1

? n

m<n

m

2

Compressive sensing

3

?

k ≤ m<n

? n

m

k

Robust compressive sensing

y=A(x+z)+eApproximate sparsity

Measurement noise

4

?

z

e

Apps: 1. Compression

5

W(x+z)

BW(x+z) = A(x+z)

M.A. Davenport, M.F. Duarte, Y.C. Eldar, and G. Kutyniok, "Introduction to Compressed Sensing,"in Compressed Sensing: Theory and Applications, Cambridge University Press, 2012. 

x+z

Apps: 2. Network tomography

Weiyu Xu; Mallada, E.; Ao Tang; , "Compressive sensing over graphs," INFOCOM, 2011M. Cheraghchi, A. Karbasi, S. Mohajer, V.Saligrama: Graph-Constrained Group Testing. IEEE Transactions on Information Theory 58(1): 248-262 (2012)

6

Apps: 3. Fast(er) Fourier Transform

7

H. Hassanieh, P. Indyk, D. Katabi, and E. Price. Nearly optimal sparse fourier transform. In Proceedings of the 44th symposium on Theory of Computing (STOC '12). ACM, New York, NY, USA, 563-578.

Apps: 4. One-pixel camera

http://dsp.rice.edu/sites/dsp.rice.edu/files/cs/cscam.gif

8

y=A(x+z)+e

9

y=A(x+z)+e

10

y=A(x+z)+e

11

y=A(x+z)+e

12

y=A(x+z)+e

(Information-theoretically) order-optimal13

(Information-theoretically) order-optimal

• Support Recovery

14

SHO(rt)-FA(st)

O(k) meas., O(k) steps

15

SHO(rt)-FA(st)

O(k) meas., O(k) steps

16

SHO(rt)-FA(st)

O(k) meas., O(k) steps

17

1. Graph-Matrix

n ck

d=3

18

A

1. Graph-Matrix

19

n ck

Ad=3

20

1. Graph-Matrix

2. (Most) x-expansion

≥2|S||S|21

3. “Many” leafs

≥2|S||S|L+L’≥2|S|

3|S|≥L+2L’

L≥|S|L+L’≤3|S|

L/(L+L’) ≥1/3L/(L+L’) ≥1/2

22

4. Matrix

23

Encoding – Recap.

24

0

1

0

1

0

Decoding – Initialization

25

Decoding – Leaf Check(2-Failed-ID)

26

Decoding – Leaf Check (4-Failed-VER)

27

Decoding – Leaf Check(1-Passed)

28

Decoding – Step 4 (4-Passed/STOP)

29

Decoding – Recap.

30

0

0

0

0

0

?

?

?0

0

0

1

0

Decoding – Recap.

28

0

1

0

1

0

32

Noise/approx. sparsity

33

Meas/phase error

34

Correlated phase meas.

35

Correlated phase meas.

36

Correlated phase meas.

37