Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth Jaggi The Chinese University of Hong Kong Venkatesh Saligrama Boston University

Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth Jaggi The Chinese University of Hong Kong Venkatesh Saligrama Boston University n-d d

Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms Chun Lam Chan, Pak Hou Che and Sidharth Jaggi The Chinese University of Hong Kong Venkatesh Saligrama Boston University n-d d

Literature  No error: [DR82], [DRR89]  With small error ϵ :  Upper bound: [AS09], [SJ10] 4

Literature  No error: [DR82], [DRR89]  With small error ϵ :  Upper bound: [AS09], [SJ10]  Lower bound: [Folklore] 5

Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms

Algorithms motivated by Compressive Sensing 7  Combinatorial Basis Pursuit (CBP)  Combinatorial Orthogonal Matching Pursuit (COMP)

Noiseless CBP 8 n-d d

Noiseless CBP 9 n-d d Discard

Noiseless CBP 10  Sample g times to form a group n-d d

Noiseless CBP 11  Sample g times to form a group n-d d

Noiseless CBP 12  Sample g times to form a group n-d d

Noiseless CBP 13  Sample g times to form a group n-d d

Noiseless CBP 14  Sample g times to form a group  Total non-defective items drawn: n-d d

Noiseless CBP 15  Sample g times to form a group  Total non-defective items drawn:  Coupon collection: n-d d

Noiseless CBP 16  Sample g times to form a group  Total non-defective items drawn:  Coupon collection:  Conclusion: n-d d

Noisy CBP 17 n-d d

Noisy CBP 18 n-d d

Noisy CBP 19 n-d d

Noisy CBP 20 n-d d

Noiseless COMP 21

Noiseless COMP 22

Noiseless COMP 23

Noiseless COMP 24

Noiseless COMP 25

Noisy COMP 26

Noisy COMP 27

Noisy COMP 28

Noisy COMP 29

Noisy COMP 30

Noisy COMP 31

Noisy COMP 32

Simulations 33

Simulations 34

Summary 35  With small error,

End Thanks 36

Noiseless COMP x My

x My x9x9 01 → Noiseless COMP 38

Noiseless COMP x My x7x7 11 →

Noiseless COMP x My x4x4 01 →

Noiseless COMP x My x4x4 00x7x7 10x9x9 (a)01 → 1(b)11 → 1(c)01 →

Noisy COMP x My ν ŷ →

Noisy COMP x My ν ŷ → x3x3 10 →

Noisy COMP x My ν ŷ → x2x2 10 →

Noisy COMP x My ν ŷ → x7x7 10 →

Noisy COMP x My ν ŷ → x2x2 01x3x3 01x7x7 (a)10 → 1(b)10 → 1(c)10 →

Noisy COMP x My ν ŷ → x2x2 01x3x3 01x7x7 (a)10 → 1(b)10 → 1(c)10 →