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

Compressive sensing meets group testing: LP decoding for non-linear (disjunctive) measurements. Venkatesh Saligrama Boston University. Chun Lam Chan, Sidharth Jaggi and Samar Agnihotri The Chinese University of Hong Kong. n-d. d. Compressive sensing. Lower bound:. What’s known. OMP:.

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

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  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

  14. n-d Noiseless CBP d

  15. n-d Noiseless CBP d Discard

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

  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 • Total non-defective items drawn: d

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

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

  23. n-d Noisy CBP d

  24. n-d Noisy CBP d

  25. n-d Noisy CBP d

  26. n-d Noisy CBP d

  27. Noiseless COMP

  28. Noiseless COMP

  29. Noiseless COMP

  30. Noiseless COMP

  31. Noiseless COMP

  32. Noisy COMP

  33. Noisy COMP

  34. Noisy COMP

  35. Noisy COMP

  36. Noisy COMP

  37. Noisy COMP

  38. Noisy COMP

  39. Simulations

  40. Simulations

  41. Summary • With small error ,

  42. Noiseless COMP

  43. Noiseless COMP

  44. Noiseless COMP

  45. Noiseless COMP

  46. Noiseless COMP

  47. Noisy COMP

  48. Noisy COMP If then =1 else =0

  49. Noisy COMP

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