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 2 ? ? n m k I. Introduction

b Compressive Phase Retrieval 2 ? ? k I. Introduction -2e iπ/3 Complex number m n 2 x → -x x →e iθ x

b Compressive Phase Retrieval ? ? k I. Introduction Applications: X-ray crystallography, Optics, Astronomical imaging… m n 2 Complex number

Compressive Phase Retrieval 2 ? ? 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: [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. b m n

x2x2 x6x6 x5x5 x4x4 x3x3 b1b1 b2b2 b3b3 b4b4 x1x1 Bipartite graph n signal nodesO(k) measurement nodes k non-zero components II. Overview/High-Level Intuition 3

x2x2 x6x6 x5x5 x4x4 x3x3 b1b1 b2b2 b3b3 b4b4 x1x1 Bipartite Graph → Measurement Matrix II. Overview/High-Level Intuition & IV. Measurement Design x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 b1b1 b2b2 b3b3 b4b4 Adjacent Matrix 4

x2x2 x6x6 x4x4 b1b1 b2b2 b3b3 b4b4 Useful Measurement Nodes n signal nodesO(k) measurement nodes II. Overview/High-Level Intuition 5

b1b1 b2b2 b3b3 b4b4 x2x2 x6x6 x4x4 Doubleton Multiton Singleton Useful Measurement Nodes II. Overview/High-Level Intuition 5

b1b1 b2b2 b3b3 b4b4 x2x2 x6x6 x4x4 Doubleton Multiton Singleton Useful Measurement Nodes Magnitude recovery Phase recovery Resolvable Δ |x 2 +x 4 | |x 2 | |x 4 | Solving a quadratic equation “Cancelling out” process: II. Overview/High-Level Intuition & V. Reconstruction Algorithm 5

Three Phases … n signal nodes … … Seeding Phase: Singletons and Resolvable Doubletons … Geometric-decay Phase: Resolvable Multitons … Cleaning-up Phase: Resolvable Multitons O(k) measurement nodes II. Overview/High-Level Intuition 6

Seeding Phase II. Overview/High-Level Intuition … n signal nodes … GIGI H 7

Seeding Phase II. Overview/High-Level Intuition … n signal nodes … GIGI “Sigma” Structure 7 x1x1 x2x2 H x2x2 x1x1

Seeding Phase II. Overview/High-Level Intuition … n signal nodes … GIGI H’ 7 H 1/2

Geometric-decay phase II. Overview/High-Level Intuition … n signal nodes … G II,l 1/4 H’ 8 H

Geometric-decay phase II. Overview/High-Level Intuition … n signal nodes … G II,l 1/8 O(k/logk) O(loglogk) stages H’ 8 H

Cleaning-up Phase II. Overview/High-Level Intuition … n signal nodes … G III |V( H ’)|=k H’ 9 H

Seeding Phase II. Overview/High-Level Intuition & III. Graph Properties … n signal nodes … ck measurement nodes … GIGI with prob. 1/k H H ’ Many Singletons Many Doubletons 10

Geometric-decay phase II. Overview/High-Level Intuition & III. Graph Properties … n signal nodes … ck/2 measurement nodes … G II,l H H ’ with prob. 2/k O(loglogk) Many Multitons 11

Geometric-decay phase II. Overview/High-Level Intuition & III. Graph Properties … n signal nodes … ck/4 measurement nodes … G II,l H H ’ with prob. 4/k O(k/logk) O(loglogk) 11 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 … G III H H ’ |V( H ’)|=k with prob. logk/k Many Multitons 12

x2x2 x6x6 x5x5 x4x4 x3x3 b1b1 b2b2 b3b3 b4b4 x1x1 Bipartite Graph → Measurement Matrix II. Overview/High-Level Intuition & IV. Measurement Design Adjacent Matrix 13 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 b1b1 b2b2 b3b3 b4b4

x2x2 x5x5 b1b1 x1x1 Bipartite Graph → Measurement Matrix II. Overview/High-Level Intuition & IV. Measurement Design x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 b1b1 b2b2 b3b3 b4b4 13

x2x2 x5x5 b1b1 x1x1 Bipartite Graph → Measurement Matrix II. Overview/High-Level Intuition & IV. Measurement Design x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 b1b1 b2b2 b3b3 b4b4 α = (π/2)/n unit phase b 1,1 b 1,2 b 1,3 b 1,4 b 1,5 13

x2x2 x5x5 b1b1 x1x1 Bipartite Graph → Measurement Matrix II. Overview/High-Level Intuition & V. Reconstruction Algorithm α = (π/2)/n unit phase b 1,1 b 1,2 b 1,3 b 1,4 b 1,5 arctan(b 1,2 /ib 1,1 )/ α = 2 Guess: x 2 ≠ 0 and |x 2 | = b 1,1 /cos2α Verify: |x 2 | = b 1,5 ? 13

Seeding Phase: Giant Connected Component VI. Parameters Design O(k) different edges in graph H’ (By Coupon Collection) O(k) right nodes Each edge appears with prob. 1/k O(k) right nodes are singletons O(k) right nodes are doubletons Size of H’ is (1-f I )k (By percolation results) H H’ EXPECTATION! 14

Geometric-decay Phase VI. Parameters Design O(f II,l-1 k) different nodes appended in graph H’ (By Coupon Collection) O(f II,l-1 k) right nodes Each edge appears with prob. 1/f II,l-1 k O(f II,l-1 k) right nodes are resolvable multitons H H’ EXPECTATION! 15

VII. Performance of Algorithm Number of Measurements … n signal nodes … … … … Seeding Phase Geometric-decay Phase Cleaning-up Phase cf II,l-1 k measurement nodes O(k) ck measurement nodes O(k) c(k/logk)log(k/logk) measurement nodes O(k) 16

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-ε II,l-1 )f II,l-1 <g II,l-1 <(1+ε II,l-1 )f lI,l-1 hold for all l. – Generalized/traditional coupon collection – Chernoff bound – Percolation results – Union bound VII. Performance of Algorithm 17

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