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

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

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

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

5. 5 CBP-LP relaxation weight positive tests negative tests

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

7. 7 “Perturbation analysis” 1.For all (“Conservation of mass”) 2. LP change under a single ρ i (Case analysis) 3. LP change under all n(n-d) ρ i s (Chernoff/union bounds) 4. LP change under all (∞) perturbations (Convexity) (5.) If d unknown but bounded, try ‘em all (“Info thry”)

8. 8 1. Perturbation vectors NCBLP feasible set x ρiρi ρjρj dn-d

9. 9 2. LP value change with ONE perturbation vector x

10. 10 3. LP value change with EACH (n(n-d)) perturbation vector Union boundChernoff bound Prob error < x

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

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

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

14. 14

15. Noiseless CBP 15 n-d d

16. Noiseless CBP 16 n-d d Discard

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

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

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

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

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

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

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

24. Noisy CBP 24 n-d d

25. Noisy CBP 25 n-d d

26. Noisy CBP 26 n-d d

27. Noisy CBP 27 n-d d

28. Noiseless COMP 28

29. Noiseless COMP 29

30. Noiseless COMP 30

31. Noiseless COMP 31

32. Noiseless COMP 32

33. Noisy COMP 33

34. Noisy COMP 34

35. Noisy COMP 35

36. Noisy COMP 36

37. Noisy COMP 37

38. Noisy COMP 38

39. Noisy COMP 39

40. Simulations 40

41. Simulations 41

42. Summary 42  With small error,

43. Noiseless COMP x001000100 My 0111000001 0001001001 0100000010 1110001101 0011011001 0000100110 0011011001 43

44. x001000100 My 0111000001 0001001001 0100000010 1110001101 0011011001 0000100110 0011011001 01 01 10x9x9 01 → 0 01 10 01 Noiseless COMP 44

45. Noiseless COMP x001000100 My 0111000001 0001001001 0100000010 1110001101 0011011001 0000100110 0011011001 00 11 00x7x7 11 → 1 11 00 11 45

46. Noiseless COMP x001000100 My 0111000001 0001001001 0100000010 1110001101 0011011001 0000100110 0011011001 11 11 00x4x4 01 → 1 11 00 11 46

47. Noiseless COMP x001000100 My 0111000001 0001001001 0100000010 1110001101 0011011001 0000100110 0011011001 110001 111101 00x4x4 00x7x7 10x9x9 (a)01 → 1(b)11 → 1(c)01 → 0 111101 000010 111101 47

48. Noisy COMP x001000100 My ν ŷ 010100000000 000100100110 010000001011 1110001111+1 → 0 011101000101 000010011000 001101100101 00 00 01 10 11 00 11 48

49. Noisy COMP x001000100 My ν ŷ 010100000000 000100100110 010000001011 1110001111+1 → 0 011101000101 000010011000 001101100101 00 00 01x3x3 10 → 1 11 00 11 49

50. Noisy COMP x001000100 My ν ŷ 010100000000 000100100110 010000001011 1110001111+1 → 0 011101000101 000010011000 001101100101 10 00 11x2x2 10 → 1 11 00 01 50

51. Noisy COMP x001000100 My ν ŷ 010100000000 000100100110 010000001011 1110001111+1 → 0 011101000101 000010011000 001101100101 00 10 01x7x7 10 → 0 01 00 11 51

52. Noisy COMP x001000100 My ν ŷ 010100000000 000100100110 010000001011 1110001111+1 → 0 011101000101 000010011000 001101100101 100000 000010 11x2x2 01x3x3 01x7x7 (a)10 → 1(b)10 → 1(c)10 → 0 111101 000000 011111 52

53. Noisy COMP x001000100 My ν ŷ 010100000000 000100100110 010000001011 1110001111+1 → 0 011101000101 000010011000 001101100101 100000 000010 11x2x2 01x3x3 01x7x7 (a)10 → 1(b)10 → 1(c)10 → 0 111101 000000 011111 53