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SUPER: Sparse signal s with Unknown Phases Efficiently Recovered

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### SUPER: Sparse signalswith Unknown Phases Efficiently Recovered

ShengCai, MayankBakshi, SidharthJaggi and Minghua Chen

The Chinese University of Hong Kong

Compressive Phase Retrieval

?

b

m

?

n

Complex number

k

Applications: X-ray crystallography, Optics, Astronomical imaging…

Compressive Phase Retrieval

?

b

m

?

n

Our contribution:

1. O(k) number of measurements (best known O(k)[1])

2. O(klogk) decoding complexity (best known O(knlogn) [2])

[1] H. Ohlsson and Y. C. Eldar, “On conditions for uniqueness in sparse phase retrieval,” e-prints, arXiv:1308.5447

[2] K. Jaganathan, S. Oymak, and B. Hassibi, “Sparse phase retrieval: Convex algorithms and limitations,” in 2013

IEEE International Symposium on Information Theory Proceedings (ISIT), 2013, pp. 1022–1026.

II. Overview/High-Level Intuition

Bipartite graphx1

b1

x2

b2

x3

k non-zero

components

x4

b3

x5

b4

x6

n signal nodes

O(k) measurement nodes

II. Overview/High-Level Intuition & IV. Measurement Design

Bipartite Graph →Measurement MatrixAdjacent Matrix

x1

b1

x1

x2

x3

x4

x5

x6

x2

b1

b2

b2

x3

b3

b4

x4

b3

x5

b4

x6

II. Overview/High-Level Intuition

Useful Measurement Nodesb1

x2

b2

x4

b3

b4

x6

n signal nodes

O(k) measurement nodes

II. Overview/High-Level Intuition & V. Reconstruction Algorithm

Useful Measurement Nodesb1

Magnitude

recovery

Singleton

x2

b2

Phase

recovery

Doubleton

Resolvable

|x2|

Δ

x4

Multiton

Resolvable

b3

|x4|

|x2+x4|

“Cancelling out” process:

b4

Solving a quadratic equation

x6

II. Overview/High-Level Intuition

Three PhasesSeeding Phase:

Singletons and Resolvable Doubletons

…

…

Geometric-decay Phase:

Resolvable Multitons

…

Cleaning-up Phase:

Resolvable Multitons

…

…

O(k) measurement nodes

n signal nodes

II. Overview/High-Level Intuition

Geometric-decay phaseGII,l

H

…

H’

O(k/logk)

1/8

…

O(loglogk) stages

n signal nodes

II. Overview/High-Level Intuition & III. Graph Properties

Seeding Phasewith prob. 1/k

GI

H

…

…

ck measurement nodes

H ’

Many Singletons

Many Doubletons

…

n signal nodes

II. Overview/High-Level Intuition & III. Graph Properties

Geometric-decay phaseO(loglogk)

GII,l

H

Many Multitons

…

with prob. 2/k

…

H ’

ck/2 measurement

nodes

…

n signal nodes

II. Overview/High-Level Intuition & III. Graph Properties

Geometric-decay phaseO(loglogk)

GII,l

H

Many Multitons

…

with prob. 4/k

…

H ’

ck/4 measurement

nodes

O(k/logk)

…

n signal nodes

II. Overview/High-Level Intuition & III. Graph Properties

Cleaning-up PhaseGIII

H

…

Many Multitons

with prob. logk/k

H ’

…

…

|V(H’)|=k

c(k/logk)log(k/logk) = O(k)

measurement nodes

n signal nodes

II. Overview/High-Level Intuition & IV. Measurement Design

Bipartite Graph →Measurement Matrixx1

b1

x1

x2

x3

x4

x5

x6

x2

b1

b2

x3

b2

b3

x4

Adjacent Matrix

b3

b4

x5

b4

x6

II. Overview/High-Level Intuition & IV. Measurement Design

Bipartite Graph →Measurement Matrixx1

b1

x1

x2

x3

x4

x5

x6

x2

b1

b2

b3

b4

x5

II. Overview/High-Level Intuition & IV. Measurement Design

Bipartite Graph →Measurement Matrixx1

b1

x1

x2

x3

x4

x5

x6

x2

b1

b2

b3

b4

b1,1

b1,2

b1,3

x5

b1,4

b1,5

α = (π/2)/n

unit phase

II. Overview/High-Level Intuition & V. Reconstruction Algorithm

Bipartite Graph →Measurement MatrixGuess:

arctan(b1,2/ib1,1)/ α = 2

x1

b1

x2 ≠ 0 and |x2|= b1,1/cos2α

x2

Verify:

|x2|= b1,5

?

b1,1

b1,2

b1,3

x5

b1,4

b1,5

α = (π/2)/n

unit phase

Seeding Phase: Giant Connected Component

O(k) right nodes

Each edge appears with prob. 1/k

H

O(k) right nodes are singletons

O(k) right nodes are doubletons

O(k) different edges in graph H’

(By Coupon Collection)

H’

EXPECTATION!

Size of H’ is (1-fI)k

(By percolation results)

Geometric-decay Phase

O(fII,l-1k) right nodes

Each edge appears with prob. 1/fII,l-1k

H

O(fII,l-1k) right nodes are resolvable

multitons

O(fII,l-1k) different nodes appended

in graph H’

(By Coupon Collection)

H’

EXPECTATION!

Number of Measurements

Seeding Phase

ck measurement nodes

…

…

O(k)

Geometric-decay

Phase

…

cfII,l-1k measurement nodes

O(k)

Cleaning-up

Phase

c(k/logk)log(k/logk)

measurement nodes

…

…

O(k)

n signal nodes

Decoding Complexity and Correctness

- BFS: O(|V|+|E|) for a graph G(V,E). O(k) in the seeding phase.
- “Cancelling out”: O(logk) for a right node. Overall decoding complexity is O(klogk).
- (1-εII,l-1)fII,l-1<gII,l-1<(1+εII,l-1)flI,l-1 hold for all l.
- Generalized/traditional coupon collection
- Chernoff bound
- Percolation results
- Union bound

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