compressive sensing meets group testing: lp decoding for non-linear (disjunctive) measurements
DESCRIPTION
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:. - PowerPoint PPT PresentationTRANSCRIPT
Compressive sensing meets group testing:LP decoding for non-linear (disjunctive)
measurementsChun Lam Chan, Sidharth Jaggi and Samar Agnihotri
The Chinese University of Hong Kong
Venkatesh Saligrama
Boston University
n-dd
Lower bound:
OMP:
What’s known
BP:
Compressive sensing
n-dd
Group testing:
1
0
0q
1q
Lower bound:
Noisy Combinatorial OMP:
What’s known
This work: Noisy Combinatorial BP:
…[CCJS11]
4
Group-testing model
p=1/D[CCJS11]
5
CBP-LP
relaxation
weight
positive tests
negative tests
6
NCBP-LP
“Slack”/noise variables
Minimum distance decoding
7
“Perturbation analysis”
1.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
NCBLP feasible set
x
ρi
ρj
d n-d
9
2. LP value change withONE perturbation vector
x
10
3. LP value change withEACH (n(n-d)) perturbation vector
Union bound Chernoff bound
Prob error <x
11
4. LP value change underALL (∞) perturbations
x
Prob error <
Convexity of
min LP = x
12
(5.) NCBP-LPs
Information-theoretic argument – just a single d “works”.
13
Bonus: NCBP-SLPs
ONLYnegative tests
ONLYpositive tests
Noiseless CBP
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Noiseless CBP
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Discard
Noiseless CBP Sample g times to form a
group
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Noiseless CBP Sample g times to form a
group
n-dd
Noiseless CBP Sample g times to form a
group
n-dd
Noiseless CBP Sample g times to form a
group
n-dd
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
Noiseless COMP
Noiseless COMP
Noiseless COMP
Noiseless COMP
Noisy COMP
Noisy COMP
Noisy COMP
Noisy COMP
Noisy COMP
Noisy COMP
Noisy COMP
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
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
Summary
CBP COMP
Noiseless
Noisy
With small error ,
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
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
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
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
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
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
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
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
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
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
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