Phase Retrieval

Gauri Jagatap

Electrical and Computer Engineering

Iowa State University

Motivation

• Signal • Magnitude

• Phase

• Fourier measurements

Magnitude |𝐹(𝑋)|

Phase ∠𝐹(𝑋)

That actress from every 90s rom-com Voldemort

Magnitude-only reconstruction Phase-only reconstruction

• Typically phase has more information about the signal than magnitude.

• What if you lose phase information?

Use phase retrieval

• NP-hard

Phase retrieval using Alternating Minimization

• Work by Praneeth Netrapalli, Prateek Jain and Sujay Sanghavi.

• Use random matrices for sensing signals.

• Requires 𝒪(𝑛 𝑙𝑜𝑔3𝑛) measurements for successful recovery.

• Two main features • Initialization

• Convergence

Measurement model • Signal 𝒙∗∈ ℝ𝑛

• Measurement vectors 𝑎𝑖 ∈ ℝ

𝑛 ,𝒩 0,1

• Measurements 𝑦𝑖, 𝑖 ∈ {1 …𝑚}

• Introduce diagonal phase matrix 𝐂∗ = 𝑑𝑖𝑎𝑔 𝐀T𝑥∗ which is the true phase of the measurements.

Signal recovery

• Non-convex optimization problem

• Not convex because entries of 𝐂 are restricted to be diagonal with ‘phases’ of form 𝑒𝑖𝜃 and hence magnitude 1.

Alternatively update 𝐂 and 𝒙

How to initialize?

• Random?

• Zeros?

oGets stuck in local optimum

• Take advantage of randomness of measurement vectors 𝑎𝑖

Ε1

𝑚 𝑦𝑖

2𝑎𝑖𝑎𝑖𝑇

𝑚

𝑖=1

= 𝕀 + 2𝑥∗𝑥∗𝑇

Top singular vector of bracketed term is a good initial estimate of 𝑥

n = 500, m = 500

n = 500, m = 500

n = 500, m = 1000

n = 500, m = 1000

n = 500, m = 2000

n = 500, m = 2000

n = 500, m = 2500

n = 500, m = 2500

Phase transition

PhaseLift (Overview)

Trace-norm relaxation

𝒜:

𝒜−1:

𝑿 = 𝒙𝒙∗ ( 𝑿 = rank 1, 𝒙 = original signal) Measurement:

Measurement operation:

Adjoint operation:

• Signal recovery from phase-less measurements: (requires 𝑚 = 𝒪(𝑛 log𝑛))

• Signal and measurement model:

Lifting up the problem of vector recovery from quadratic constraints into that of recovering a rank-one matrix from affine constraints via semidefinite programming.

Scalability Issues

• Dependence of 𝑚 on 𝑛 when 𝑛 is large ~104

𝑛 log 𝑛~105 , 𝑛 (log 𝑛)3~107

• Use signal’s structure to reduce the number of measurements

Compressive phase retrieval 𝑚 = 𝒪( 𝑘 log

𝑛

𝑘 ) where 𝑘 is the sparsity of signal

If 𝑛~104, 𝑘~102 then 𝑘 log𝑛

𝑘 ~102

Efficient Compressive Phase Retrieval with Constrained Sensing Vectors

• Work by Sohail Bahmani, Justin Romberg

• Combines two key points of discussion so far • Lifting

• Sparsity

Measurement model

n = 500, m = 100

Comparison

Method Sample complexity (m)*

AltMinPhase 𝑛 log3 𝑛

PhaseLift 𝑛 log 𝑛

Efficient CPR 𝑘 log𝑛

𝑘

*for n-length k-sparse signal

References

• Netrapalli, Praneeth, Prateek Jain, and Sujay Sanghavi. "Phase retrieval

using alternating minimization." Advances in Neural Information Processing Systems. 2013.

• Candes, Emmanuel J., Thomas Strohmer, and Vladislav Voroninski. "Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming." Communications on Pure and Applied Mathematics 66.8 (2013): 1241-1274.

• Bahmani, Sohail, and Justin Romberg. "Efficient compressive phase retrieval with constrained sensing vectors." Advances in Neural Information Processing Systems. 2015.