An algebraic approach to phase retrieval Cynthia Vinzant University - PowerPoint PPT Presentation

An algebraic approach to phase retrieval Cynthia Vinzant University of Michigan joint with Aldo Conca, Dan Edidin, and Milena Hering. Cynthia Vinzant An algebraic approach to phase retrieval

I learned about frame theory from . . . Frame theory intersects geometry July 29 to August 2, 2013 at the American Institute of Mathematics, Palo Alto, California organized by Bernhard Bodmann, Gitta Kutyniok, and Tim Roemer This workshop, sponsored by AIM and the NSF, will be devoted to outstanding problems Cynthia Vinzant An algebraic approach to phase retrieval

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . (a “redundant basis”) Cynthia Vinzant An algebraic approach to phase retrieval

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . (a “redundant basis”) A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 φ ∗ k xx ∗ φ k = for k = 1 , . . . , n . Cynthia Vinzant An algebraic approach to phase retrieval

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . (a “redundant basis”) A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 φ ∗ k xx ∗ φ k = for k = 1 , . . . , n . Phase Retrieval: Recover x from its measurements |� φ k , x �| 2 . Cynthia Vinzant An algebraic approach to phase retrieval

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . (a “redundant basis”) A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 φ ∗ k xx ∗ φ k = for k = 1 , . . . , n . Phase Retrieval: Recover x from its measurements |� φ k , x �| 2 . Some Questions: How do we recover the signal x ? Cynthia Vinzant An algebraic approach to phase retrieval

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . (a “redundant basis”) A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 φ ∗ k xx ∗ φ k = for k = 1 , . . . , n . Phase Retrieval: Recover x from its measurements |� φ k , x �| 2 . Some Questions: How do we recover the signal x ? When is recovery of signals in C d possible? Cynthia Vinzant An algebraic approach to phase retrieval

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . (a “redundant basis”) A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 φ ∗ k xx ∗ φ k = for k = 1 , . . . , n . Phase Retrieval: Recover x from its measurements |� φ k , x �| 2 . Some Questions: How do we recover the signal x ? When is recovery of signals in C d possible? When is recovery of signals in C d stable? Cynthia Vinzant An algebraic approach to phase retrieval

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . (a “redundant basis”) A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 φ ∗ k xx ∗ φ k = for k = 1 , . . . , n . Phase Retrieval: Recover x from its measurements |� φ k , x �| 2 . Some Questions: How do we recover the signal x ? When is recovery of signals in C d possible? When is recovery of signals in C d stable? Cynthia Vinzant An algebraic approach to phase retrieval

Motivation and Applications In practice the signal is some structure that is too small (DNA, crystals) or far away (astronomical phenomena) or obscured (medical images) to observe directly. (picture from Cand´ es-Eldar-Strohmer-Voroninski 2013) Cynthia Vinzant An algebraic approach to phase retrieval

Motivation and Applications In practice the signal is some structure that is too small (DNA, crystals) or far away (astronomical phenomena) or obscured (medical images) to observe directly. If some measurements are possible, then one hopes to reconstruct this structure. (picture from Cand´ es-Eldar-Strohmer-Voroninski 2013) Cynthia Vinzant An algebraic approach to phase retrieval

Motivation and Applications In practice the signal is some structure that is too small (DNA, crystals) or far away (astronomical phenomena) or obscured (medical images) to observe directly. If some measurements are possible, then one hopes to reconstruct this structure. Here our signal x lies in a finite- dimensional space ( C d ), and its measurements are modeled by |� φ k , x �| 2 for φ k ∈ C d . (picture from Cand´ es-Eldar-Strohmer-Voroninski 2013) Cynthia Vinzant An algebraic approach to phase retrieval

Phase Retrieval: recovering a vector from its measurements When do the frame measurements |� φ k , x �| 2 determine x ∈ C d ? Cynthia Vinzant An algebraic approach to phase retrieval

Phase Retrieval: recovering a vector from its measurements When do the frame measurements |� φ k , x �| 2 determine x ∈ C d ? (Never: |� φ k , x �| 2 invariant under x �→ e i θ x ) Cynthia Vinzant An algebraic approach to phase retrieval

Phase Retrieval: recovering a vector from its measurements When do the frame measurements |� φ k , x �| 2 determine x ∈ C d ? (Never: |� φ k , x �| 2 invariant under x �→ e i θ x ) The frame measurements define a map � |� x , φ k �| 2 � M Φ : ( C d / S 1 ) → R n by x �→ k Cynthia Vinzant An algebraic approach to phase retrieval

Phase Retrieval: recovering a vector from its measurements When do the frame measurements |� φ k , x �| 2 determine x ∈ C d ? (Never: |� φ k , x �| 2 invariant under x �→ e i θ x ) The frame measurements define a map � |� x , φ k �| 2 � M Φ : ( C d / S 1 ) → R n by x �→ or k � � � rank-1 Hermitian → R n M Φ : by X �→ ( trace ( X · A k )) k . d × d matrices where X = xx ∗ , A k = φ k φ ∗ k Cynthia Vinzant An algebraic approach to phase retrieval

Phase Retrieval: recovering a vector from its measurements When do the frame measurements |� φ k , x �| 2 determine x ∈ C d ? (Never: |� φ k , x �| 2 invariant under x �→ e i θ x ) The frame measurements define a map � |� x , φ k �| 2 � M Φ : ( C d / S 1 ) → R n by x �→ or k � � � rank-1 Hermitian → R n M Φ : by X �→ ( trace ( X · A k )) k . d × d matrices where X = xx ∗ , A k = φ k φ ∗ k Better question: When is the map M Φ injective? Cynthia Vinzant An algebraic approach to phase retrieval

Question: How many measurements? We need n ≈ 4 d measurements to recover vectors in C d . Cynthia Vinzant An algebraic approach to phase retrieval

Question: How many measurements? We need n ≈ 4 d measurements to recover vectors in C d . ◮ (Balan-Casazza-Edidin, 2006): For n ≥ 4 d − 2, M Φ is injective for generic Φ ∈ C d × n . Cynthia Vinzant An algebraic approach to phase retrieval

Question: How many measurements? We need n ≈ 4 d measurements to recover vectors in C d . ◮ (Balan-Casazza-Edidin, 2006): For n ≥ 4 d − 2, M Φ is injective for generic Φ ∈ C d × n . ◮ (Heinosaari-Mazzarella-Wolf, 2011): For n < 4 d − 2 α − 3, M Φ is not injective, where α = # of 1’s in binary expansion of d − 1. Cynthia Vinzant An algebraic approach to phase retrieval

Question: How many measurements? We need n ≈ 4 d measurements to recover vectors in C d . ◮ (Balan-Casazza-Edidin, 2006): For n ≥ 4 d − 2, M Φ is injective for generic Φ ∈ C d × n . ◮ (Heinosaari-Mazzarella-Wolf, 2011): For n < 4 d − 2 α − 3, M Φ is not injective, where α = # of 1’s in binary expansion of d − 1. Conjecture (Bandeira-Cahill-Mixon-Nelson, 2013) (a) If n < 4 d − 4, then M Φ is not injective. (b) If n ≥ 4 d − 4, then M Φ is injective for generic Φ. Cynthia Vinzant An algebraic approach to phase retrieval

Question: How many measurements? We need n ≈ 4 d measurements to recover vectors in C d . ◮ (Balan-Casazza-Edidin, 2006): For n ≥ 4 d − 2, M Φ is injective for generic Φ ∈ C d × n . ◮ (Heinosaari-Mazzarella-Wolf, 2011): For n < 4 d − 2 α − 3, M Φ is not injective, where α = # of 1’s in binary expansion of d − 1. Conjecture (Bandeira-Cahill-Mixon-Nelson, 2013) (a) If n < 4 d − 4, then M Φ is not injective. (b) If n ≥ 4 d − 4, then M Φ is injective for generic Φ. We prove (b) by writing injectivity as an algebraic condition . Cynthia Vinzant An algebraic approach to phase retrieval

A nice reformulation of non-injectivity Observation (Bandeira et al ., among others): ∃ a nonzero matrix Q ∈ C d × d M Φ is non-injective ⇔ Herm with φ ∗ rank( Q ) ≤ 2 and k Q φ k = 0 for each 1 ≤ k ≤ n . Cynthia Vinzant An algebraic approach to phase retrieval

A nice reformulation of non-injectivity Observation (Bandeira et al ., among others): ∃ a nonzero matrix Q ∈ C d × d M Φ is non-injective ⇔ Herm with φ ∗ rank( Q ) ≤ 2 and k Q φ k = 0 for each 1 ≤ k ≤ n . Why? Cynthia Vinzant An algebraic approach to phase retrieval

A nice reformulation of non-injectivity Observation (Bandeira et al ., among others): ∃ a nonzero matrix Q ∈ C d × d M Φ is non-injective ⇔ Herm with φ ∗ rank( Q ) ≤ 2 and k Q φ k = 0 for each 1 ≤ k ≤ n . Why? φ ∗ k xx ∗ φ k = φ ∗ k yy ∗ φ k M Φ ( x ) = M Φ ( y ) ⇔ for 1 ≤ k ≤ n Cynthia Vinzant An algebraic approach to phase retrieval