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Spectral Initialization for High-Dimensional Phase Retrieval with Biased Spatial Directions

Pierre Bousseyroux, Marc Potters

Abstract

We explore a spectral initialization method that plays a central role in contemporary research on signal estimation in nonconvex scenarios. In a noiseless phase retrieval framework, we precisely analyze the method's performance in the high-dimensional limit when sensing vectors follow a multivariate Gaussian distribution for two rotationally invariant models of the covariance matrix C. In the first model C is a projector on a lower dimensional space while in the second it is a Wishart matrix. Our analytical results extend the well-established case when C is the identity matrix. Our examination shows that the introduction of biased spatial directions leads to a substantial improvement in the spectral method's effectiveness, particularly when the number of measurements is less than the signal's dimension. This extension also consistently reveals a phase transition phenomenon dependent on the ratio between sample size and signal dimension. Surprisingly, both of these models share the same threshold value.

Spectral Initialization for High-Dimensional Phase Retrieval with Biased Spatial Directions

Abstract

We explore a spectral initialization method that plays a central role in contemporary research on signal estimation in nonconvex scenarios. In a noiseless phase retrieval framework, we precisely analyze the method's performance in the high-dimensional limit when sensing vectors follow a multivariate Gaussian distribution for two rotationally invariant models of the covariance matrix C. In the first model C is a projector on a lower dimensional space while in the second it is a Wishart matrix. Our analytical results extend the well-established case when C is the identity matrix. Our examination shows that the introduction of biased spatial directions leads to a substantial improvement in the spectral method's effectiveness, particularly when the number of measurements is less than the signal's dimension. This extension also consistently reveals a phase transition phenomenon dependent on the ratio between sample size and signal dimension. Surprisingly, both of these models share the same threshold value.
Paper Structure (13 sections, 11 theorems, 141 equations, 4 figures)

This paper contains 13 sections, 11 theorems, 141 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. The orthogonal case
  4. Favoring exploitation over exploration
  5. The theoretical overlap $\rho$ for a Wishart matrix $\vb{C}$
  6. Around the phase transition
  7. Proof of results
  8. Proof of theorem \ref{['orthogonalcase']}
  9. Proof of theorem \ref{['theorem1']}
  10. Proof of theorem \ref{['generalisation']}
  11. Proof of theorem \ref{['universal']}
  12. REMINDER ON TRANSFORMS IN RMT
  13. Some useful lemmas

Key Result

Theorem 2.1

Let $\vb{a}_1, ..., \vb{a}_T$ represent the first $T$ columns of an $N\times N$ Haar distributed random orthogonal matrix. Let $\vb{M}$ be with $f$ an increasing function and $m_k:=\left< \vb{a}_k, \vb{x} \right>^2$. Then, in the large $N$ and $T$ limit while $q :=\frac{N}{T}$ remains finite, where $\vb{y}$ the eigenvector corresponding to the largest eigenvalue of $\vb{M}$.

Figures (4)

  • Figure 1: The function $q\mapsto \rho_{cc}(q)q$ with its maximum.
  • Figure 2: The plot of the $\rho$ improved \ref{['improvedequation']}. The dashed line corresponds to the classical case.
  • Figure 3: Overlap $\rho = \frac{\left< \vb{x}, \vb{y} \right>^2}{||\vb{x}||^2 ||\vb{y}||^2}$ between the largest eigenvector $\vb{y}$ of $\vb{M}$ and the true signal as a function of $q = \frac{N}{T}$ for the function $f(y) = 1 - 1/y$. Each dot correspond to a single matrix $\vb{M}$ of aspect ratio $q$ and $NT = 10^7$ where the columns were independently drawn from $\mathcal{N}(0, \vb{C})$ where $\vb{C}$ is a Wishart matrix with parameter $p$. The theory curves are given by equations \ref{['grosysteme']}. The dashed line corresponds to the classical case when the covariance matrix $\vb{C}$ is the identity studied in the erratum erratum.
  • Figure 4: Overlap $\rho = \frac{\left< \vb{x}, \vb{y} \right>^2}{||\vb{x}||^2 ||\vb{y}||^2}$ between the largest eigenvector $\vb{y}$ of $\vb{M}$ and the true signal as a function of $q = \frac{N}{T}$ for the function $f(y) = 1 - 1/y$. Each dot correspond to a single matrix $\vb{M}$ of aspect ratio $q$ and $NT = 10^7$ where the columns were independently drawn from $\mathcal{N}(0, \vb{C})$ where $\vb{C} = \vb{\hbox{o}rigin=c]{180}{W}}_{p = 1.5}$ is an inverse-Wishart matrix with parameter $p = 1.5$. The theory curves are given by theorem \ref{['theorem_limited']} for the model $\vb{P}_{1 + 1.5}$ and theorem \ref{['generalisation']} if $\vb{C}$ is a Wishart matrix of parameter $p = 1.5$.

Theorems & Definitions (19)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • proof
  • proof
  • proof
  • Lemma B.1
  • ...and 9 more