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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.
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