Blind Deconvolutional Phase Retrieval via Convex Programming Ali Ahmed, Alireza Aghasi, Paul Hand Funding provided in part by the National Science Foundation December 5, 2018 Ahmed, Aghasi, Hand Blind Deconvolutional Phase Retrieval December 5, 2018 1 / 7
Motivation: Blind Deconvolutional Phase Retrieval Free Space Propagation Inverse Problem recovered measurements y = | F ( w ⊛ x ) | 2 Observe: ˆ x ∈ R L , w ∈ R L Find: Assumption: w = Bh , x = Cm , B ∈ R L × K , C ∈ R L × N Ahmed, Aghasi, Hand Blind Deconvolutional Phase Retrieval December 5, 2018 2 / 7
Blind Deconvolutional Phase Retrieval (BDPR): Lifting ℓ h | 2 · | c ∗ ℓ m | 2 y [ ℓ ] = | b ∗ Observe: ˆ b ∗ ℓ is ℓ th row of FB c ∗ ℓ is ℓ th row of FC h ∈ R K , m ∈ R N Find: � h � 2 + � m � 2 Solve: minimize h , m subject to � b ℓ b ∗ ℓ , X 1 �� c ℓ c ∗ ℓ , X 2 � = ˆ y [ ℓ ] X 1 = hh ∗ , X 2 = mm ∗ Ahmed, Aghasi, Hand Blind Deconvolutional Phase Retrieval December 5, 2018 3 / 7
Novel Convex Relaxation via BranchHull minimize trace( X 1 ) + trace( X 2 ) X 1 , X 2 subject to � b ℓ b ∗ ℓ , X 1 �� c ℓ c ∗ ℓ , X 2 � = ˆ y [ ℓ ] X 1 � 0 , X 2 � 0 Hyperbolic constraint set Ahmed, Aghasi, Hand Blind Deconvolutional Phase Retrieval December 5, 2018 4 / 7
Novel Convex Relaxation via BranchHull minimize trace( X 1 ) + trace( X 2 ) X 1 , X 2 subject to � b ℓ b ∗ ℓ , X 1 �� c ℓ c ∗ ℓ , X 2 � ≥ ˆ y [ ℓ ] X 1 � 0 , X 2 � 0 Hyperbolic constraint set Ahmed, Aghasi, Hand Blind Deconvolutional Phase Retrieval December 5, 2018 4 / 7
Cartoon of the BranchHull Geometry Blue: PSD Cone, Red: Boundary of Hyperbolic Constraint Point in intersection with smallest trace lives along the ridge where hyperbolic constraints are satisfied with equalities. Ahmed, Aghasi, Hand Blind Deconvolutional Phase Retrieval December 5, 2018 5 / 7
Main Result: Exact Recovery Convex program for Blind Deconvolutional Phase Retrieval minimize trace( X 1 ) + trace( X 2 ) X 1 , X 2 subject to � b ℓ b ∗ ℓ , X 1 �� c ℓ c ∗ ℓ , X 2 � ≥ ˆ y [ ℓ ] X 1 � 0 , X 2 � 0 . Theorem [Ahmed, Aghasi, Hand]: Choose B and C to have i.i.d. standard normal entries. Then, h ∈ R K and m ∈ R N can be exactly recovered (up to global rescaling) with high probability if L � ( K + N ) log 2 L . Ahmed, Aghasi, Hand Blind Deconvolutional Phase Retrieval December 5, 2018 6 / 7
Phase Portrait for an ADMM Implementation Successful Recovery L N + K = 2 . 5 L log − 2 L Unsuccessful Recovery N + K Convex BDPR succeeds for reasonable constants in sample complexity. Ahmed, Aghasi, Hand Blind Deconvolutional Phase Retrieval December 5, 2018 7 / 7
Blind Deconvolutional Phase Retrieval via Convex Programming Ali Ahmed, Alireza Aghasi, Paul Hand Funding provided in part by the National Science Foundation December 5, 2018 Ahmed, Aghasi, Hand Blind Deconvolutional Phase Retrieval December 5, 2018 1 / 7
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