SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

1. Faster Higher Stronger SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits SHO-FA: compressive sensing with Robust measurements, and bits order-optimal complexity, MayankBakshi, SidharthJaggi, ShengCaiand MinghuaChen The Chinese University of Hong Kong

2. Compressive sensing ? m ? n k k ≤ m<n

3. Robust compressive sensing ? e z Random y=A(x+z)+e Approximate sparsity Measurement noise

5. # of measurements Lower bound °CM’06 °GSTV’06 °TG’07 °SBB’06 °C’08 °IR’08 Lower bound °RS’60 This work  °DJM’11 °MV’12,KP’12 Decoding complexity

6. SHO(rt)-FA(st) Good Bad Good Bad

7. High-Level Overview 3 3 4 4 4 4 ck ck n n k=2 k=2

8. High-Level Overview How to guarantee the existence of leaf node How to find the leaf nodes and utilize the leaf nodes to do decoding 3 3 4 4 4 ck n k=2

9. Q1: How to guarantee the existence of leaf node? Left-regular Bipartite Graph d=3 A ck n

10. Q1: How to guarantee the existence of leaf node? Existence of leaf nodes e.g., existence of 2-core in d-uniform hypergraph Sharp transition M. T. Goodrich and M. Mitzenmacher, “Invertible bloom lookup tables,” ArXiv.org e-Print archive, arXiv:1101.2245 [cs.DS], 2011.

11. Q1: How to guarantee the existence of leaf node? Existence of “Many” leafs L+L’≥2|S| ≥2|S| |S| 3|S|≥L+2L’ (L+L’)/(L+2L’) ≥2/3 L/(L+L’) ≥1/2

12. Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding? Bipartite Graph → Sensing Matrix d=3 Distinct weights A ck n

13. Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding? Bipartite Graph → Sensing Matrix A ck n

14. Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding? Encoding

15. Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding? Decoding

16. Decoding – First Iteration

17. Decoding – Second Iteration Verification Measurements

18. Decoding – Third Iteration

19. Decoding – Fourth Iteration

20. SHO-FA v.s. Pick-Up-Sticks Peeling process: Iterative Decoding Observation: Identification Check: Verification Picking up a “top” stick: Leaf-based decoding

21. Robust Compressive Sensing Phase error Propagation error …… …… Pawar, Sameer and Ramchandran, Kannan, “A Hybrid DFT-LDPC Framework for Fast and Robust Compressive Sensing”

22. Truncated Reconstruction Threshold

23. Correlated Measurements Phase quantization

24. Correlated Measurements (First bit) Phase quantization

25. Correlated Measurements (Second bit)

26. Correlated Measurements (Third bit)

27. Additional Properties • Other works • Group Testing • Network Tomography • Reduce the number of measurements • Combine Identification and verification • More noise models • Sparse in different bases • Database query • ……

28. Thank you謝謝

29. 2-core in d-uniform hypergraph • The 2-core is the largest sub-hypergraph that has minimum degree at least 2. • The standard “peeling process” finds the 2-core: while there exists a vertex with degree 1, delete it and the corresponding hyperedges. Node degree 1 hyperedge

30. (Almost) S(x)-expansion ck n ≥2|S| |S|