Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global convergence of gradient descent for phase retrieval

Théodore Fougereux, Cédric Josz, Xiaopeng Li

TL;DR

A tensor-based criterion for benign landscape in phase retrieval and boundedness of gradient trajectories implies that gradient descent will converge to a global minimum for almost every initial point.

Abstract

We propose a tensor-based criterion for benign landscape in phase retrieval and establish boundedness of gradient trajectories. This implies that gradient descent will converge to a global minimum for almost every initial point.

Global convergence of gradient descent for phase retrieval

TL;DR

A tensor-based criterion for benign landscape in phase retrieval and boundedness of gradient trajectories implies that gradient descent will converge to a global minimum for almost every initial point.

Abstract

We propose a tensor-based criterion for benign landscape in phase retrieval and establish boundedness of gradient trajectories. This implies that gradient descent will converge to a global minimum for almost every initial point.

Paper Structure

This paper contains 23 sections, 11 theorems, 60 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem formulation and related work
  3. Problem formulation
  4. Related work
  5. Main results
  6. Equivalent formulation of the problem
  7. Landscape results
  8. Convergence of gradient descent
  9. Experiments
  10. Conclusion and discussion
  11. Appendix
  12. Proof of equivalent formulation results
  13. Proof of \ref{['prop:formulaf']}
  14. Proof of landscape results
  15. Proof of \ref{['prop:expectation-f']}
  16. ...and 8 more sections

Key Result

Theorem 3.1

Assume $\|{\bm{\mathsfit{T}}}-{\bm{\mathsfit{S}}}\|_{\rm{op}}\le \delta_0$ for some constant $\delta_0>0$ small enough. The only local minima of $f$ defined in eq:realPR are global minima ${\bm{x}}^{\natural}e^{i\theta}$ and all saddle points of $f$ are strictA strict saddle point is a critical poin

Figures (3)

  • Figure 1: Overview of the regions considered in the proof
  • Figure 2: Values of $\|{\bm{\mathsfit{T}}}-{\bm{\mathsfit{S}}}\|_{\rm{op}}$ for different values of $m$ and $n$
  • Figure 3: Loss of $20$ trajectories for $n=4$ and $m\in [10,20,30,40]$

Theorems & Definitions (20)

  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Proposition 3.8
  • Proposition 3.9
  • Proposition 3.10
  • ...and 10 more