Chun Lam Chan , Pak Hou Che and Sidharth Jaggi The Chinese University of Hong Kong

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

bounds with efficient algorithmsChun Lam Chan, Pak Hou Che and Sidharth Jaggi

The Chinese University of Hong KongVenkatesh Saligrama

Boston University




Boston University

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Boston University

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Literature No error: [DR82], [DRR89] With small error ϵ:

Upper bound: [AS09], [SJ10]

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Literature No error: [DR82], [DRR89] With small error ϵ:

Upper bound: [AS09], [SJ10]

Lower bound: [Folklore]


bounds with efficient algorithms

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

(COMP)

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Noiseless CBP

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Noiseless CBP

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Discard

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Noiseless CBP Sample g times to form a

group

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11


group

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12


group

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13


group

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14


group

Total non-defective items drawn:

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15


group


Coupon collection:

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16


group


Coupon collection:

Conclusion:

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Noisy CBP

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Noisy CBP

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19

Noisy CBP

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Noisy CBP

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21

Noiseless COMP

22

Noiseless COMP

23

Noiseless COMP

24

Noiseless COMP

25

Noiseless COMP

26

Noisy COMP

27

Noisy COMP

28

Noisy COMP

29

Noisy COMP

30

Noisy COMP

31

Noisy COMP

32

Noisy COMP

33

Simulations

0 100 200 300 400 500 600 700 8000

1

Experimental; q=0

Theoretical-lower; q=0

Theoretical-upper;q=0

number of tests (T)

succ

ess r

ate

34

Simulations

0 500 1000 1500 2000 2500 30000

1

Experimental; q=0Experimental; q=0.1Experimental; q=0.2Theoretical-lower; q=0Theoretical-lowerl; q=0.1Theoretical-lower; q=0.2Theoretical-upper;q=0Theoretical-lower; q=0.1Theoretical-lower; q=0.2

number of tests (T)

succ

ess r

ate

35

Summary

CBP COMPNoiselessNoisy

With small error ,

36

End

Thanks

37

Noiseless COMP

x 0 0 1 0 0 0 1 0 0

M y0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 00 0 1 1 0 1 1 0 0 1

38

x 0 0 1 0 0 0 1 0 0

M y0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 00 0 1 1 0 1 1 0 0 1

0 10 11 0 x90 1 → 00 11 00 1

Noiseless COMP

39

Noiseless COMP

x 0 0 1 0 0 0 1 0 0

M y0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 00 0 1 1 0 1 1 0 0 1

0 01 10 0 x71 1 → 11 10 01 1

40

Noiseless COMP

x 0 0 1 0 0 0 1 0 0

M y0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 00 0 1 1 0 1 1 0 0 1

1 11 10 0 x40 1 → 11 10 01 1

41

Noiseless COMP

x 0 0 1 0 0 0 1 0 0

M y0 1 1 1 0 0 0 0 0 10 0 0 1 0 0 1 0 0 10 1 0 0 0 0 0 0 1 01 1 1 0 0 0 1 1 0 10 0 1 1 0 1 1 0 0 10 0 0 0 1 0 0 1 1 00 0 1 1 0 1 1 0 0 1

1 1 0 0 0 11 1 1 1 0 10 0 x4 0 0 x7 1 0 x9

(a) 0 1 → 1 (b) 1 1 → 1 (c) 0 1 → 01 1 1 1 0 10 0 0 0 1 01 1 1 1 0 1

42

Noisy COMPx 0 0 1 0 0 0 1 0 0

M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

0 00 00 11 01 10 01 1

43

Noisy COMPx 0 0 1 0 0 0 1 0 0

M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

0 00 00 1 x31 0 → 11 10 01 1

If then =1 else =0

44

Noisy COMPx 0 0 1 0 0 0 1 0 0

M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

1 00 01 1 x21 0 → 11 10 00 1

45

Noisy COMPx 0 0 1 0 0 0 1 0 0

M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

0 01 00 1 x71 0 → 00 10 01 1

46

Noisy COMPx 0 0 1 0 0 0 1 0 0

M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

1 0 0 0 0 00 0 0 0 1 01 1 x2 0 1 x3 0 1 x7

(a) 1 0 → 1 (b) 1 0 → 1 (c) 1 0 → 01 1 1 1 0 10 0 0 0 0 00 1 1 1 1 1

47

Noisy COMPx 0 0 1 0 0 0 1 0 0

M y ν ŷ0 1 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 1 0 0 1 1 00 1 0 0 0 0 0 0 1 0 1 11 1 1 0 0 0 1 1 1 1 + 1 → 00 1 1 1 0 1 0 0 0 1 0 10 0 0 0 1 0 0 1 1 0 0 00 0 1 1 0 1 1 0 0 1 0 1

1 0 0 0 0 00 0 0 0 1 01 1 x2 0 1 x3 0 1 x7

(a) 1 0 → 1 (b) 1 0 → 1 (c) 1 0 → 01 1 1 1 0 10 0 0 0 0 00 1 1 1 1 1

Chun Lam Chan , Pak Hou Che and Sidharth Jaggi The Chinese University of Hong Kong

Text of Chun Lam Chan , Pak Hou Che and Sidharth Jaggi The Chinese University of Hong Kong

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Chun Lam Chan , Pak Hou Che and Sidharth Jaggi The Chinese University of Hong Kong

Text of Chun Lam Chan , Pak Hou Che and Sidharth Jaggi The Chinese University of Hong Kong