LP decoding for nonlinear disjunctive measurements Chun Lam Chan Sidharth Jaggi and Samar Agnihotri The Chinese University of Hong Kong Venkatesh Saligrama Boston University 2 ID: 598082
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Slide1
Compressive sensing meets group testing:LP decoding for non-linear (disjunctive) measurements
Chun Lam Chan, Sidharth Jaggi and Samar AgnihotriThe Chinese University of Hong Kong
Venkatesh
SaligramaBoston UniversitySlide2
2
n-d
d
Lower bound:OMP:
What’s known
BP:
Compressive sensingSlide3
3
n-d
d
Group testing:
1
0
0
q
1
q
Lower bound:
Noisy Combinatorial OMP:
What’s known
This work:
Noisy Combinatorial BP:
…[CCJS11]Slide4
4
Group-testing model
p
=1/D[CCJS11]Slide5
5
CBP-LPrelaxationweight
positive tests
negative testsSlide6
6
NCBP-LP
“Slack”/noise variables
Minimum distance
decodingSlide7
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”) Slide8
8
1. Perturbation vectors
NCBLP
feasible setxρi
ρj
d
n
-d
Slide9
9
2. LP value change withONE perturbation vector
xSlide10
10
3. LP value change withEACH (n(n-d)) perturbation vector
Union bound
Chernoff boundProb error <
xSlide11
11
4. LP value change underALL (∞) perturbations
x
Prob
error <
Convexity of
m
in LP = xSlide12
12
(5.) NCBP-LPs
Information-theoretic argument –
just a single d “works”.Slide13
13
Bonus: NCBP-SLPs
ONLYnegative tests
ONLYpositive testsSlide14
14Slide15
Noiseless CBP15
n-d
dSlide16
Noiseless CBP16
n-d
d
DiscardSlide17
Noiseless CBP17
Sample g times to form a group
n-d
dSlide18
Noiseless CBP18
Sample g times to form a group
n-d
dSlide19
Noiseless CBP19
Sample g times to form a group
n-d
dSlide20
Noiseless CBP20
Sample g times to form a group
n-d
dSlide21
Noiseless CBP21
Sample g times to form a groupTotal non-defective items drawn:
n-d
dSlide22
Noiseless CBP22
Sample g times to form a groupTotal non-defective items drawn:Coupon collection:
n-d
dSlide23
Noiseless CBP23
Sample g times to form a groupTotal non-defective items drawn:Coupon collection:Conclusion:
n-d
dSlide24
Noisy CBP24
n-d
dSlide25
Noisy CBP25
n-d
dSlide26
Noisy CBP26
n-d
dSlide27
Noisy CBP27
n-d
dSlide28
Noiseless COMP
28Slide29
Noiseless COMP
29Slide30
Noiseless COMP
30Slide31
Noiseless COMP
31Slide32
Noiseless COMP
32Slide33
Noisy COMP
33Slide34
Noisy COMP
34Slide35
Noisy COMP
35
Slide36
Noisy COMP
36Slide37
Noisy COMP
37Slide38
Noisy COMP
38Slide39
Noisy COMP
39Slide40
Simulations40Slide41
Simulations41Slide42
Summary42
CBP
COMP
NoiselessNoisy
With small error , Slide43
Noiseless COMP
x
0
01
0
0
0
1
0
0
M
y
0
1
1
1
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
1
0
1
0
0
0
0
0
0
1
0
1
1
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
1
1
0
1
1
0
0
1
43Slide44
x
0
0
1
0
0
0
1
0
0
M
y
0
1
1
1
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
1
0
1
0
0
0
0
0
0
1
0
1
1
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
1
1
0
1
1
0
0
1
0
1
0
1
1
0
x
9
0
1
→
0
0
1
1
0
0
1
Noiseless COMP
44Slide45
Noiseless COMP
x
0
01
0
0
0
1
0
0
M
y
0
1
1
1
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
1
0
1
0
0
0
0
0
0
1
0
1
1
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
1
1
0
1
1
0
0
1
0
0
1
1
0
0
x
7
1
1
→
1
1
1
0
0
1
1
45Slide46
Noiseless COMP
x
0
01
0
0
0
1
0
0
M
y
0
1
1
1
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
1
0
1
0
0
0
0
0
0
1
0
1
1
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
1
1
0
1
1
0
0
1
1
1
1
1
0
0
x
4
0
1
→
1
1
1
0
0
1
1
46Slide47
Noiseless COMP
x
0
01
0
0
0
1
0
0
M
y
0
1
1
1
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
1
0
1
0
0
0
0
0
0
1
0
1
1
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
1
1
0
1
1
0
0
1
1
1
0
0
0
1
1
1
1
1
0
1
0
0
x
4
0
0
x
7
1
0
x
9
(a)
0
1
→
1
(b)
1
1
→
1
(
c)
0
1
→
0
1
1
1
1
0
1
0
0
0
0
1
0
1
1
1
1
0
1
47Slide48
Noisy COMP
x
0
01
0
0
0
1
0
0
M
y
ν
ŷ
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
1
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
1
0
0
0
1
1
1
1
+
1
→
0
0
1
1
1
0
1
0
0
0
1
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
0
0
0
0
0
1
1
0
1
1
0
0
1
1
48Slide49
Noisy COMP
x
0
01
0
0
0
1
0
0
M
y
ν
ŷ
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
1
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
1
0
0
0
1
1
1
1
+
1
→
0
0
1
1
1
0
1
0
0
0
1
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
0
0
0
0
0
1
x
3
1
0
→
1
1
1
0
0
1
1
49
If
then
=1
else
=0
Slide50
Noisy COMP
x
0
01
0
0
0
1
0
0
M
y
ν
ŷ
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
1
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
1
0
0
0
1
1
1
1
+
1
→
0
0
1
1
1
0
1
0
0
0
1
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
1
0
0
0
1
1
x
2
1
0
→
1
1
1
0
0
0
1
50Slide51
Noisy COMP
x
0
01
0
0
0
1
0
0
M
y
ν
ŷ
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
1
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
1
0
0
0
1
1
1
1
+
1
→
0
0
1
1
1
0
1
0
0
0
1
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
0
0
1
0
0
1
x
7
1
0
→
0
0
1
0
0
1
1
51Slide52
Noisy COMP
x
0
01
0
0
0
1
0
0
M
y
ν
ŷ
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
1
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
1
0
0
0
1
1
1
1
+
1
→
0
0
1
1
1
0
1
0
0
0
1
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
1
0
1
1
x
2
0
1
x
3
0
1
x
7
(a)
1
0
→
1
(b)
1
0
→
1
(
c)
1
0
→
0
1
1
1
1
0
1
0
0
0
0
0
0
0
1
1
1
1
1
52Slide53
Noisy COMP
x
0
01
0
0
0
1
0
0
M
y
ν
ŷ
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
1
1
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
1
0
0
0
1
1
1
1
+
1
→
0
0
1
1
1
0
1
0
0
0
1
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
1
0
0
0
0
0
0
0
0
0
1
0
1
1
x
2
0
1
x
3
0
1
x
7
(a)
1
0
→
1
(b)
1
0
→
1
(
c)
1
0
→
0
1
1
1
1
0
1
0
0
0
0
0
0
0
1
1
1
1
1
53