compressive sensing sho-fa: robust compressive sensing with order-optimal complexity, measurements,...
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compressive sensingSHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits
Mayank Bakshi, Sidharth Jaggi, Sheng Cai and Minghua ChenThe Chinese University of Hong Kong
Faster Higher Stronger
order-optimal complexity, measurements, and bitswithRobustSHO-FA:
Compressive sensing
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?
k ≤ m<n
? n
m
k
Robust compressive sensing
y=A(x+z)+eApproximate sparsity
Measurement noise
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z
e
Random
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Decoding complexity
# of
mea
sure
men
ts
°RS’60
°TG’07
°CM’06
°C’08°IR’08
°SBB’06°GSTV’06
°MV’12,KP’12 °DJM’11This work
Lower bound
Lower bound
Sparsity (k)
Nu
mb
er
of
Me
as
ure
me
nts
(m
)
Probability of Successful Reconstruction, n=1000
20 40 60 80 100 120 140
100
200
300
400
500
0
0.2
0.4
0.6
0.8
10.98
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SHO(rt)-FA(st)
Length of Signal (log(n))
Nu
mb
er
of
Me
as
ure
me
nts
(m
)
Probability of Successful Reconstruction, k=20
2 3 4 515
30
45
60
75
90
0
0.2
0.4
0.6
0.8
1
Good
Bad
Good
Bad
High-Level Overview
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43
4
n ck k=2
43
4
n ck k=2
High-Level Overview
8
43
4
3
4
n ck k=2
How to find the leaf nodes and utilize the leaf nodes to do decoding
How to guarantee the existence of leaf node
Left-regular Bipartite Graph
n ck
d=3
9
A
Q1: How to guarantee the existence of leaf node?
Existence of leaf nodes
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e.g., existence of 2-core in d-uniform hypergraph
M. T. Goodrich and M. Mitzenmacher, “Invertible bloom lookup tables,” ArXiv.org e-Print archive, arXiv:1101.2245 [cs.DS], 2011.
Sparsity (k)
Nu
mb
er
of
Me
as
ure
me
nts
(m
)
Probability of Successful Reconstruction, n=1000
20 40 60 80 100 120 140
100
200
300
400
500
0
0.2
0.4
0.6
0.8
10.98
Sharp transition
Q1: How to guarantee the existence of leaf node?
Existence of “Many” leafs
≥2|S||S|L+L’≥2|S|
3|S|≥L+2L’
(L+L’)/(L+2L’) ≥2/3
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L/(L+L’) ≥1/2
Q1: How to guarantee the existence of leaf node?
Bipartite Graph → Sensing Matrix
n ck
d=3
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ADistinct weights
Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding?
Bipartite Graph → Sensing Matrix
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n ck
A
Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding?
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Encoding
Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding?
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Q2: How to find the leaf nodes and utilize the leaf nodes to do decoding?
Decoding
Decoding – First Iteration
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Decoding – Second Iteration
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Verification Measurements
Decoding – Third Iteration
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Decoding – Fourth Iteration
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SHO-FA v.s. Pick-Up-SticksPeeling process: Iterative Decoding Observation: Identification
Check: Verification Picking up a “top” stick: Leaf-based decoding
Robust Compressive Sensing
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Phase error
Propagation error
……
……
Pawar, Sameer and Ramchandran, Kannan, “A Hybrid DFT-LDPC Framework for Fast and Robust Compressive Sensing”
Truncated Reconstruction
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Threshold
Correlated Measurements
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Phase quantization
Correlated Measurements (First bit)
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Phase quantization
Correlated Measurements (Second bit)
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Correlated Measurements (Third bit)
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Additional Properties
• Other works– Group Testing– Network Tomography
• Reduce the number of measurements– Combine Identification and verification
• More noise models• Sparse in different bases• Database query• ……
THANK YOU謝謝
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2-core in d-uniform hypergraph
• The 2-core is the largest sub-hypergraph that has minimum degree at least 2.
• The standard “peeling process” finds the 2-core: while there exists a vertex with degree 1, delete it and the corresponding hyperedges.
hyperedgeNode degree 1
(Almost) S(x)-expansion
≥2|S||S|30
n ck