non-adaptive probabilistic group testing with noisy measurements: near-optimal bounds with efficient...
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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 Kong
Venkatesh Saligrama
Boston University
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 Kong
Venkatesh Saligrama
Boston University
n-dd
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 Kong
Venkatesh Saligrama
Boston University
n-dd
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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]
Non-adaptive probabilistic group testing with noisy measurements: Near-optimal
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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Noiseless CBP Sample g times to form a
group
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Noiseless CBP Sample g times to form a
group
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Noiseless CBP Sample g times to form a
group
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Noiseless CBP Sample g times to form a
group
Total non-defective items drawn:
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Noiseless CBP Sample g times to form a
group
Total non-defective items drawn:
Coupon collection:
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Noiseless CBP Sample g times to form a
group
Total non-defective items drawn:
Coupon collection:
Conclusion:
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Noisy CBP
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Noisy CBP
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Noisy CBP
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Noisy CBP
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Noiseless COMP
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Noiseless COMP
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Noiseless COMP
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Noiseless COMP
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Noiseless COMP
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Noisy COMP
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Noisy COMP
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Noisy COMP
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Noisy COMP
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Noisy COMP
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Noisy COMP
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Noisy COMP
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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
rate
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Simulations
0 500 1000 1500 2000 2500 30000
1
Experimental; q=0
Experimental; q=0.1
Experimental; q=0.2
Theoretical-lower; q=0
Theoretical-lowerl; q=0.1
Theoretical-lower; q=0.2
Theoretical-upper;q=0
Theoretical-lower; q=0.1
Theoretical-lower; q=0.2
number of tests (T)
succ
ess
rate
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Summary
CBP COMP
Noiseless
Noisy
With small error ,
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End
Thanks
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Noiseless COMP
x 0 0 1 0 0 0 1 0 0
M y
0 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 0
0 0 1 1 0 1 1 0 0 1
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x 0 0 1 0 0 0 1 0 0
M y
0 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 0
0 0 1 1 0 1 1 0 0 1
0 10 11 0 x90 1 → 00 11 00 1
Noiseless COMP
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Noiseless COMP
x 0 0 1 0 0 0 1 0 0
M y
0 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 0
0 0 1 1 0 1 1 0 0 1
0 01 10 0 x71 1 → 11 10 01 1
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Noiseless COMP
x 0 0 1 0 0 0 1 0 0
M y
0 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 0
0 0 1 1 0 1 1 0 0 1
1 11 10 0 x40 1 → 11 10 01 1
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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
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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
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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
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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 x2
1 0 → 1
1 10 00 1
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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
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Noisy COMP
x 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
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Noisy COMP
x 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
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