

TL;DR

The paper establishes theoretical guarantees for phase retrieval under semi-algebraic priors, showing that the power spectrum uniquely identifies signals up to sign when the ambient dimension $N$ exceeds a multiple of the prior dimension $M$ (e.g., $N \ge 2M$, or $N \ge 4M$ for universal recovery). It extends these results to generic rotations and to multiplicity-free multi-reference alignment models, deriving corresponding dimension thresholds and providing sample-complexity implications in the high-noise regime. A key contribution is the unification of the second-moment approach with fiber-dimension counting, enabling precise identifiability statements for diverse priors, including neural-network generative models with ReLU activations and sparse representations. The work also presents concrete extensions to band-limited functions on the sphere and outlines future directions for cryo-EM and analytic activation functions, highlighting the practical impact on imaging modalities and generative-prior phase retrieval.

Abstract

The key ingredient to retrieving a signal from its Fourier magnitudes, namely, to solve the phase retrieval problem, is an effective prior on the sought signal. In this paper, we study the phase retrieval problem under the prior that the signal lies in a semi-algebraic set. This is a very general prior as semi-algebraic sets include linear models, sparse models, and ReLU neural network generative models. The latter is the main motivation of this paper, due to the remarkable success of deep generative models in a variety of imaging tasks, including phase retrieval. We prove that almost all signals in R^N can be determined from their Fourier magnitudes, up to a sign, if they lie in a (generic) semi-algebraic set of dimension N/2. The same is true for all signals if the semi-algebraic set is of dimension N/4. We also generalize these results to the problem of signal recovery from the second moment in multi-reference alignment models with multiplicity free representations of compact groups. This general result is then used to derive improved sample complexity bounds for recovering band-limited functions on the sphere from their noisy copies, each acted upon by a random element of SO(3).

Phase retrieval with semi-algebraic and ReLU neural network priors

TL;DR

The paper establishes theoretical guarantees for phase retrieval under semi-algebraic priors, showing that the power spectrum uniquely identifies signals up to sign when the ambient dimension exceeds a multiple of the prior dimension (e.g., , or for universal recovery). It extends these results to generic rotations and to multiplicity-free multi-reference alignment models, deriving corresponding dimension thresholds and providing sample-complexity implications in the high-noise regime. A key contribution is the unification of the second-moment approach with fiber-dimension counting, enabling precise identifiability statements for diverse priors, including neural-network generative models with ReLU activations and sparse representations. The work also presents concrete extensions to band-limited functions on the sphere and outlines future directions for cryo-EM and analytic activation functions, highlighting the practical impact on imaging modalities and generative-prior phase retrieval.

Abstract

The key ingredient to retrieving a signal from its Fourier magnitudes, namely, to solve the phase retrieval problem, is an effective prior on the sought signal. In this paper, we study the phase retrieval problem under the prior that the signal lies in a semi-algebraic set. This is a very general prior as semi-algebraic sets include linear models, sparse models, and ReLU neural network generative models. The latter is the main motivation of this paper, due to the remarkable success of deep generative models in a variety of imaging tasks, including phase retrieval. We prove that almost all signals in R^N can be determined from their Fourier magnitudes, up to a sign, if they lie in a (generic) semi-algebraic set of dimension N/2. The same is true for all signals if the semi-algebraic set is of dimension N/4. We also generalize these results to the problem of signal recovery from the second moment in multi-reference alignment models with multiplicity free representations of compact groups. This general result is then used to derive improved sample complexity bounds for recovering band-limited functions on the sphere from their noisy copies, each acted upon by a random element of SO(3).
Paper Structure (28 sections, 18 theorems, 69 equations)

This paper contains 28 sections, 18 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries and notation
  3. Main results
  4. Applications
  5. Organization of the paper
  6. Common strategy for both proofs
  7. The power spectrum and the second moment
  8. Reduction to fiber dimension counting
  9. Proof of Theorem \ref{['thm.main']}
  10. Implications for neural generative models: Proof of Corollary \ref{['cor.deepnetwork']}
  11. Proof of Theorem \ref{['thm.rotation']}
  12. Proof of Proposition \ref{['prop.crux']}
  13. Multi-reference alignment with irreducibles with multiplicity one
  14. Introduction
  15. Main results of MRA for multiplicity-free representations
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\mathcal{M}$ be a semi-algebraic subset of $\mathbb{R}^N$ of dimension $M$. Then, for a generic invertible matrix $A \in \mathop{\mathrm{GL}}\nolimits(N)$, the following hold:

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3: Deep neural networks
  • Corollary 1.4: Sparsity
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem.RNN']}
  • Proposition 4.1
  • ...and 20 more