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SUPER: Sparse signal s with Unknown Phases Efficiently Recovered. Sheng Cai , Mayank Bakshi , Sidharth Jaggi and Minghua Chen The Chinese University of Hong Kong. I. Introduction. Compressive Sensing. ?. b. m. ?. n. k. I. Introduction. Compressive Phase Retrieval. ?. b. m. ?.

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super sparse signal s with unknown phases efficiently recovered

SUPER: Sparse signalswith Unknown Phases Efficiently Recovered

ShengCai, MayankBakshi, SidharthJaggi and Minghua Chen

The Chinese University of Hong Kong

compressive phase retrieval

I. Introduction

Compressive Phase Retrieval

?

b

m

?

n

Complex number

-2eiπ/3

k

2

x → -x

x →eiθx

compressive phase retrieval1

I. Introduction

Compressive Phase Retrieval

?

b

m

?

n

Complex number

k

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

compressive phase retrieval2

I. Introduction

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.

bipartite graph

II. Overview/High-Level Intuition

Bipartite graph

x1

b1

x2

b2

x3

k non-zero

components

x4

b3

x5

b4

x6

n signal nodes

O(k) measurement nodes

useful measurement nodes

II. Overview/High-Level Intuition

Useful Measurement Nodes

b1

x2

b2

x4

b3

b4

x6

n signal nodes

O(k) measurement nodes

useful measurement nodes1

II. Overview/High-Level Intuition

Useful Measurement Nodes

b1

Singleton

x2

b2

Doubleton

x4

Multiton

b3

b4

x6

useful measurement nodes2

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

Useful Measurement Nodes

b1

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

three phases

II. Overview/High-Level Intuition

Three Phases

Seeding Phase:

Singletons and Resolvable Doubletons

Geometric-decay Phase:

Resolvable Multitons

Cleaning-up Phase:

Resolvable Multitons

O(k) measurement nodes

n signal nodes

seeding phase1

II. Overview/High-Level Intuition

Seeding Phase

x1

GI

x2

H

x1

x2

“Sigma” Structure

n signal nodes

geometric decay phase1

II. Overview/High-Level Intuition

Geometric-decay phase

GII,l

H

H’

O(k/logk)

1/8

O(loglogk) stages

n signal nodes

cleaning up phase

II. Overview/High-Level Intuition

Cleaning-up Phase

GIII

H

H’

|V(H’)|=k

n signal nodes

seeding phase3

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

Seeding Phase

with prob. 1/k

GI

H

ck measurement nodes

H ’

Many Singletons

Many Doubletons

n signal nodes

geometric decay phase2

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

Geometric-decay phase

O(loglogk)

GII,l

H

Many Multitons

with prob. 2/k

H ’

ck/2 measurement

nodes

n signal nodes

geometric decay phase3

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

Geometric-decay phase

O(loglogk)

GII,l

H

Many Multitons

with prob. 4/k

H ’

ck/4 measurement

nodes

O(k/logk)

n signal nodes

cleaning up phase1

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

Cleaning-up Phase

GIII

H

Many Multitons

with prob. logk/k

H ’

|V(H’)|=k

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

measurement nodes

n signal nodes

bipartite graph measurement matrix3

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

Bipartite Graph →Measurement Matrix

x1

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

bipartite graph measurement matrix4

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

Bipartite Graph →Measurement Matrix

Guess:

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

VI. Parameters Design

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 phase4

VI. Parameters Design

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

VII. Performance of Algorithm

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

VII. Performance of Algorithm

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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