SUPER: Sparse signals with Unknown Phases Efficiently Recovered

Sheng Cai, Mayank Bakshi, Sidharth Jaggi and Minghua Chen

The Chinese University of Hong Kong

b

Compressive Sensing

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? n

m

k

I. Introduction

b

Compressive Phase Retrieval

2

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k

I. Introduction

-2eiπ/3

Complex number

m

n

2 x → -x

x →eiθx

b

Compressive Phase Retrieval?

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k

I. Introduction

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

m

n

2

Complex number

Compressive Phase Retrieval

2

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I. Introduction

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.

bm

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x2

x6

x5

x4

x3

b1

b2

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x1

Bipartite graph

n signal nodes O(k) measurement nodes

k non-zerocomponents

II. Overview/High-Level Intuition

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x2

x6

x5

x4

x3

b1

b2

b3

b4

x1

Bipartite Graph → Measurement Matrix

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

x1 x2 x3 x4 x5 x6

b1

b2

b3

b4

Adjacent Matrix

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x2

x6

x4

b1

b2

b3

b4

Useful Measurement Nodes

n signal nodes O(k) measurement nodes

II. Overview/High-Level Intuition

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b1

b2

b3

b4

x2

x6

x4

Doubleton

Multiton

Singleton

Useful Measurement Nodes

II. Overview/High-Level Intuition

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b1

b2

b3

b4

x2

x6

x4

Doubleton

Multiton

Singleton

Useful Measurement Nodes

Magnitude recovery

Phase recovery

Resolvable

Resolvable

Δ

|x2+x4|

|x2|

|x4|Solving a quadratic equation

“Cancelling out” process:

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

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Three Phases

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n signal nodes

… …

Seeding Phase: Singletons and Resolvable Doubletons

… Geometric-decay Phase:Resolvable Multitons

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Cleaning-up Phase:Resolvable Multitons

O(k) measurement nodes

II. Overview/High-Level Intuition

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Seeding Phase

II. Overview/High-Level Intuition

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n signal nodes

…

GI

H

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Seeding Phase

II. Overview/High-Level Intuition

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n signal nodes

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GI

“Sigma” Structure

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x1 x2Hx2

x1

Seeding Phase

II. Overview/High-Level Intuition

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n signal nodes

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GI

H’

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H

1/2

Geometric-decay phase

II. Overview/High-Level Intuition

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n signal nodes

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GII,l

1/4

H’

8

H

Geometric-decay phase

II. Overview/High-Level Intuition

…

n signal nodes

…

GII,l

1/8 O(k/logk)

O(loglogk) stages

H’

8

H

Cleaning-up Phase

II. Overview/High-Level Intuition

…

n signal nodes

…

GIII

|V(H’)|=k

H’

9

H

Seeding Phase

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

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n signal nodes

…

ck measurement nodes

…

GI

with prob. 1/k

H

H ’Many SingletonsMany Doubletons

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Geometric-decay phase

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

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n signal nodes

…

ck/2 measurement nodes

…

GII,lH

H ’

with prob. 2/k

O(loglogk)

Many Multitons

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Geometric-decay phase

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

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n signal nodes

…

ck/4 measurement nodes

…

GII,lH

H ’

with prob. 4/k

O(k/logk)

O(loglogk)

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Many Multitons

Cleaning-up Phase

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

…

n signal nodes

…

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

…

GIIIH

H ’

|V(H’)|=k

with prob. logk/k

Many Multitons

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x2

x6

x5

x4

x3

b1

b2

b3

b4

x1

Bipartite Graph → Measurement Matrix

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

Adjacent Matrix

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x1 x2 x3 x4 x5 x6

b1

b2

b3

b4

x2

x5

b1

x1

Bipartite Graph → Measurement Matrix

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

x1 x2 x3 x4 x5 x6

b1

b2

b3

b4

13

x2

x5

b1

x1

Bipartite Graph → Measurement Matrix

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

x1 x2 x3 x4 x5 x6

b1

b2

b3

b4

α = (π/2)/nunit phase

b1,1

b1,2

b1,3

b1,4

b1,5

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x2

x5

b1

x1

Bipartite Graph → Measurement Matrix

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

α = (π/2)/nunit phase

b1,1

b1,2

b1,3

b1,4

b1,5

arctan(b1,2/ib1,1)/ α = 2Guess:

x2 ≠ 0 and |x2| = b1,1/cos2α

Verify: |x2| = b1,5 ?

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Seeding Phase: Giant Connected Component

VI. Parameters Design

O(k) different edges in graph H’(By Coupon Collection)

O(k) right nodesEach edge appears with prob. 1/k

O(k) right nodes are singletons O(k) right nodes are doubletons

Size of H’ is (1-fI)k(By percolation results)

H

H’

EXPECTATION!

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Geometric-decay Phase

VI. Parameters Design

O(fII,l-1k) different nodes appendedin graph H’ (By Coupon Collection)

O(fII,l-1k) right nodesEach edge appears with prob. 1/fII,l-1k

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

H

H’

EXPECTATION!

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VII. Performance of Algorithm

Number of Measurements

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n signal nodes

… ……

…

Seeding Phase

Geometric-decayPhase

Cleaning-upPhase

cfII,l-1k measurement nodes

O(k)

ck measurement nodes

O(k)

c(k/logk)log(k/logk) measurement nodes

O(k)

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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

VII. Performance of Algorithm

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