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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