Compressive sensing meets group testing: LP decoding for non-linear (disjunctive) measurements

1. Compressive sensing meets group testing:LP decoding for non-linear (disjunctive) measurements VenkateshSaligrama Boston University Chun Lam Chan, SidharthJaggi and Samar Agnihotri The Chinese University of Hong Kong

2. n-d d Compressive sensing Lower bound: What’s known OMP: BP:

3. n-d q d 0 1 q 0 1 Group testing: Lower bound: What’s known …[CCJS11] Noisy Combinatorial OMP: This work: Noisy Combinatorial BP:

4. Group-testing model p=1/D [CCJS11]

5. CBP-LP negative tests positive tests weight relaxation

6. NCBP-LP Minimum distance decoding “Slack”/noise variables

7. “Perturbation analysis” For all (“Conservation of mass”) 2. LP change under a single ρi(Case analysis) 3. LP change under all n(n-d) ρis(Chernoff/union bounds) 4. LP change under all (∞) perturbations (Convexity) (5.) If d unknown but bounded, try ‘em all (“Info thry”)

8. 1. Perturbation vectors d n-d NCBLP feasible set ρi ρj x

9. 2. LP value change withONE perturbation vector x

10. 3. LP value change withEACH (n(n-d)) perturbation vector x Prob error < Union bound Chernoff bound

11. 4. LP value change underALL (∞) perturbations x Prob error < min LP = x Convexity of

12. (5.) NCBP-LPs Information-theoretic argument – just a single d “works”.

13. Bonus: NCBP-SLPs ONLY positive tests ONLY negative tests

15. n-d Noiseless CBP d

16. n-d Noiseless CBP d Discard

17. Noiseless CBP n-d • Sample g times to form a group d

18. Noiseless CBP n-d • Sample g times to form a group d

19. Noiseless CBP n-d • Sample g times to form a group d

20. Noiseless CBP n-d • Sample g times to form a group d

21. Noiseless CBP n-d • Sample g times to form a group • Total non-defective items drawn: d

22. Noiseless CBP n-d • Sample g times to form a group • Total non-defective items drawn: • Coupon collection: d

23. Noiseless CBP n-d • Sample g times to form a group • Total non-defective items drawn: • Coupon collection: • Conclusion: d

24. n-d Noisy CBP d

25. n-d Noisy CBP d

26. n-d Noisy CBP d

27. n-d Noisy CBP d

28. Noiseless COMP

29. Noiseless COMP

30. Noiseless COMP

31. Noiseless COMP

32. Noiseless COMP

33. Noisy COMP

34. Noisy COMP

35. Noisy COMP

36. Noisy COMP

37. Noisy COMP

38. Noisy COMP

39. Noisy COMP

40. Simulations

41. Simulations

42. Summary • With small error ,

43. Noiseless COMP

44. Noiseless COMP

45. Noiseless COMP

46. Noiseless COMP

47. Noiseless COMP

48. Noisy COMP

49. Noisy COMP If then =1 else =0

50. Noisy COMP