Injectivity of Hermitian frame measurements Cynthia Vinzant North - PowerPoint PPT Presentation

Injectivity of Hermitian frame measurements Cynthia Vinzant North Carolina State University Frames and Algebraic & Combinatorial Geometry July 31, 2015 Cynthia Vinzant Injectivity of Hermitian frame measurements

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . Cynthia Vinzant Injectivity of Hermitian frame measurements

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 tr( φ k φ ∗ k xx ∗ ) = for k = 1 , . . . , n . Cynthia Vinzant Injectivity of Hermitian frame measurements

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 tr( φ k φ ∗ k xx ∗ ) = for k = 1 , . . . , n . Phase Retrieval: Recover xx ∗ from tr( φ k φ ∗ k xx ∗ ). Cynthia Vinzant Injectivity of Hermitian frame measurements

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 tr( φ k φ ∗ k xx ∗ ) = for k = 1 , . . . , n . Phase Retrieval: Recover xx ∗ from tr( φ k φ ∗ k xx ∗ ). Some Questions: How do we recover the signal x ? Cynthia Vinzant Injectivity of Hermitian frame measurements

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

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 tr( φ k φ ∗ k xx ∗ ) = for k = 1 , . . . , n . Phase Retrieval: Recover xx ∗ from tr( φ k φ ∗ k xx ∗ ). 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 Injectivity of Hermitian frame measurements

Frames and intensity measurements A frame is a collection of vectors Φ = { φ 1 , . . . , φ n } spanning C d . A frame defines intensity measurements of a signal x ∈ C d : |� φ k , x �| 2 tr( φ k φ ∗ k xx ∗ ) = for k = 1 , . . . , n . Phase Retrieval: Recover xx ∗ from tr( φ k φ ∗ k xx ∗ ). 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 Injectivity of Hermitian frame measurements

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 Injectivity of Hermitian frame measurements

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 Injectivity of Hermitian frame measurements

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 Injectivity of Hermitian frame measurements

Phase Retrieval: recovering a vector from its measurements k xx ∗ ) determine xx ∗ ∈ C d × d When do the measurements tr( φ k φ ∗ Herm ? Cynthia Vinzant Injectivity of Hermitian frame measurements

Phase Retrieval: recovering a vector from its measurements k xx ∗ ) determine xx ∗ ∈ C d × d When do the measurements tr( φ k φ ∗ Herm ? That is, for what collections of vectors Φ = ( φ 1 . . . φ n ) is the map � � rank-1 Hermitian → R n given by X �→ (tr( φ k φ ∗ M Φ : k · X )) k d × d matrices injective? Cynthia Vinzant Injectivity of Hermitian frame measurements

How many measurements for injectivity? About 4 d . (Heinosaari–Mazzarella–Wolf, 2011): For n < 4 d − 2 α − 4, M Φ is not injective, where α = # of 1’s in binary expansion of d − 1. Cynthia Vinzant Injectivity of Hermitian frame measurements

How many measurements for injectivity? About 4 d . (Heinosaari–Mazzarella–Wolf, 2011): For n < 4 d − 2 α − 4, 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 Injectivity of Hermitian frame measurements

How many measurements for injectivity? About 4 d . (Heinosaari–Mazzarella–Wolf, 2011): For n < 4 d − 2 α − 4, 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. (Conca–Edidin–Hering–V., 2014) For n ≥ 4 d − 4, M Φ is injective for generic Φ ∈ C d × n . If d = 2 k + 1 and n < 4 d − 4, M Φ is not injective. Cynthia Vinzant Injectivity of Hermitian frame measurements

A nice reformulation of non-injectivity Observation (Bandeira-Cahill-Mixon-Nelson): ∃ 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 Injectivity of Hermitian frame measurements

A nice reformulation of non-injectivity Observation (Bandeira-Cahill-Mixon-Nelson): ∃ 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 Injectivity of Hermitian frame measurements

A nice reformulation of non-injectivity Observation (Bandeira-Cahill-Mixon-Nelson): ∃ 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 . (*) M Φ ( x ) = M Φ ( y ) ⇔ φ ∗ k xx ∗ φ k = φ ∗ k yy ∗ φ k Why? ∀ k Cynthia Vinzant Injectivity of Hermitian frame measurements

A nice reformulation of non-injectivity Observation (Bandeira-Cahill-Mixon-Nelson): ∃ 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 . (*) M Φ ( x ) = M Φ ( y ) ⇔ φ ∗ k xx ∗ φ k = φ ∗ k yy ∗ φ k Why? ∀ k k ( xx ∗ − yy ∗ ) φ k = 0 ⇔ φ ∗ ∀ k Cynthia Vinzant Injectivity of Hermitian frame measurements

A nice reformulation of non-injectivity Observation (Bandeira-Cahill-Mixon-Nelson): ∃ 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 . (*) M Φ ( x ) = M Φ ( y ) ⇔ φ ∗ k xx ∗ φ k = φ ∗ k yy ∗ φ k Why? ∀ k k ( xx ∗ − yy ∗ ⇔ φ ∗ ) φ k = 0 ∀ k � �� � rank 2 Cynthia Vinzant Injectivity of Hermitian frame measurements

A nice reformulation of non-injectivity Observation (Bandeira-Cahill-Mixon-Nelson): ∃ 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 . (*) M Φ ( x ) = M Φ ( y ) ⇔ φ ∗ k xx ∗ φ k = φ ∗ k yy ∗ φ k Why? ∀ k k ( xx ∗ − yy ∗ ⇔ φ ∗ ) φ k = 0 ∀ k � �� � rank 2 When does ( span R { φ 1 φ ∗ 1 , . . . , φ n φ ∗ n } ) ⊥ More algebraic question: intersect the rank-2 locus of C d × d Herm ? Cynthia Vinzant Injectivity of Hermitian frame measurements

Getting (Real) Algebraic Consider the incidence set � � (Φ , Q ) ∈ P ( C d × n ) × P ( C d × d Herm ) : rank( Q ) ≤ 2 and φ ∗ k Q φ k = 0 ∀ k . Cynthia Vinzant Injectivity of Hermitian frame measurements

Getting (Real) Algebraic Consider the incidence set � � (Φ , Q ) ∈ P ( C d × n ) × P ( C d × d Herm ) : rank( Q ) ≤ 2 and φ ∗ k Q φ k = 0 ∀ k . Φ ∈ C d × n A + i B where A , B ∈ R d × n − → Cynthia Vinzant Injectivity of Hermitian frame measurements

Getting (Real) Algebraic Consider the incidence set � � (Φ , Q ) ∈ P ( C d × n ) × P ( C d × d Herm ) : rank( Q ) ≤ 2 and φ ∗ k Q φ k = 0 ∀ k . Φ ∈ C d × n A + i B where A , B ∈ R d × n − → Q ∈ C d × d X + i Y where X ∈ R d × d sym , Y ∈ R d × d − → Herm skew Cynthia Vinzant Injectivity of Hermitian frame measurements