order-optimal compressive sensing for approximately k -sparse signals:
DESCRIPTION
Order-optimal Compressive Sensing for Approximately k -sparse Signals: O(k) measurements and O(k) decoding steps Mayank Bakshi , Sidharth Jaggi , Sheng Cai , Minghua Chen. Overview. Exactly - sparse . A - sparse. Measurement operation: Unknowns : Signal “Noise” - PowerPoint PPT PresentationTRANSCRIPT
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Order-optimal Compressive Sensing for Approximately k-sparse Signals: O(k) measurements and O(k) decoding steps
Mayank Bakshi, Sidharth Jaggi, Sheng Cai, Minghua Chen
Exactly - sparse A- sparse
SHO-FA decoder:
[𝑦 1(𝐼 )
𝑦1(𝑉 )
𝑦 2(𝐼 )
𝑦2(𝑉 )
𝑦 3(𝐼 )
𝑦3(𝑉 )
𝑦 4(𝐼 )
𝑦4(𝑉 )
]identification
verification
Measurement design:
for some ?
?
Y
Y
N
N
is not a leaf
is not a leaf
is a leaf
Check if leaf
Identify
Verify
Input: Node Output: Is a leaf?
Previous design fails: - “small” noise in => possibly large noise in phase of and => identification/verification error- Estimation error propagates (and amplifies) over iterations
Three new ideas:
1. Truncation:
2. Repeated identification/verification measurements
3. Concatenation and Coupon Collection
Figure 5
Figure 6
Figure 7
Figure 10
Figure 9
Figure 8
- Noise does not change phase much
- Most of the norm of captured
….
..
…...…
...
…...
𝐿1𝐿2
- Represent each node on left as a sequence of digits
- Separate identification/verification measurements for each digit
1st digit 3rd digit……
- Run SHO-FA independently on chunks, each of size , recover most of the signal
- Reconstruct the failed locations by looking for leafs in random linear combinations - like coupon collection
References
[1] Accompanying short writeup available at http://personal.ie.cuhk.edu.hk/~mayank/CS/writeup.pdf[2] M. Bakshi, S. Jaggi, S. Cai, M. Chen, “Order-optimal compressive sensing for k-sparse signals with noisy tails: O(k) measurements, O(k) steps”, pre-print available at http://personal.ie.cuhk.edu.hk/~sjaggi/CS_)1.pdf, Video at http://youtu.be/UrTsZX7-fhI
at all right nodes;
Pick
a. Identify signal coordinate, s.t.
b. Output
Subtract contribution of from
at each neighbour of ;Update
?
N
Y
Declare failure
steps
Check if leaf
Check if leaf
# outputs = ?
N
Y Declare success
At m
ost
itera
tions
Overview
Key tool: “Almost” Expanders
Settings: a. Exactly -sparse b. Approximately k-sparse with for some .
Information Theoretically order-optimal
Our Result: a. measurements suffice b. “SHO(rt)-FA(st)” algorithm: steps suffice c. processed “bits”/operations
1. High probability of vertex expansion: - Every set S of size at most k (and all its subsets) have expansion at least with a high probability over the construction of
Figure 2
Figure 3
Figure 4
ck
deg=31
2
5
4
3
1
3
4
2
n
Figure1
Expands Does not expand
Sparsity (k)
Num
ber o
f Mea
sure
men
ts (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
Length of Signal (log(n))
Num
ber o
f Mea
sure
men
ts (m
)
Probability of Successful Reconstruction, k=20
2 3 4 520
40
60
80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
- Bipartite, left regular - uniformly chosen neighbours of each left node
S :support of
≥2|S||S|
?
? n
m<n
m
Key tool: “Almost” Expanders
Measurement operation:
Unknowns: Signal
“Noise”
Objective: Design , decoder s.t. estimation error “small”, i.e.,
w.h.p.
Construction of graph :
2. “Many” S-leaf nodes