Phase Retrieval Gauri Jagatap Electrical and Computer Engineering Iowa State University
Phase Retrieval
Gauri Jagatap
Electrical and Computer Engineering
Iowa State University
Motivation
β’ Signal β’ Magnitude
β’ Phase
β’ Fourier measurements
Magnitude |πΉ(π)|
Phase β πΉ(π)
That actress from every 90s rom-com Voldemort
Magnitude-only reconstruction Phase-only reconstruction
β’ Typically phase has more information about the signal than magnitude.
β’ What if you lose phase information?
Use phase retrieval
β’ NP-hard
Phase retrieval using Alternating Minimization
β’ Work by Praneeth Netrapalli, Prateek Jain and Sujay Sanghavi.
β’ Use random matrices for sensing signals.
β’ Requires πͺ(π πππ3π) measurements for successful recovery.
β’ Two main features β’ Initialization
β’ Convergence
Measurement model β’ Signal πββ βπ
β’ Measurement vectors ππ β β
π ,π© 0,1
β’ Measurements π¦π, π β {1 β¦π}
β’ Introduce diagonal phase matrix πβ = ππππ πTπ₯β which is the true phase of the measurements.
Signal recovery
β’ Non-convex optimization problem
β’ Not convex because entries of π are restricted to be diagonal with βphasesβ of form πππ and hence magnitude 1.
Alternatively update π and π
How to initialize?
β’ Random?
β’ Zeros?
oGets stuck in local optimum
β’ Take advantage of randomness of measurement vectors ππ
Ξ1
π π¦π
2πππππ
π
π=1
= π + 2π₯βπ₯βπ
Top singular vector of bracketed term is a good initial estimate of π₯
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
π:
πβ1:
πΏ = ππβ ( πΏ = rank 1, π = original signal) Measurement:
Measurement operation:
Adjoint operation:
β’ Signal recovery from phase-less measurements: (requires π = πͺ(π logπ))
β’ 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
β’ Dependence of π on π when π is large ~104
π log π~105 , π (log π)3~107
β’ Use signalβs structure to reduce the number of measurements
Compressive phase retrieval π = πͺ( π log
π
π ) where π is the sparsity of signal
If π~104, π~102 then π logπ
π ~102
Efficient Compressive Phase Retrieval with Constrained Sensing Vectors
β’ Work by Sohail Bahmani, Justin Romberg
β’ Combines two key points of discussion so far β’ Lifting
β’ Sparsity
Measurement model
n = 500, m = 100
Comparison
Method Sample complexity (m)*
AltMinPhase π log3 π
PhaseLift π log π
Efficient CPR π logπ
π
*for n-length k-sparse signal
References
β’ Netrapalli, Praneeth, Prateek Jain, and Sujay Sanghavi. "Phase retrieval
using alternating minimization." Advances in Neural Information Processing Systems. 2013.
β’ 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.
β’ Bahmani, Sohail, and Justin Romberg. "Efficient compressive phase retrieval with constrained sensing vectors." Advances in Neural Information Processing Systems. 2015.