SUPER: Sparse signal s with Unknown Phases Efficiently Recovered

1. SUPER: Sparse signalswith Unknown Phases Efficiently Recovered ShengCai, MayankBakshi, SidharthJaggi and Minghua Chen The Chinese University of Hong Kong

2. I. Introduction Compressive Sensing ? b m ? n k

3. I. Introduction Compressive Phase Retrieval ? b m ? n Complex number -2eiπ/3 k 2 x → -x x →eiθx

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

5. 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.

6. 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

7. II. Overview/High-Level Intuition & IV. Measurement Design Bipartite Graph →Measurement Matrix Adjacent Matrix x1 b1 x1 x2 x3 x4 x5 x6 x2 b1 b2 b2 x3 b3 b4 x4 b3 x5 b4 x6

8. II. Overview/High-Level Intuition Useful Measurement Nodes b1 x2 b2 x4 b3 b4 x6 n signal nodes O(k) measurement nodes

9. II. Overview/High-Level Intuition Useful Measurement Nodes b1 Singleton x2 b2 Doubleton x4 Multiton b3 b4 x6

10. 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

11. 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

12. II. Overview/High-Level Intuition Seeding Phase GI H … … n signal nodes

13. II. Overview/High-Level Intuition Seeding Phase x1 GI x2 H x1 x2 … “Sigma” Structure … n signal nodes

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

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

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

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

18. 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

19. 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

20. 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

21. 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

22. II. Overview/High-Level Intuition & IV. Measurement Design Bipartite Graph →Measurement Matrix x1 b1 x1 x2 x3 x4 x5 x6 x2 b1 b2 x3 b2 b3 x4 Adjacent Matrix b3 b4 x5 b4 x6

23. 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 x5

24. 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

25. 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

26. 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)

27. 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!

28. 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

29. 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

30. Thank you謝謝